Sec o al composi ion and mac oeconomic dynamics
Jaime Alonso-Ca e a
Depa amen o de Fundamen os del Análisis Económico and RGEA
Uni e sidade de Vigo
Jo di Caballé
Uni a de Fonamen s de l’Anàlisi Economica and MOVE
Uni e si a Au ònoma de Ba celona
Xa ie Rau ich
Depa amen de Teo ia Econòmica and CREB
Uni e si a de Ba celona
Ap il 5, 2011
Abs ac
We analyze he ansi ional dynamics o a model wi h he e ogeneous consump ion
goods. In his model, con e gence is d i en by wo di¤e en o ces: he ypical
diminishing e u ns o capi al and he sec o al change inducing he a ia ion in
ela i e p ices. We show ha his second o ce a¤ec s he g ow h a e i he wo
consump ion goods a e no Edgewo h independen and i hese wo goods a e p o-
duced wi h echnologies exhibi ing di¤e en capi al in ensi ies. Because he a o e-
men ioned dynamic sec o al change a ises only unde he e ogeneous consump ion
goods, he ansi ional dynamics o his model exhibi s s iking di¤e ences wi h he
g ow h model wi h a single consump ion good. We also show ha hese di¤e ences
in he ansi ional dynamics can gi e aise o la ge disc epancies in he wel a e cos
o shocks be ween he economy wi h a unique consump ion good and he economy
wi h mul iple consump ion goods.
JEL classi…ca ion codes: O41, O47.
Keywo ds: mul i-sec o g ow h models, ansi ional dynamics, consump ion g ow h.
Financial suppo om he Go e nmen o Spain h ough g an s ECO2009-09847, ECO2009-
06953 and ECO2008-02752; PR2009-0162 and SM2009-0001; he Gene ali a o Ca alonia h ough
he Ba celona GSE Resea ch Ne wo k and g an s SGR2009-00350 and SGR2009-1051; and he Xun a
de Galicia h ough g an 10PXIB300177PR is g a e ully acknowledged. Alonso-Ca e a also hanks he
Resea ch School o Economics (Aus alian Na ional Uni e si y) o i s hospi ali y. Caballé also hanks
he …nancial suppo om he ICREA Academia p og am. The pape has bene… ed om commen s
by pa icipan s in he Wo ld Cong es o he Econome ic Socie y (Shanghai), DEGIT (Los Angeles),
ESEM (Milan), SAEe (G anada), ASSET (Pado a), Aus alasian Wo kshop in Mac oeconomic Dynam-
ics and semina s in UPV (Bilbao), IAE-CSIC (Ba celona), Aus alian Na ional Uni e si y, Uni e si y
o Melbou ne, Monash Uni e si y, Macqua ie Uni e si y, Uni e si y o Wollongong, Na ional Uni e -
si y o I eland (Maynoo h), Gea y Ins i u e (UCD), Uni e sidad de Mu cia, Uni e sidad de las Islas
Balea es, Uni e si a Ro i a i Vi gili and Uni e sidade do Minho.
Co esponding Au ho : Jo di Caballé. Uni e si a Au ònoma de Ba celona. Depa amen d’Economia
i d’His ò ia Econòmica. Edi…ci B. 08193 Bella e a (Ba celona). Spain. E-mail: jo [email protected]
1. In oduc ion
The li e a u e on economic g ow h has gene ally aken he s anda d model o capi al
accumula ion wi h a single …nal consump ion good as he canonical amewo k o s udy
he g ow h pa e n o an economy. In pa icula , his model has been widely used o
he analysis o he dynamic e¤ec s o shocks in undamen s and o he no ma i e and
posi i e cha ac e iza ion o mac oeconomic policy. The main ea u e o his model is
ha he economic dynamics is ully d i en by he e olu ion o he e u n o capi al. As
he seminal con ibu ion o Ramsey (1928) s a ed, he op imal in e empo al alloca ion
o consump ion and in es men leads he g ow h o consump ion expendi u e o depend
on he ne in e es a e only. In his pape , we claim ha his esul does no apply o
models ha allow o se e al he e ogeneous consump ion goods. Mo e p ecisely, he
a o emen ioned benchma k model may be unsui able o s udy he dynamic e¤ec s o
hose shocks ea u ing a pe manen e¤ec on he sec o al composi ion o consump ion.
To illus a e his poin , we cha ac e ize he p ope ies o he ansi ional dynamics o a
g ow h model whe e indi iduals de i e u ili y om consump ion o wo he e ogeneous
goods.
The ecen g owing in e es o he analysis o s uc u al change and in e na ional
ade has made popula he use o mul i-sec o g ow h models wi h he e ogeneous
consump ion goods.1A ypical by-p oduc o his li e a u e is ha he dynamics o
he agg ega e a iables a e iden ical o hose p edic ed by he model wi h a single
consump ion good: he g ow h a e o consump ion expendi u e only depends on he
ma ginal p oduc o capi al. Acco ding o his esul , he p ocess o con e gence
would be only de e mined by he e u n o capi al wi h independence o he numbe o
consump ion goods. We a gue ins ead ha his isomo phism be ween he wo ypes o
models is a consequence o some es ic i e assump ions imposed on hese mul i-sec o
models, namely, ei he he u ili y unc ion is addi i ely sepa able in he amoun s o
consump ion o he di¤e en goods o hese consump ion goods a e p oduced by means
o echnologies wi h iden ical capi al in ensi ies. By elaxing hese assump ions, we
… s p o e ha he a e o g ow h o expendi u e depends no only on he in e es
a e, bu also on he g ow h a e o ela i e p ices o goods. The e o e, he p ocess o
con e gence in a gene al mul i-sec o g ow h model is d i en by wo o ces: he e u n
o capi al and he dynamic adjus men o ela i e p ices a ising om he change in he
sec o al composi ion. Ou main pu pose in his pape is o analyze how he p esence
o he la e o ce modi…es he dynamic beha io o he economy.
The e¤ec o he in e es a e on consump ion g ow h is measu ed by he
in e empo al elas ici y o subs i u ion (IES, hence o h). On he con a y, he g ow h
e¤ec o he a ia ion in he ela i e p ice o goods is join ly de e mined by he IES and
he Edgewo h elas ici y be ween goods.2The e o e, he ela i e impo ance o hese
wo o ces in de e mining he in e empo al alloca ion o consump ion expendi u e
c ucially depends on his Edgewo h elas ici y. In ac , we show ha he g ow h a e
o ela i e p ices inc eases (dec eases) he a e o g ow h o expendi u e when he
1Examples include, among many o he s, Eche a ia (1997), Konsamung e al. (2001), Ngai and
Pissa ides (2008), o Pe ez and Guillo (2010).
2The Edgewo h elas ici y be ween wo goods is de…ned as he elas ici y o he ma ginal u ili y o
one good wi h espec he consump ion le el o he o he good.
2
wo consump ion goods a e Edgewo h subs i u e (complemen a y). The in ui ion o
his esul is ha he inc ease in he ela i e p ice o one good educes he demand
o his good, which inc eases (dec eases) he demand o he co esponding subs i u e
(complemen a y) goods.
