InformationTechnologies, EconomicG rowth
andP roductivityShocks
ClaudioM attalia
¤September2002
A bstract
T his paperdevelops amulti-sectoralendogenous growth modelin ordertoreproducesomeoftheessentialcharacteristicsoftheso-called ”ICT R evolution”. T heeconomyconsists offoursectors andthemost importantfeatures aretheembodiednatureoftechnologicalprogress, the horizontal di¤ erentiation and the ”lab-equipment” speci…cation ofR &D . A fterthedescriptionofthedi¤ erentsectors, theequilibrium conditionsareobtained, thebalancedgrowthpathischaracterizedan-alyticallyandthecorrespondingsteadystatesystem is derived. From this system some analyticalresults can be obtained, in particularit turns outthatshocks ontheproductivityofthedi¤ erentsectors have permanente¤ ects onlong-term growth(contrarytotheversionofthe modelwithoutthe ”lab-equipment” assumption, where only a shock on the productivity oftheR &D sectorin‡uences long-term growth). T hese results are con…rmed, in the last part of the paper, by the numerical simulation ofthe model, that allows also to analyse the short-run response ofthe system tothe di¤ erentshocks thatcan hit theeconomyandtostudytherobustness ofthemodel.
Keywords: Information T echnology, Endogenous G rowth. JournalofEconomicL iterature: E22, O 40, C63.
¤D ipartimento di Statistica e M atematica A pplicata alle Scienze U mane ”D iego de
Castro”, U niversita’ degli Studi di Torino (Italy) and IR ES, U niversite’ catholique de L ouvain, L ouvain-la-N euve (B elgium). E-mail: claudio.mattalia@ unito.it. T his paperis partofthe A R C P rogramme ”G rowth and Incentive D esign”. T he authorthanks R aouf B oucekkineforveryhelpfulcomments and suggestions.
1 Introduction
T hesectorofInformationandCommunicationT echnologies (ICT ) has been recentlyconsideredoffundamentalimportanceintheexplanationoftheeco-nomicperformanceofseveralcountries. Inparticular, thestrongproductivity growthregisteredinthecomputersector(i.e. intheproductionofhardware) has ledsomeanalysts toconcludethattheeraofa”N ewEconomy” has be-gun, an erathatcan be considered asortof”T hird IndustrialR evolution” inwhichinformationandcommunicationtechnologiescanbecomparedwith thegreatinventions ofthepastthatcharacterizedthetraditionalIndustrial R evolution.
Forallthesereasons, agreatattention has been devotedtothestudyof whathasbeencalledthe”ICT R evolution” andofitse¤ ectsontheeconomy, andthedebateislargelyopen, bothfrom anempiricalandfrom atheoretical pointofview.
O ntheempiricalside, themainstudies (G ordon(19 9 9 , 2000), Jorgenson and Stiroh (2000), O liner and Sichel (2000), W helan (2000)) outline the strongproductivitygrowthinthecomputersector(particularlyintheyears 19 9 5-19 9 9 ), butatthesametimeevidencealsoproblems ofmeasurementof therealcontributionofICT tothegrowthandproductivityoftheeconomy, togetherwith the factthatthe productivity growth in thecomputersector has notbeen accompanied by spillovers from this sectortothe restofthe economy.
O nthetheoreticalside(themostimportantcontributionsareG reenwood and Y orukoglu (19 9 7 ), G reenwood and Jovanovic(19 9 8, 19 9 9 ), H obijn and Jovanovic (19 9 9 ), Jovanovic and R ousseau (2000)) ithas been underlined the importance ofembodimentoftechnologicalprogress (i.e. the factthat only the newmachines incorporate the latesttechnologicaladvances), and atthesametimethefactthatthe”ICT R evolution” has beenaccompanied bysome”puzzlingphenomena”.
In fact, itis possible toobserve thatthis ”revolution” has been charac-terized bye¤ ects both on therealand on the…nancialsideoftheeconomy, with the emergence ofsome ”puzzling” aspects. In particular, on the real sidetherehasbeenaninitialstrongdecreaseintheproductivityofthewhole economy(theso-called ”productivityslowdown”) immediatelyafterthebe-ginningofthe”ICT R evolution” (intheearly’7 0s - themicroprocessor, that can be considered the ”starting point” ofthis revolution, was invented in 19 7 1 -), followed onlyrecentlyby arise (as outlined above, in the late’9 0s
the rise ofproductivity in the computersectorhas been very strong, more than 40% in the U SA ). A tthe same time, on the …nancialside there has beenaninitialstrongdecreaseintheratiobetweenstockmarketcapitaliza-tion and G D P (from about1 atthe beginning ofthe ’7 0s to 0.45 in 19 7 4 fortheleadingO ECD countries), followedonlyrecentlybyarise(todaythis ratiois closeto2).