As was men ioned be o e, p e ious mul i-sec o g ow h models ound in he
li e a u e impose assump ions ha p e en he ela i e p ices o consump ion goods
om displaying he a o emen ioned g ow h e¤ec s. Some au ho s assume ha he
consump ion goods a e Edgewo h independen (see, e.g., Eche a ia, 1997; Lai ne ,
2000; o Pe ez and Guillo, 2010) o use a echnology yielding a cons an ela i e p ice
be ween goods (Kongsamun e al., 2001; o S ege , 2006). Two excep ions a e he
mul i-sec o g ow h models conside ed in Rebelo (2001) and Ngai and Pissa ides (2007).
In he la e model, he g ow h o p ices a¤ec s he a e o g ow h o expendi u e.
Howe e , since he capi al in ensi ies a e iden ical ac oss sec o s, he a ia ion in p ices
a ises only om exogenous, unbiased echnological changes in sec o al p oduc i i ies.
On he con a y, in ou model he dynamics o p ices is endogenous as we conside
di¤e en capi al in ensi ies ac oss sec o s. In his way, he dynamic adjus men o p ices
di ec ly de e mines he esponse o he economy o changes in undamen als. While
Rebelo (2001) does also conside a model whe e p ices a e endogenous and he goods
a e no Edgewo h independen , he does no analyze he co esponding ansi ional
dynamics. The e o e, o he bes o ou knowledge, he p esen pape is he … s
analyzing he ansi ional dynamics o a g ow h model wi h he e ogeneous consump ion
goods when he wo a o emen ioned o ces d i ing he ansi ion a e ope a i e.
In o de o s udy he ansi ional dynamics when he a ia ion o p ices displays
he a o emen ioned g ow h e¤ec s, we analyze a h ee sec o g ow h model wi h a
homo he ic u ili y unc ion whose a gumen is a composi e good combining wo di¤e en
consump ion goods. These goods a e p oduced by means o cons an e u ns o scale
echnologies ha use physical and human capi al as inpu s. Fu he mo e, echnologies
exhibi di¤e en capi al in ensi ies ac oss sec o s. As was explained be o e, he las
assump ion makes he ela i e p ice be ween he wo consump ion goods no cons an
along he ansi ion. To gain some in ui ion abou his esul , suppose ha human
capi al becomes ela i ely sca ce han physical capi al. Then, he consump ion good
p oduced in he physical capi al in ensi e sec o becomes less cos ly and he ela i e
p ice o his consump ion good dec eases. No e ha i he consump ion goods we e
p oduced wi h echnologies wi h he same capi al in ensi y hen he imbalances be ween
he wo capi al s ocks would no modi y he ela i e p ice be ween hese consump ion
goods. Finally, we assume in ou analysis ha he wo consump ion goods a e no
Edgewo h independen so ha his dynamic adjus men o he ela i e p ice esul s in
a modi…ca ion o he g ow h a e o consump ion expendi u e.
As occu s in mul i-sec o g ow h models wi h wo ypes o capi al, he ansi ional
dynamics will be go e ned by he imbalances be ween he wo s ocks o capi al.
Howe e , he exis ence o wo di¤e en o ces go e ning he ansi ion yields wo
in e es ing di¤e ences wi h espec o he ansi ional dynamics ob ained in he
s anda d g ow h model wi h a unique consump ion good. Fi s , in g ow h models
wi h a unique consump ion good, con e gence in he consump ion g ow h a e occu s
om below (abo e) i he ini ial alue o he a io o physical o human capi al is la ge
(smalle ) han i s s a iona y alue. We will show ha his beha io may be e e sed by
3
in oducing he e ogeneous consump ion goods. In pa icula , we p o ide a condi ion
ha implies ha con e gence is om abo e when he ini ial alue o he capi al a io
is la ge han i s s a iona y alue and om below o he wise. I should be no iced ha ,
when his condi ion is sa is…ed, he ini ial e¤ec on consump ion g ow h o a shock in
one o he capi al s ocks will be he opposi e o he one ob ained in a model wi h a
single consump ion good. As an example, conside an economy su¤e ing a nega i e
shock in human capi al. Then, i he e is a unique consump ion good, his economy
will expe ience a dec ease in he g ow h a e o consump ion. In con as , in ou model
wi h he e ogeneous consump ion goods, he economy will display an inc ease in he
g ow h a e o consump ion expendi u e.
Second, while he g ow h a e o consump ion expendi u e exhibi s a mono onic
beha io when he diminishing e u ns o capi al is he only o ce go e ning he
ansi ion, i may exhibi ins ead a non-mono onic beha io in ou model. Al a ez-
Cuad ado e al. (2004) men ion e idence o non-mono onic beha io o he consump ion
g ow h a e. S ege (2000), among o he s, has accoun ed o his non-mono onic
beha io by means o he in oduc ion o a minimum consump ion le el ha makes
p e e ences non-homo he ic. In con as , in ou model he non-mono onic beha io
is explained by he p esence o he a o emen ioned wo di¤e en o ces ac ing on he
ansi ional dynamics. In ac , he g ow h a e exhibi s a non-mono onic beha io when
hese wo o ces exhibi opposi e g ow h e¤ec s.
The wo di¤e ences we ha e jus men ioned imply ha he pa e ns o g ow h along
he ansi ion c ucially depend on he pa ame e s alues o ou model. Mo e p ecisely,
we show ha he capi al in ensi y anking ac oss sec o s and he alue o he Edgewo h
elas ici y de e mine he na u e o he ansi ion. We will simula e he economy in o de
o analyze he ansi ional dynamics and show ha he wo o ces go e ning he a e o
g ow h o expendi u e ha e opposi e g ow h e¤ec s. As a consequence, in he simula ed
economy his g ow h a e exhibi s a non-mono onic con e gence owa ds he s eady-
s a e and, mo eo e , he sign o he g ow h e¤ec s o a shock in one o he capi al s ocks
depends on he alue o he Edgewo h elas ici y. We also use he simula ed model o
s udy he g ow h and wel a e e¤ec s o echnological shocks. This analysis allows us
o compa e he e¤ec s o hese shocks in he economy wi h a single consump ion good
wi h he e¤ec s in he economy wi h he e ogeneous consump ion goods. Rega ding
he wel a e cos o shocks, we show ha hey will s ongly depend on he sec o al
composi ion o he composi e consump ion good when hese shocks cause la ge e¤ec s
on he uni a y cos o his composi e good. These la ge e¤ec s occu when we conside
shocks ha modi y he long- un alue o ela i e p ices. In his case, he shocks esul in
a la ge dis o ion in he in a empo al decision conce ning he sec o al composi ion o
consump ion, which ansla es in u n in o sizeable addi ional wel a e e¤ec s. We hen
conclude ha he exis ing li e a u e, by conside ing speci…c models whe e he o ce
linked o he dynamics o he ela i e p ices be ween goods is no ope a i e, ob ain
biased esul s abou he e¤ec s o hose shocks.
The pape is o ganized as ollows. Sec ion 2 p esen s he ing edien s o he model.
Sec ions 3 and 4 cha ac e ize he equilib ium dynamics o ela i e p ices and o he
g ow h a e o expendi u e, espec i ely. Sec ion 5 de elops he nume ical analysis
conce ning he ansi ional dynamics and he e¤ec s o echnological shocks. Sec ion 6
p esen s some concluding ema ks, while he Appendix con ains he p oo s o all he
4
esul s o he pape .
2. The economy
Le us conside a h ee-sec o g ow h model in which he ou pu in each sec o is
ob ained om combining amoun s o wo ypes o capi al, kand h, which we dub
physical and human capi al, espec i ely. The … s sec o p oduces an amoun y1o
commodi y using he ollowing p oduc ion unc ion:
y1=A1(s1k)(u1h)1=A1u1hz
1;
whe e s1and u1a e he sha es o physical and human capi al alloca ed o his sec o ,
z1=s1k/u1his he physical o human capi al a io, A1>0is he sec o al o al ac o
p oduc i i y (TFP), and 2(0;1) measu es he in ensi y o physical capi al in his
sec o . We in e p e his sec o as he one p oducing manu ac u es and assume ha
he commodi y y1can be ei he consumed o added o he s ock o physical capi al.
The law o mo ion o he physical capi al s ock is hus gi en by
_
k=A1u1hz
1c1k; (2.1)
whe e c1is he amoun o good y1de o ed o consump ion, and 2[0;1] is he
dep ecia ion a e o he physical capi al s ock. To ease he no a ion we omi he ime
a gumen o all he a iables. The second sec o p oduces a consump ion good y2by
means o he p oduc ion unc ion
y2=A2(s2k)(u2h)1=A2u2hz
2;(2.2)
whe e s2and u2a e he sha es o physical and human capi al alloca ed o his sec o ,
espec i ely, z2=s2k/u2his he physical o human capi al a io, A2>0is he sec o al
TFP, and 2(0;1) measu es he in ensi y o physical capi al in his sec o . We
in e p e his sec o as he one p oducing ood and se ices de o ed o consump ion,
such as cul u al o en e ainmen goods. Thus, he ou pu o his sec o can only be
de o ed o consump ion, which we deno e by c2;so ha y2=c2in equilib ium. Finally,
he hi d sec o p oduces a commodi y y3by means o he p oduc ion unc ion
y3=A3[(1 s1s2)k][(1 u1u2)h]1=A3(1 u1u2)hz
3;
whe e z3= (1 s1s2)k/(1 u1u2)his he physical o human capi al a io,
A3>0is he sec o al TFP, and 2(0;1) measu es he in ensi y o physical capi al
in his sec o . This commodi y is de o ed exclusi ely o inc ease he s ock o
human capi al and, he e o e, we iden i y his sec o wi h he educa ion sec o . The
accumula ion o he human capi al s ock is hus gi en by
_
h=A3(1 u1u2)hz
3h; (2.3)
whe e 2[0;1] is he dep ecia ion a e o human capi al.
The economy is popula ed by an in…ni ely li ed ep esen a i e agen cha ac e ized
by he ins an aneous u ili y unc ion
U(c1; c2) = c
1c1
21
1;(2.4)
5

whe e he pa ame e 2[0;1] measu es he sha e o good c1in he composi e
consump ion good, m=c
1c1
2;and  > 0is he (cons an ) elas ici y o he ma ginal
u ili y o his composi e consump ion good. No e ha his u ili y unc ion is
homo he ic, s ic ly conca e, and inc easing. The ep esen a i e agen is endowed
wi h kuni s o physical capi al and huni s o human capi al. Le wbe he a e o
e u n on human capi al (i.e., he eal wage pe uni o human capi al) and he a e
o e u n on physical capi al (i.e., he eal in e es a e). We assume pe ec sec o al
mobili y so ha he wage and in e es a e a e independen o he sec o whe e he
ep esen a i e agen alloca es he uni s o physical and human capi al. The e o e, he
budge cons ain o he consume is gi en by
wh + k = (c1+pc2)+(Ik+phIh);(2.5)
whe e pis he ela i e p ice o good c2measu ed in uni s o good c1,phis he ela i e
p ice o human capi al measu ed in uni s o physical capi al (o consump ion good c1).
Finally, Ihand Ika e he g oss in es men in human and physical capi al, espec i ely,
Ik=_
k+k; (2.6)
and
Ih=_
h+h: (2.7)
3. Dynamics o ela i e p ices
In his sec ion we … s sol e he p oblems o consume s and … ms and hen we de i e
he sys em o di¤e en ial equa ions cha ac e izing he compe i i e equilib ium. We use
hese equa ions o …nd he long- un equilib ium and o s udy how he in oduc ion o
a second consump ion good modi…es he equilib ium dynamics o ela i e p ices.
The ep esen a i e agen maximizes
Z1
0
e U(c1; c2)d ; (3.1)
subjec o (2.5), (2.6), and (2.7), whe e  > 0is he subjec i e discoun a e. The
solu ion o his op imiza ion p oblem is gi en by he ollowing equa ions de i ed in he
Appendix:
p=1
c1
c2;(3.2)
_ph
ph
= w
ph
+; (3.3)
_c1
c1
= 
(1 ) (1 )
_p
p;(3.4)
and he ans e sali y condi ions
lim
!1e p(1)(1)ck= 0;(3.5)
and
lim
!1e p(1)(1)ch= 0:(3.6)
6
Equa ion (3.2) ells us ha he p ice a io pis equal o he ma ginal a e o
subs i u ion be ween he wo consump ion goods. Equa ion (3.3) shows ha he g ow h
o he p ice phis de e mined by he s anda d non-a bi age condi ion be ween he
in es men s in physical and human capi al. Finally, equa ion (3.4) cha ac e izes he
g ow h a e o consump ion good c1:F om his equa ion we can easily ob ain he
g ow h a e o o al consump ion expendi u e, which is de…ned as c=c1+pc2. No e
ha equa ion (3.2) implies ha
c=c1
=pc2
1:(3.7)
Hence, he g ow h a e o consump ion expendi u e ccoincides wi h he g ow h a e
o c1( he consump ion expendi u e in he good y1;which is he nume ai e). We hen
ob ain om (3.4) ha
_c
c= 
(1 ) (1 )
_p
p:(3.8)
Equa ion (3.8) ells us ha he g ow h a e o consump ion expendi u e is d i en
by bo h he in e es a e and by he change in he ela i e p ice o he wo consump ion
goods. The e¤ec o a ise in he in e es a e on he a e o g ow h o cis summa ized
by he in e empo al elas ici y o subs i u ion IES = 1=: On he con a y, he g ow h
e¤ec o a ise in he g ow h a e o he ela i e p ice is join ly de e mined by he IES
and Edgewo h elas ici y (i.e., he elas ici y o he ma ginal u ili y o he consump ion
good c1wi h espec o he consump ion good c2) which is gi en by
" c2@2U=@c1@c2
@U=@c1=(1 ) (1 ):
By using (3.8), we see ha he g ow h a e o he ela i e p ice pdi ec ly a¤ec s he
g ow h a e o consump ion expendi u e cwhen "6= 0;i.e., when he wo consump ion
goods a e no Edgewo h independen . Unde he ins an aneous u ili y unc ion (2.4),
he Edgewo h elas ici y "is de e mined by he pa ame e s and : In pa icula , he
wo consump ion goods a e Edgewo h independen when = 1 because in his case
he u ili y unc ion is addi i ely sepa able in he wo goods c1and c2. The p e ious
li e a u e on mul isec o al g ow h models commonly uses a loga i hmic speci…ca ion o
p e e ences and his explains why i does no ob ain he g ow h e¤ec o he a ia ion
in ela i e p ices.