T he main explanations that have been proposed to this situation are basedontheideathattheinitialdrop inproductivity(ontherealsideofthe economy) is duetoanadoptionperiodofthenewtechnologies (becausethe pre-existing…rms arenotabletouseimmediatelythesenewtechnologies at theirfullpotential);this period is characterized by learningcosts and slow di¤usion (and itis preciselyin this phasethatthe”productivityslowdown” takes place), and itis followed byan ageofmaturityduringwhich theICT sectorstarts drivingthewholeeconomy. Furthermore, accordingtothecon-tributionsproposed, theinitialdrop inthestockmarketcapitalization/G D P ratio(on the …nancialside ofthe economy) should be due tothe factthat amajortechnologicalinnovation determines atemporaryundervaluation of the stock market. In fact, again, old …rms are notable to implementthe newtechnologies and new…rms are created to use these technologies; ini-tially these…rms donot”appear” in the stockmarket(forinstance ittook 10 years toM icrosofttogopublic), andthis determines thedrop intheratio considered, whilewhen thesenew…rms areIP O ’dtheclaims totheirfuture dividends enterthe stock market, and this determines a rise in the same ratio.
T hemodelpresentedheretakesadi¤ erentviewandtriestoexplainsome oftheessentialfeatures ofthe”ICT R evolution” consideringtheframework ofendogenousgrowththeories. Inparticular, itisaR omer-likemodel(19 9 0) inordertocapturetheR &D e¤ortofthe…rms operatingintheICT sector, andinadditionitconsidersembodiedtechnologicalprogress. M oreprecisely, itis based on the originalcontribution developed by B oucekkine and de la Croix (2001), with the main di¤ erence represented by the ”lab-equipment” speci…cation fortheR &D sector(seeR ivera-B atiz and R omer(19 9 1)), and itis a multi-sectoralmodelofendogenous growth thatreproduces some of theessentialcharacteristics oftheICT -basedeconomy, inparticulartheem-bodied natureoftechnologicalprogress (sincethe technologicalinnovations thatcharacterize the ICT sectorare typically embodied in the newcapital goods), thepreminentroleoftheR &D sector(sincetheamountofresources
devoted to research is particularly high, especially in the U SA ), and the linkbetween innovationand marketpower(sinceICT markets aretypically non-competitive).
T he paperis organized as follows. Section 2 provides the illustration of themodelwith thedescription ofthedi¤ erentsectors thatcharacterizethe economy. Section 3 describes the equilibrium conditions, characterizes an-alytically the balanced growth path and derives the corresponding steady state system. From this system it is possible to …nd some analytical re-sults concerning the e¤ ects on growth ofdi¤ erentshocks thatcan interest the economy. In particular, itturns outthatshocks on the productivity of the…nalgood sector, oftheequipmentsectorand oftheintermediategood sectorhavepermanente¤ ects on long-term growth. Section 4 considers the numericalsimulation ofa calibrated version ofthe model, con…rming the analytical…ndings and giving some furtherinsights, concerning the short-run response ofthe system tothe shocks and the robustness ofthe model. Section 5 concludes.
2 T he model
T he modeldeveloped is based on B oucekkine and de la Croix (2001) and R omer(19 9 0), and is a multi-sectoralmodelwritten in discrete time with in…nitehorizon(timegoes from 0 to1 ), endogenous growth andhorizontal di¤erentiation. T heeconomyischaracterizedby4sectors: the…nalgoodsec-tor, theequipmentsecthe…nalgoodsec-tor, theintermediategoodsectorandtheR &D sector. Inparticular, the…nalgoodsectorproduces acompositegood(usedtocon-sumeortoinvestinphysicalcapital) usinge¢cientcapital(boughtfrom the equipmentsector) andlabor;theequipmentsectorproduces e¢cientcapital (soldtothe…nalgoodsector) usingphysicalcapital(thatcanbeinterpreted as hardware) boughtfrom the …nalgood producers and immaterialcapital (thatcanbeinterpretedassoftware) boughtfrom theintermediategoodpro-ducers;the intermediate good sectorproduces the immaterialcapital(sold to the equipmentsector) using only labor; the R &D sectorresearches for newvarieties ofimmaterialcapitaland in this way increases the range of softwares (horizontaldi¤erentiation).