The in ui ion on he a o emen ioned g ow h e¤ec o he dynamic adjus men o
ela i e p ices is as ollows. Equa ion (3.8) is he Eule equa ion equa ing he ma ke
e u n om in es ing one uni o he nume ai e y1and he g ow h o he ma ginal
u ili y a ising om consuming one addi ional uni o his commodi y. When he
wo consump ion goods a e Edgewo h independen , hen he ma ginal u ili y o one
consump ion good does no depend on he o he consump ion good. In his case, he
g ow h a e o o al consump ion expendi u e only depends on he in e es a e. In
con as , when he wo consump ion goods a e no Edgewo h independen a change
in he consump ion o good c2al e s he ma ginal u ili y o consump ion good c1:
Thus, in his case, he g ow h o he ma ginal u ili y o one good will depend on he
7
g ow h o bo h consump ion goods. As ollows om equa ion (3.2), he consump ion o
hese goods depends on he ela i e p ice. Ac ually, he conca i y o he u ili y unc ion
implies ha an inc ease in he ela i e p ice p educes he amoun consumed o good c2.
This educ ion implies an inc ease ( educ ion) in he ma ginal u ili y o consump ion
good c1and in he amoun o good c1consumed when he wo goods a e Edgewo h
subs i u e (complemen a y).3
A e ha ing p esen ed he equilib ium condi ions on he demand side o ou
economy, we will now mo e o he supply side and we will cha ac e ize how he dynamics
o ela i e p ices is de e mined. This dynamics depends on he echnologies used by
he di¤e en sec o s and on he ma ke s uc u e. In pa icula , … ms maximize p o… s
in each sec o and, hus, he compe i i e ac o s paymen mus sa is y simul aneously
he ollowing equa ions:
=A1z1
1;(3.9)
=pA2z1
2;(3.10)
=phA3z1
3;(3.11)
w= (1 )A1z
1;(3.12)
w=p(1 )A2z
2;(3.13)
and
w=ph(1 )A3z
3:(3.14)
Combining he sys em o equa ions (3.9) o (3.14) when 6=, we ob ain
zi= ip1
; o i= 1;2;3;(3.15)
whe e
1=

1
11
A2
A11

;
2=
11
 1;(3.16)
and
3=
11
 1:(3.17)
F om he p e ious se o equilib ium condi ions we ob ain he ollowing well-known
esul , which has impo an consequences o he equilib ium dynamics o ou economy.
P oposi ion 3.1. The ela i e p ice po consump ion goods is cons an o e ime o
all ini ial alues o he capi al a io z=k=h i and only i a leas one o he ollowing
condi ions holds: (i) =, (ii) =:
3No e ha he e¤ec o ela i e p ices on expendi u e g ow h appea s because only he good c1can
be used as physical capi al. I he equilib ium mix o he wo consump ions goods could be de o ed o
in es men in physical capi al, hen he ela i e p ice would no a¤ec he g ow h a e o consump ion
expendi u e c(see Acemoglu and Gue ie i, 2008).
8
Le us … s conside he condi ion =; which means ha he wo consump ion
goods c1and c2a e p oduced by means o echnologies wi h he same capi al in ensi y.
We see ha unde his condi ion, equa ion (3.16) implies ha 2= 1when 6=
and hen, om equa ion (3.15), we ge z1=z2. The e o e, by combining equa ions
(3.9) and (3.10), i ollows ha he ela i e p ice be ween he wo consump ion goods
emains cons an and equal o p=A1
A2:This ob iously means ha he g ow h a e
o consump ion expendi u e only depends on he in e es a e (see equa ion (3.8)).
The e o e, he ansi ional dynamics o ou model when =coincides wi h he
ansi ional dynamics o he wo-sec o g ow h model wi h a unique consump ion good,
which was analyzed by Uzawa (1965) and Lucas (1988).
Le us now conside he condi ion =: Unde his condi ion he wo capi al
goods kand ha e p oduced by means o echnologies wi h he same capi al in ensi y.
Obse e ha in his case condi ions (3.9), (3.11), (3.12) and (3.14) imply ha z1=z3
and, hus, he ela i e p ice be ween he wo capi al s ocks is cons an and gi en by
ph=A1
A3:Equa ion (3.3) implies ha he wage o in e es a e a io w= emains
cons an when phis cons an . Then, om combining (3.9) and (3.12) we immedia ely
see ha z1is cons an when phis cons an . The e o e, bo h he in e es a e and z2
a e cons an as ollows om (3.9) and (3.11). Finally, equa ion (3.10) shows ha in
his case he ela i e p ice pbe ween he wo consump ion goods emains cons an . In
ac , i is easy o see ha he h ee sec o s a e using Ak echnologies when =:4
The e o e, he ansi ion dynamics in his case coincides wi h he ansi ion in Ak
g ow h models wi h se e al consump ion goods (see, e.g., Rebelo, 1991).
We ha e jus es ablished he condi ions unde which he g ow h a e o consump ion
expendi u e depends no only on he in e es a e, bu also on he g ow h a e
o he ela i e p ice p: This new dependence equi es ha he consump ion goods
be no Edgewo h independen and o be p oduced by means o echnologies wi h
di¤e en capi al in ensi ies. These a gumen s hen explain why he p e ious mul i-
sec o g ow h models do no …nd a di ec e¤ec o ela i e p ices on consump ion
g ow h. Some o hese models conside loga i hmic p e e ences so ha hey implici ly
assume ha consump ion goods a e Edgewo h independen . O he models assume
ha consump ion goods a e p oduced wi h echnologies ha sha e he same capi al
in ensi y. Ob iously, in his case he a ia ion o ela i e p ices could s ill a¤ec di ec ly
he g ow h a e o consump ion expendi u e unde exogenous and biased echnological
change, ha is, when he sec o al TFPs g ow a exogenous g ow h a es ha a e
di¤e en ac oss sec o s (see, e.g., Ngai and Pissa ides, 2007). Howe e , i echnologies
exhibi di¤e en capi al in ensi ies, he ela i e p ice be ween consump ion goods
appea as an endogenous channel o he p opaga ion o shocks in undamen als. In
he es o he pape , we will illus a e he consequences o his endogenous mechanism
and, hence, we will assume ha 6=and 6=:
No e ha ela i e p ices would also a¤ec he g ow h a e o consump ion
expendi u e when =; ha is, when se ices and human capi al a e p oduced wi h
4No e ha he echnology ha p oduces commodi y y1can be ew i en as ollows y1=b
A1u1h;
whe e b
A1=A1(z
1)is cons an . The echnology ha p oduces commodi y y2can be ew i en as
y2=b
A2u2h; whe e b
A2=A2z
2is cons an and, …nally, he echnology ha p oduces commodi y y3
can be ew i en as y3=b
A3(1 u1u2)h; whe e b
A3=A3(z
1)is cons an . Since goods y1and y2
a e p oduced wi h linea echnologies, hei ela i e p ices a e cons an and gi en by p=b
A1
b
A2
:
9
wo dynamic o ces o a gi en capi al in ensi y anking ac oss sec o s and expendi u e
sha e (see he exp ession o in equa ion (4:3)). We hen conside h ee di¤e en
alues o ": 0:7;0:95 and 1:2:We se he alues o and  ha join ly eplica e hose
alues o "and a long- un g ow h a e equal o 2%:In he low elas ici y economy we
ob ain = 2 and = 0:016;whe eas we ge = 2:357 and = 0:0089 o he economy
wi h "= 0:95, and …nally we ge = 2:7143 and = 0:0017 o he high elas ici y
economy. Obse e ha his calib a ion implies easonable alues o he IES:0:5,
0.4243 and 0:3684.
We nex simula e he esponse o each o he h ee pa ame e ized economies o
imbalances in he capi al a io, i.e., when z06=z:In o de o show how impo an is
he g ow h e¤ec o p ice a ia ion, we compa e he esponse o hese baseline economies
wi h he esponse o he co esponding economy wi h a unique consump ion good. In
o de wo ds, we compa e he dynamic beha io s o he economy wi h = 0:3and he
economy wi h = 1:
5.1. T ansi ional dynamics
The exp ession o in equa ion (4:3) implies ha i akes posi i e alues when  < 
and " > 0:Thus, he alue o is posi i e unde ou empi ically plausible alues o
he undamen al pa ame e s. In his case, he wo a o emen ioned o ces go e ning
he ansi ion display opposi e g ow h e¤ec s. In ou nume ical examples, we show
ha , i he o ce associa ed wi h he a ia ion o p ices is he domina ing hen he
ansi ion is going o be di¤e en om ha o models wi h a single consump ion
good. Figu es 2, 3 and 4 show ha his is he case when he Edgewo h elas ici y
is high (i.e., when he alue o is high). These …gu es show he dynamic esponse
o some ele an a iables o imbalances in he capi al a io. In pa icula , each o
hese …gu es con ains six panels. Panels (i), (i ), ( ) and ( i) display, espec i ely, he
g ow h a e o consump ion expendi u e, he g ow h a e o GDP, he ela i e p ice o
consump ion goods and he speed o con e gence o he s a e a iable zas a unc ion
o he de ia ions o he capi al a io wi h espec o i s s a iona y alue. No e ha ,
ollowing Reiss (2000),we de…ne he non-asymp o ic speed o con e gence o he a io
o capi als as _z/(zz). Panels (ii) and (iii) display, espec i ely, he ime pa h o
he g ow h a e o consump ion expendi u e when he s a e a iable is ini ially below
i s long- un alue and when i is ini ially abo e. Fu he mo e, all panels compa e
he ansi ional dynamics o he baseline economy wi h he e ogeneous consump ion
goods (con inuous line) wi h he ansi ion in an equi alen economy wi h a unique
consump ion good, i.e., wi h = 1 (dashed line). We pa ame ize he coun e ac ual
economy wi h = 1 so ha i eplica es he same empi ical ac s used o calib a e
ou benchma k economy wi h wo he e ogenous consump ion goods.We obse e ha
he di¤e ences be ween he wo economies unde conside a ion a e qui e signi…can in
he h ee pa ame ic scena ios. Hence, he di ec e¤ec o he p ice adjus men on he
in e empo al alloca ion o consump ion expendi u e also has impo an quan i a i e
consequences o mac oeconomic dynamics.
[Inse Figu es 2, 3 and 4]
The … s h ee panels o Figu es 2, 3 and 4 illus a e nume ically he esul s in
16

P oposi ion 4.4. We obse e ha he dynamic adjus men o consump ion expendi u e
is non mono onic unde he highe alues o in he economy wi h wo consump ion
goods (= 0:3):Mo eo e , when is high, he in oduc ion o he e ogeneous
consump ion goods e e ses he ansi ion. This occu s because de e mines he alue
o he Edgewo h elas ici y "p o ided a alue  o he consump ion sha e:When he
Edgewo h elas ici y "is high, he g ow h e¤ec o changes in he in e es a e is low
in compa ison wi h he g ow h e¤ec s o changes in he g ow h o he ela i e p ice.
In his case, e en i he ini ial alues o he economy a e close o he co esponding
s eady-s a e alues, he ansi ion is di¤e en om he one a ising in an economy whe e
he ansi ion is go e ned only by he diminishing e u ns o capi al.
The signi…can e¤ec s o he p ice a ia ion on he in e empo al alloca ion o
consump ion expendi u e and sa ings ha e impo an quan i a i e consequences o he
dynamic beha io o he o he mac oeconomic a iables. As an illus a ion, Figu es 2,
3 and 4 shows ha he pa hs o he GDP g ow h a e, he ela i e p ice o goods and
he speed o con e gence also depend on he alue o he pa ame e : This pa ame e
measu es he weigh o he human capi al in ensi e good in he composi e consump ion
good. Thus, a educ ion in makes he composi e good mo e in ensi e in physical
capi al, which explains he esul s displayed in hese h ee …gu es. In ui i ely, he e
a e wo non-compe ing ways o inc easing in ela i e e ms he s ock o he sca ce
capi al and, hus, o adjus ing he imbalances in he capi al a io: (i) To dec ease he
accumula ion o he ela i ely abundan capi al; and (ii) o dec ease he consump ion
expendi u e. The mo e in ensi e in physical capi al is he composi e consump ion good,
he la ge is he ela i e impo ance o he second way when z < z. The g ow h a e
o GDP is hen a dec easing unc ion o i z < z:On he con a y, he mo e in ensi e
in physical capi al is he composi e good, he la ge is he ela i e impo ance o he
… s p ocedu e when z > z:This implies ha he g ow h a e o GDP is an inc easing
unc ion o i z > z:The e o e, he dynamic adjus men o any imbalance in he
capi al a io is as e when he composi e consump ion good is mo e physical in ensi e.
This ac explains why he non-asymp o ic speed o con e gence always dec eases wi h
(see Panel ( i)).
We …nally illus a e he implica ions o he di¤e ences in he ansi ional dynamics
ac oss he al e na i e pa ame ic scena ios by compu ing he wel a e e¤ec s o he ini ial
imbalances in he capi al a io.11 Table 1 epo s he ime-in a ian inc ease (dec ease)
in consump ion equi ed o compensa e he wel a e cos s (gains) o ha ing an ini ial
capi al a io smalle (la ge ) han he s a iona y a io. We again show he esul s o
ou baseline economy wi h = 0:3and o he economy wi h a single consump ion
good (i.e., = 1):The las column o his able compa es he di¤e ences in wel a e
cos s be ween hese wo economies and shows ha hey a e la ge. In pa icula , he
wel a e cos is app oxima ely 20% la ge in he economy wi h wo consump ion goods,
whe eas he wel a e gain is 17% la ge . These esul s ollow again om he ac ha he
composi e consump ion good in he economies wi h a low alue o is mo e in ensi e
in physical capi al. Ob iously, in hese economies he uni a y cos o he composi e
good is mo e sensi i e o he ela i e endowmen o physical capi al.