Inthis modeltechnologicalprogress is mainlyembodied(theideais that thenewsoftwarescanonlyberunonthemostrecenthardware) andtheinno-vators haveamarketpowerrepresentedbycopyrights, in ordertostimulate
innovation and growth (in particular, innovation corresponds toan expan-sion in the varieties ofsoftwares thatare available). A llthese elements are importanttoreproducetheessentialcharacteristics oftheICT sector.
2.1 T he …nalgoodsector
T he …nalgood sectorproduces a composite good (used to consume orto invest in physical capital) using e¢cient capital (bought from the equip-mentsector) andlabor. P roductionis obtainedthroughthefollowingCobb-D ouglas technology(as inSolow(19 60)):
Yt= z tKt®L1¡®t ® 2 [0 ;1] (1)
wherez trepresentstotalfactorproductivity(disembodiedtechnologicalprogress).
T hestockofcapitalis de…ned as: Kt=
t
X
s=¡1
Es(1¡±)t¡s (2)
where Es represents the e¢cientcapitalboughtfrom the equipmentsector
attimes and ± is thephysicaldepreciationrate(constant).
T hediscountedpro…ts ofinvestingEtine¢cientcapitalaregivenby:
¼t= 1 X s= t [Ys¡wsLs]R st¡dtEt where: Rt t= 1 Rts= Qs ¿= t+ 1 ³ 1 1+ r¿ ´
representthediscountfactors attimetand attimes respectively, r¿ is the
interestrateattime¿, wsis thewageforlaborinputattimes anddtis the
price ofe¢cientcapitalattime t. T he representative …rm chooses e¢cient capitaland laborinputin orderto maximize its discounted pro…ts taking prices as given andsubjecttoits technologicalconstraints:
m ax
Et;fLsg1s= t ¼t
T he…rstorderconditions characterizingan interiormaximum for¼tare: ® E®¡1 t 1 X s= t Rs tz s(1¡±)® (s¡t)L1¡®s = dt (3) (1¡® )z sKs®L¡®s = ws 8s ¸t (4)
and from (4 )weobtain:
Lt= µ (1¡® )z t wt ¶1 ® Kt (5)
thatis thedemand forlaborbythe…nalgood sector.
2.2 T he equipmentsector
T heequipmentsectorproducese¢cientcapital(soldtothe…nalgoodsector) usingphysicalcapital(hardware) boughtfrom the…nalgood producers and immaterialcapital(software) boughtfrom theintermediategoodproducers. E¢cientcapitalis producedwithaconstantreturntoscaletechnology:
Et= etQ¸tIt1¡¸ ¸ 2 (0 ;1) (6)
whereetis aproductivityvariable, Itrepresents physicalcapital(hardware)
and Qtrepresents immaterialcapital(software). T he immaterialcapitalis
builtfrom a series ofspecialized intermediate goods, accordingtoa D ixit-Stiglitz CES function:
Q t= µZnt 0 x¾¡1¾ i;t d i ¶ ¾ ¾¡1 (7 ) wherentis the numberofvarieties ofintermediateinputavailablein t, xi;t
is thequantity ofintermediateinputofvarietyiused in tand ¾ > 1 is the elasticityofsubstitutionbetweentwovarieties. T hepro…ts oftheequipmentsectorattimetare: ¼0t= dtEt¡It¡ Znt 0 pi;txi;td i
wherepi;tisthepriceofsoftwareofvarietyiattimet. T herepresentative…rm
chooses theinvestmentinphysicalcapitalandinimmaterialcapitalinorder to maximize pro…ts taking prices as given and subjectto its technological constraints:
m ax
It;xi;t ¼0t
s:t: (6);(7) T he…rstorderconditions forthis problem are:
(1¡¸)dtetqt¸ = 1 (8) ¸ dtetqt¸¡1 µ Qt xi;t ¶1 ¾ = pi;t 8i2 [0 ;nt] (9 )
whereqt= QIttisthesoftware-hardwareratio, andfrom (8) and(9 ) weobtain:
xi;t= µ Á qt ¶¾ Qtp¡¾i;t (10)
thatis the demand forintermediate inputiby the …rms ofthe equipment
sectorattimet(herewehavede…nedÁ = ¸
1¡¸).
2.3 T he intermediate goodsector
T heintermediategoodsectorproduces immaterialcapital(software, soldto theequipmentsector) anditresearches fornewvarieties, inordertoexpand therangeofsoftware.