11 As in Lucas (1987), we measu e he wel a e cos o he imbalances in he capi al a io by he
pe cen age inc ease in composi e consump ion good mnecessa y o ob ain he same discoun ed sum o
u ili y as in he si ua ion whe e he capi al a io is ini ially equal o i s s a iona y alue.
17
[Inse Table 1]
By epea ing he p e ious nume ical exe cises we ob ain ha he epo ed
di¤e ences in wel a e be ween he wo economies a e ex emely obus o bo h he
size o shocks and he alue o . The insigni…can e¤ec o is explained by analyzing
he dynamic beha io o he composi e good m=c
1c1
2;which is he undamen al
a iable o wel a e analysis. By using condi ions (3.2), (3.7) and (3.27), we ob ain
_m
m=1
A1z1
1(1 )(p):(5.1)
Ob iously, he g ow h a e o malso depends on he o ces d i ing he in e empo al
alloca ion o consump ion expendi u e c: he diminishing e u ns o capi al and he
g ow h a e o p ices. Howe e , obse e ha he ne e¤ec o hese wo o ces does
no depend in his case on he alue o : This occu s because he di ec e¤ec o
he a ia ion in he ela i e p ice on he g ow h a e o mdoes no depend on he
Edgewo h elas ici y ". This hen explains he insigni…can e¤ec o on he wel a e
compa ison be ween he economy wi h = 0:3and he economy wi h = 1:
Nex , we complemen he analysis in his subsec ion by s udying how he esponse o
he economy o shocks in undamen als depends on he alue o : Gi en he p e ious
conclusion abou he independence o wel a e e¤ec s on ; we will only p esen he
esul s o he case o = 2;which is associa ed wi h he alue "= 0:7 o he
Edgewo h elas ici y:
5.2. Compa a i e dynamics and wel a e
We now p oceed o s udy he dynamic adjus men s and he wel a e cos s om
wo di¤e en shocks: a sec o al biased echnological shock and a sec o al unbiased
echnological shock. Fo ha pu pose, we assume ha he economy is ini ially in a
BGP and, unexpec edly, one o hese shocks is in oduced in a pe manen basis. The
aim o his analysis is o compa e he e¤ec s o hese shocks in he baseline economy
wi h wo consump ion goods (= 0:3) wi h he e¤ec s in he economy wi h a unique
consump ion good (= 1):
We … s analyze he e¤ec s o a biased echnological shock ha consis s o educing
he TFP o he manu ac u ing sec o A1by a 15%. We explain hese e¤ec s by using
Figu e 5, which summa izes how he economy esponds o he shock; and Table 2,
which p o ides he wel a e cos o his shock. Obse e ha he a e o g ow h o
expendi u e ini ially su¤e s a s ong decline and hen i inc eases un il i con e ges
o i s new long- un, which is smalle han he one be o e he shock. In he economy
wi h a single consump ion good, he g ow h a e only depends on he in e es a e,
which ins an aneously alls due o he echnological shock. This educes in es men
and, as a consequence, he s ock o physical capi al declines du ing he ansi ion.
The educ ion in he s ock o physical capi al implies ha he in e es a e inc eases
du ing he ansi ion. No e ha he beha io o he in e es a e ully explains he
ini ial s ong educ ion in he a e o g ow h o expendi u e and also i s pos e io
inc ease du ing he ansi ion. On he con a y, in he economy wi h wo consump ion
goods, he a e o g ow h o expendi u e also depends on he g ow h o he ela i e
18
p ice po consump ion goods. This p ice dec eases ins an aneously because he shock
di ec ly a¤ec s he sec o p oducing manu ac u es, whe eas i inc eases du ing he
ansi ion because he con inuous educ ion in he s ock o physical capi al ises he
cos o p oducing se ices, which is ela i ely in ensi e in his capi al. This beha io
o he ela i e p ice phas a posi i e e¤ec on he a e o g ow h o expendi u e as he
Edgewo h elas ici y in he benchma k economy sa is…es " > 0. The p esence o his
posi i e g ow h e¤ec in he economy wi h wo consump ion goods explains bo h he
smalle ini ial educ ion in he a e o g ow h o expendi u e and i s la ge alues along
he ansi ion.
[Inse Figu e 5 and Table 2]
The … s ow o Table 2 epo s he wel a e cos o he conside ed pe manen
educ ion in he TFP o he manu ac u ing sec o . The main esul is ha he wel a e
cos is a 45:6% la ge in he economy wi h a unique consump ion good. This la ge
di¤e ence a ises om he ac ha he esponse o he composi e good m o he shock
is la ge , he la ge is he sha e o manu ac u es in he composi e good. Figu e 5
illus a es he dynamic adjus men o ha good. Panel (iii) epo s de ia ions o he
composi e good o physical capi al a io m=k om i s ini ial s a iona y alue. F om
his panel we conclude ha he ini ial educ ion in he alue o mis smalle in he
economy wi h = 0:3:The in a empo al subs i u ion be ween goods in his economy
educes he impac o he shock in he le el o he composi e good. On he con a y,
as Panel (i ) shows, he g ow h a e o composi e good inc eases du ing he ansi ion
and, wha is mo e in e es ing, i is smalle in he economy wi h = 0:3due o he
nega i e e¤ec o he inc ease in he ela i e p ice p(see equa ion (5:1)). Howe e ,
he la ge eco e y o he amoun o he composi e good in he economy wi h = 1 is
no enough o ou weigh i s la ge ins an aneous educ ion. In o he wo ds, he ini ial
di¤e ence in he esponse o he composi e good in he wo economies explains he
la ge wel a e cos in he economy wi h = 1.
Figu e 6 displays he dynamic e¤ec s o an unbiased echnological shock consis ing
o a 5% dec ease in he TFP in each sec o . We obse e ha he dynamic adjus men
in his case is quali a i ely simila o he one acc uing om a biased echnological
shock when = 1. Mo eo e , he di¤e ences be ween he wo economies a e now
quan i a i ely insigni…can because o he smalle incidence o he p ice adjus men on
he a e o g ow h o expendi u e. Since each sec o al TFP alls in he same p opo ion,
he esponses o he ela i e p ice pand o consump ion composi ion a e bo h smalle
when he shock is unbiased. This explains he small disc epancies be ween he wo
economies unde conside a ion conce ning he dynamic esponse o he a e o g ow h
o expendi u e and he le el o composi e consump ion. Finally, his implies ha he
wel a e cos associa ed wi h he unbiased shock is e y simila in he wo economies.
As he second ow o Table 2 shows, he wel a e cos in he economy wi h = 0:3is
less han 2% la ge han in he economy wi h = 1:
A his poin , we should also men ion ha he di¤e ences in he e¤ec s o he
unbiased shock be ween he wo economies only a ise because he dep ecia ion a es o
bo h capi al s ocks a e di¤e en , which makes he shock dis o he op imal alloca ion
o capi al among sec o s. I =, hen he s a iona y alue o pis no a¤ec ed by he
19
unbiased shock as i can be de i ed om (3.15) and (3.26). Mo eo e , in his case we
ob ain ha he wel a e cos in he wo economies would coincide. As can be seen om
Figu e 6, e en i some di¤e ences a ise in he dynamic adjus men o bo h he g ow h
a e o expendi u e and he amoun o composi e good be ween he wo economies, he
la ge eco e y o he composi e good in he economy wi h = 1 will ully o¤se i s
la ge ins an aneous educ ion. The e o e, in spi e o displaying iden ical wel a e cos s,
he ime-pa h o he wel a e cos associa ed wi h a shock is di¤e en ac oss he wo
economies e en i he echnological shock is unbiased. We can hus conclude ha he
disc epancy in he wel a e cos o shocks be ween he wo economies unde conside a ion
only a ises when hese shocks ha e pe manen e¤ec s on he ela i e p ices and on he
sec o al composi ion o consump ion in he economy wi h wo goods.
[Inse Figu e 6]
6. Concluding ema ks
We ha e analyzed he ansi ional dynamics o an endogenous g ow h model wi h wo
consump ion goods. We ha e shown ha he g ow h a e o expendi u e no only
depends on he in e es a e, bu also on he g ow h a e o he ela i e p ice o
consump ion goods. Con e gence in his case may be de e mined by wo di¤e en
o ces: he diminishing e u ns o capi al and he g ow h o p ices. In pa icula ,
his esul a ises when he wo consump ion goods a e no Edgewo h independen
and he echnologies p oducing he wo consump ion goods ha e di¤e en capi al
in ensi ies. These g ow h e¤ec s o ela i e p ices yield in e es ing di¤e ences wi h
espec o he ansi ional dynamics ob ained in he s anda d g ow h model wi h a
unique consump ion good. We illus a e hese di¤e ences using a g ow h model wi h
wo capi al s ocks ha we iden i y wi h human and physical capi al. Fi s , we show ha
in con as wi h he s anda d g ow h model, con e gence in he g ow h a e may occu
om abo e i he ini ial alue o he a io o physical o human capi al is la ge han
i s s a iona y alue and may occu om below o he wise. Second, we show ha he
g ow h a e o consump ion expendi u e may exhibi a non-mono onic beha io when
he wo a o emen ioned dynamic o ces ha e opposi e g ow h e¤ec s. These di¤e ences
in he ansi ion ha e o he no ewo hy implica ions.
Fi s , economies wi h he same in e es a e may exhibi di¤e en g ow h a es
o consump ion along he ansi ion. The e o e, ou model p o ides an addi ional
explana ion o he c oss-coun y di¤e ences in he g ow h a es. Rebelo (1992) shows
ha he in oduc ion o a minimum consump ion equi emen also implies ha he
g ow h a es do no equalize. This occu s because he minimum consump ion makes
p e e ences non-homo he ic so ha he IES is no longe cons an along he ansi ion.
In his amewo k, con e gence is d i en by he in e es a e and by he ime- a ying
IES. Mo e ecen ly, S ege (2006) shows ha , i he e a e he e ogeneous consump ion
goods and a unique capi al s ock, hen he IES is no cons an and he g ow h a es do
no equalize. Ob iously, he de i es his esul when p e e ences a e non-homo he ic. In
con as , we show ha , when he e a e he e ogeneous consump ion goods, he g ow h
a es a e di¤e en e en wi h a cons an IES because o he e¤ec o he g ow h o he
ela i e p ices along he ansi ion.
20
The p e ious ema k can be illus a ed in a di¤e en way. By combining (3.8), (3.3),
(3.19) and (4.3) we ob ain ha he a e o g ow h o consump ion expendi u e sa is…es
_c
c=(ph) = 1
 +w
ph+
():
This equa ion shows ha he a e o g ow h o o al expendi u e depends bo h on
he in e es a e and on he wage a e when 6= 0:This implies ha c oss-coun y
di¤e ences in he g ow h a es will also be explained by wage di¤e en ials when 6= 0
(i.e., when he e a e se e al consump ion goods ha a e Edgewo h dependen and
p oduced by echnologies wi h di¤e en capi al in ensi y). Mo eo e , o alues o 
close o he IES ;in e es a e di¤e en ials will no explain c oss coun y di¤e ences in
he g ow h a es.
Acco ding o ou esul s, he wel a e cos o shocks will also depend on he
sec o al composi ion o he composi e consump ion good. The ela ionship be ween
he wel a e cos o shocks and he sec o al composi ion o consump ion expendi u e
will be pa icula ly s ong when he shocks pe manen ly modi y he alue o ela i e
p ices. In his case, he e¤ec o hese shocks on he cos o he composi e consump ion
good will depend on i s sec o al composi ion. We ha e shown ha biased echnological
shocks ha inc ease he gap be ween he e u n on physical and human capi al cause
la ge and pe manen e¤ec s on p ices. We ha e also shown ha he wel a e cos o
hese shocks depends on he in ensi y o he di ec g ow h e¤ec o dynamic p ice
adjus men . The e o e, his g ow h e¤ec o ela i e p ice is an unexplo ed channel
a¤ec ing he pe sis ence and p opaga ion o shocks.
We summa ize ou analysis by saying ha he esul s ob ained in agg ega e g ow h
models wi h a single consump ion good canno be gene alized o mo e disagg ega ed
models wi h he e ogeneous consump ion goods. In hese disagg ega ed models, he
wel a e cos s o shocks depend on he alue o he pa ame e s measu ing he sec o al
composi ion o consump ion and on he physical capi al in ensi ies o he sec o s
p oducing hese consump ion goods. The e o e, he empi ical es ima ion o he sec o al
composi ion pa ame e s should be an impo an conce n o u u e esea ch on he
assessmen o he wel a e cos o mac oeconomic shocks.
A na u al ex ension o ou pape is o in oduce a minimum consump ion
equi emen in one o he consump ion goods. The p ice o his good will be high in he
ini ial s ages o de elopmen since he minimum consump ion equi emen will induce
a high ma ginal u ili y o his good. Then, as he economy de elops, he p ice will all
sha ply un il con e gence is a ained. The e o e, i seems ha he in oduc ion o a
minimum consump ion may accele a e he change o p ices and, hence, he in oduc ion
o his consump ion equi emen may inc ease he e¤ec o he g ow h o he ela i e
p ice on bo h he g ow h a e o consump ion expendi u es and on he wel a e cos o
shocks.
21

Re e ences
[1] Acemoglu, D. and Gue ie i V. (2008). “Capi al Deepening and Nonbalanced
Economic G ow h,”Jou nal o Poli ical Economy 116, 467-498.
[2] Al a ez-Cuad ado, F., Mon e io, G. and Tu no sky, S. (2004). “Habi Fo ma ion,
Ca ching-up wi h he Joneses, and Economic G ow h,” Jou nal o Economic
G ow h 9, 47-80.
[3] Bond E., Wang P. and Yip C. (1996). “A Gene al Two-Sec o Model o Endogenous
G ow h wi h Human and Physical Capi al: Balanced G ow h and T ansi ional
Dynamics,”Jou nal o Economic Theo y 68, 149-173.
[4] Caballé J. and San os M. (1993). “On Endogenous G ow h wi h Physical and
Human Capi al,”Jou nal o Poli ical Economy 101, 1042-1067.