T he variety i ofsoftware is produced according to a lineartechnology thatuses laboras theonlyinput:
xi;t= ¿tLi;t (11)
where Li;tis the labor employed in the intermediate good sector and ¿t
represents laborproductivity. T heproducerbehaves monopolistically(since marketpoweris given by the presence ofcopyrights which have an in…nite
lifetime- i.e. theinventorofanewvarietyofsoftwareobtainsthesecopyrights forever-) andits pro…tis:
¼00i;t= pi;txi;t¡wtLi;t=
µ pi;t¡ wt ¿t ¶ xi;t
T he price ofoutputis chosen so as to maximize this pro…tsubjectto the demand formulated by the equipmentsector, hence the problem solved by the…rm is:
m ax
pi;t ¼00i;t s:t: (10 ) T he…rstorderconditionforthis problem is:
µ Á qt ¶¾ Qtp¡¾i;t = ¾ µ pi;t¡ wt ¿t ¶ µ Á qt ¶¾ Qtp¡¾¡1i;t
from which weget: pi;t= µ ¾ ¾¡1 ¶ wt ¿t 8i2 [0 ;nt ] (12)
i.e. theoutputpriceis amark-up overunitlaborcost.
2.4 T he R &D sector
B esidesproducingsoftwares, theintermediategoodsectorresearchesfornew varietiesofimmaterialcapital, inordertoexpandtheirrange. Inthisversion ofthemodelweassumetheso-called”lab equipment” speci…cationofR &D (seeR ivera-B atiz and R omer(19 9 1)), accordingtowhich thecosttocreate anewtypeofproduct(i.e. anewvariety ofsoftware) is …xed at´ units of Y . T here willbe entry ofnew…rms in the economy untilthis costis equal tothediscounted‡owofpro…tslinkedtooneinvention, andthisequilibrium conditioncanbewrittenas: ´ = 1 X z = t R zt¼00i;z
and since: ¼00i;t= µ pi;t¡ wt ¿t ¶ xi;t= 1 ¾ ¡1 wt ¿t xi;t thefree-entrycondition is: ´ = 1 X z = t Rz t 1 ¾ ¡1 wz ¿t xi;z
2.5 H ouseholdbehavior
A fterthe4 sectors thatcharacterizethe economy itis alsopossibletocon-siderthe household presentin this economy. W ith referencetothis aspect, the representative household consumes, saves forfuture consumption and supplies labor. T heutilityoftherepresentativehousehold attime0 is:
u0 = 1
X
t= 0
½tlnCt
i.e. itis thediscounted sum ofinstantaneous utilities from 0 to1 , where½ is thepsychologicaldiscountfactorand theinstantaneous utilityfunctionis assumedlogarithmic. T hecorrespondingbudgetconstraintis:
At+ 1 = (1 + rt+ 1)At+ wtL¡Ct (13)
whereAtrepresents theassets detained bythehouseholdattimet.
T herepresentativehouseholdchoosestheassetsdetainedinordertomax-imizeits discountedutilitysubjecttothebudgetconstraint: m ax fAt+ 1g1t= 0 u0 s:t: (13)
and the…rstordercondition forthis problem leads to: Ct+ 1
Ct
= (1 + rt+ 1)½ (14)
that, togetherwith the usual transversality condition, is su¢cient foran optimum.
3 T he equilibrium
Itisnowpossibletocharacterizetheequilibrium oftheeconomyinthemodel considered, thatis determined by the equilibrium on the labormarketand on the…nalgoodmarket.
Firstofall, equilibrium on thelabormarketimplies thatthelaborforce isemployedeitherinthe…nalgoodsectororintheintermediategoodsector:
L = Lt+
Znt
0
Li;td i (15)
wherethesupplyoflaborcan benormalizedto1 (i.e. L = 1). Equilibrium onthe…nalgood market, then, implies:
Yt= Ct+ It+ ´4 nt (16)
where´ 4 ntis thecostofresearch fornewvarieties.
3.1 T he equilibrium conditions
W ecannowderivethedi¤erentequilibrium conditions, thatsummarizethe …rst orderoptimality conditions and the market equilibrium relationships derived above.