[5] Eche a ia, C. (1997). “Changes in Sec o al Composi ion Associa ed wi h
Economic G ow h,”In e na ional Economic Re iew 38, 431-452.
[6] Kongsamun , P., Rebelo, S. and Xie, D. (2001). “Beyond Balanced G ow h,”
Re iew o Economic S udies 68, 869-882.
[7] Lucas, R. E. (1987). “Models o Business Cycles,”Basil Blackwell.
[8] Lucas, R. (1988). “On he Mechanics o Economic De elopmen ,” Jou nal o
Mone a y Economics 22, 3-42.
[9] Mulligan C. and Sala-i-Ma ín X. (1993). “T ansi ional Dynamics in Two-Sec o
Models o Endogenous G ow h,”Qua e ly Jou nal o Economics 108, 737-773.
[10] Ngai, R. and Pissa ides, C. (2007). “S uc u al Change in a Mul i-sec o Model o
G ow h,”Ame ican Economic Re iew 97, 429-443.
[11] Pe ez, F. and Guillo, D. (2010). “Reexamining he Role o Land in Economic
G ow h,”Manusc ip .
[12] Pe li R. and Sakella is P. (1998). “Human Capi al Fo ma ion and Business Cycle
Pe sis ence,”Jou nal o Mone a y Economics 42, 67-92.
[13] Ramsey, F.P. (1928). “A Ma hema ical Theo y o Sa ing,”Economic Jou nal 38,
543–559.
[14] Rebelo, S. (1991). “Long- un Policy Analysis and Long- un G ow h,”Jou nal o
Poli ical Economy 99, 500-521.
[15] Rebelo, S. (1992). “G ow h in Open Economies,”Ca negie-Roches e Con e ence
Se ies on Public Policy 36, 5-46.
[16] Reiss, J. P. (2000). “On he Con e gence Speed in G ow h Models,” FEMM
Wo king. Pape 22/2000.
22
[17] S ege , T.M., (2000). “Economic G ow h wi h Subsis ence Consump ion,”Jou nal
o De elopmen Economics 62, 343-361.
[18] S ege , T.M., (2006). “He e ogeneous Consump ion Goods, Sec o al Change and
Economic G ow h,”S udies in Nonlinea Dynamics and Econome ics 10, No. 1,
A icle 2.
[19] Uzawa, H. (1965). “Op imum Technical Change in an Agg ega i e Model o
Economic G ow h,”In e na ional Economic Re iew 60, 12-31.
23
A. Appendix
Solu ion o he consume ’s op imiza ion p oblem.
The Hamil onian unc ion associa ed wi h he maximiza ion o (3.1) subjec o
(2.5), (2.6) and (2.7) is
H=e U(c1; c2) +
(wh + k c1pc2IkphIh) + 1(Ikk) + 2(Ihh);
whe e ,1, and 2a e he co-s a e a iables co esponding o he cons ain s (2.5),
(2.6) and (2.7), espec i ely. The … s o de condi ions a e
e 2
6
4
c
1c1
21
c13
7
5= 0;(A.1)
e 2
6
4
(1 )c
1c1
21
c23
7
5p = 0;(A.2)
=1;(A.3)
ph=2;(A.4)
 1=_1;(A.5)
w 2=_2:(A.6)
Combining (A.1) and (A.2), we ob ain (3.2) and
_c2
c2
=_c1
c1
_p
p:(A.7)
Using (A.3) and (A.4), we ob ain
ph1=2;
which implies ha _ph
ph
+_1
1
=_2
2
;
and (3.3) ollows om using (A.5) and (A.6). Combining (A.1), (A.3) and (A.5), we
ob ain
 +=+ [(1 )1] _c1
c1+ (1 ) (1 )_c2
c2;
and (3.4) ollows om using (A.7). Finally, he ans e sali y condi ions (3.5) and (3.6)
ollow om combining (A.1) and (3.2).
P oo o P oposi ion 3.2. The uniqueness o p ollows om he mono onici y o
(p), which can be shown using (3.26),
0(p) = "(1 )A1 1
1p1

#"+  1
'
!p1+
#>(<) 0 i  < (>);
24
and he ac ha lim
p!0(p) = 1(1)and lim
p!1(p) = 1(1)when  < (>):
Combining (3.20), (3.21) and (3.22), we ob ain
u1=z3z
z3z1
+ 1
pA2z
2!z2z3
z3z1qz (A.8)
and
1u1u2=zz1
z3z1
+ 1
pA2z
2!z1z2
z3z1qz: (A.9)
In a s eady s a e, equa ions (3.25) and (3.24) simpli y o
1u
1u
2=g+
A3(z
3);
A1u
1(z
1)
zq=g+:
By using (A.8) and (A.9), he p e ious wo equa ions can be ew i en as he ollowing
sys em o wo equa ions:
z+ 1
pA2(z
2)!
| {z }
1
(z
1z
2)qz=g+
A3(z
3)(z
3z
1) + z
1
| {z }
2
;
z
3+
1(z
2z
3)(z
3z
1)
A1(z
1)qz=(z
3z
1)g+
A1(z
1)+ 1
| {z }
3
z:
The s eady s a e alues o zand qa e he unique solu ion o his sys em o equa ions
and hey a e equal o
z=
12(z
2z
3) + 1(z
1z
2)z
32(z
3z
1)
A1(z
1)
1(z
2z
3) + 13(z
1z
2)z
3z
1
A1(z
1)
;
and
q=23z
3
12(z
2z
3)2(z
3z
1)
A1(z
1)+1(z
1z
2)z
3;
whe e he s eady-s a e alues o zi; i = 1;2;3g;sa is y z
i= i(p)1
as ollows om
(3.15).
25
Figu e 3. T ansi ional dynamics wi h = 2:357
—Economy wi h = 0:3- - - Economy wi h = 1
32

Figu e 4. T ansi ional dynamics wi h = 2:7143
—Economy wi h = 0:3- - - Economy wi h = 1
33
Figu e 5. Dynamic e¤ec s o a biased echnological shock when = 2
—Economy wi h = 0:3- - - Economy wi h = 1
34
Figu e 6. Dynamic e¤ec s o an unbiased echnological shock when = 2
—Economy wi h = 0:3- - - Economy wi h = 1
35
Table 1. Wel a e cos o imbalances in he capi al a io
z0= (0:75) z
  = 0:3(a) = 1 (b) a=b
2 7:0608% 5:8959% 1:1976
2:357 7:0619% 5:8954% 1:1979
2:7143 7:0634% 5:8959% 1:1980
z0= (1=0:75) z
  = 0:3(a) = 1 (b) a=b
27:6300% 6:5049% 1:1730
2:357 7:6284% 6:5039% 1:1729
2:7143 7:6275% 6:5031% 1:1729
Table 2. Wel a e cos o echnological shocks (= 2)
Type o shock = 0:3(a) = 1 (b) a=b
Sec o al biased: A1=0:15A114:3821% 26:4386% 0:5440
Sec o al unbiased: A1
A1
=A2
A2
=A3
A3
=0:05 13:5843% 13:3788% 1:0154
36