Firstofall, the demand forlaborby the …nalgood sectoris given by equation (5), while the demand forlaborby the intermediate good sector canbeobtainedfrom equations (10 ), (11) and(12 ) and is givenby:
Znt 0 Li;td i= nt µ ¾ ¡1 ¾ ¶¾ µ Á wt ¶¾ q1¡¾t It¿¾t¡1
A s a consequence, the equilibrium on the labormarket, given by equation (15), is: µ (1¡® )z t wt ¶1 ® Kt+ nt µ ¾ ¡1 ¾ ¶¾µ Á wt ¶¾ qt1¡¾It¿¾t¡1 = 1 (17 )
T heequilibrium onthe…nalgoodmarketcanbeobtainedfrom equations (1), (5) and (16) and is:
z ®1 tKt µ 1¡® wt ¶1¡® ® = Ct+ It+ ´4 nt (18)
T helawofmotionforqtcanbeobtainedsubstitutingtheexpression(5)
into(3), usingthede…nitionofKsandreplacingthevalueofdtfrom (8), so
thatweget: ® z 1 ® t(1¡¸)etqt¸ µ 1¡® wt ¶1¡® ® = 1¡ µ 1¡± 1 + rt+ 1 ¶ µ et et+ 1 ¶ µ qt qt+ 1 ¶¸ (19 ) T heoptimalconsumption is then givenbyequation (14 ):
Ct+ 1
Ct
= (1 + rt+ 1)½ (20)
whiletheaccumulationruleofcapitalis obtainedfrom equations (2 ) and(6) and is:
Kt= (1¡±)Kt¡1+ etq¸tIt (21)
whereetq¸tcan beseen as ameasureofembodied technologicalprogress (in
contrastto the variable z tthatappears in the …nalgood sectorand that
measures disembodied orneutraltechnologicalprogress).
T he value ofQtcan be determined using equations (7), (10 ) and (12 )
thatleadto: wtqt ¿tÁ = n 1 ¾¡1 t µ ¾ ¡1 ¾ ¶ (22) thatlinks theembodiedtechnologicalprogress totheexpansion in thevari-eties ofintermediateproducts.
Finally, thefree-entryconditioncan bewritten as:
´ = Á ¾ ¿1¡¾t (¾ ¡1)1¡¾¾¾ 1 X z = t R ztwz1¡¾qz1¡¾Iz and then: ´Rtt+ 1= Á ¾ ¿1¡¾t (¾ ¡1)1¡¾¾¾ 1 X z = t+ 1 R ztwz1¡¾q1z ¡¾Iz
T hetwoexpressions fortand t+ 1 aretherefore: ¿1t¡¾(¾ ¡1)1¡¾¾¾ Á¾ ´ = 1 X z = t Rztw1¡¾z q1¡¾z Iz
¿1¡¾t (¾ ¡1) 1¡¾¾¾ Á¾ ´R t+ 1 t = 1 X z = t+ 1 Rztw1¡¾z qz1¡¾Iz
and subtractingthesecond from the…rst: ¿1t¡¾(¾ ¡1)1¡¾ ¾¾
Á¾
´rt+ 1
1 + rt+ 1
= wt1¡¾qt1¡¾It (23)
T heseresults can besummarizedinthefollowingP roposition:
P roposition1 G iven the initialconditions K¡1 and n¡1 an equilibrium is
apath:
fwt;qt;Ct;It;Kt;nt;rt+ 1gt¸0
thatsatis…es theequations (17) ¡(2 3) illustratedabove.
3.2 T he balancedgrowthpath
A fterthe characterization ofthe equilibrium, the nextstep is the analysis of the balanced growth path ofthe model. In this case we assume that the exogenous productivity variables z t, etand ¿tand the interestrate rt
are constantin the long term, while each endogenous variable grows ata constantrate alongabalanced growth path. In general, ifgx is thegrowth
factorofthevariablex and x is theinitiallevelofthevariable, wehave:
xt= xgxt (24)
Sinceabalancedgrowthpathmustsatisfytheequations(17)¡(2 3), wehave sevenrestrictions amongthevarious growth factors, thatare:
gw;gq;gC;gI;gK;gn
In particular, itis possibletorewriteeachoftheequations (17) ¡(2 3) sub-stitutingeachvariablewiththecorrespondingexpressionsimilarto(2 4 ). In this way, from equation (17) weget:
gK(gw)¡
1
® = 1 = gn (gw)¾
From equation(18) weobtain:
gC = gI = gn = gK(gw)¡
1¡®
® (26)
(and from the good marketequilibrium also gY = gC = gI = gn). From
equation(19 ) weget:
(gq)¸(gw)¡
1¡®
® = 1 (27 )
From equation(2 0 ) wehave:
gC = (1 + r)½ (28)
From equation(2 1) weobtain:
gK = (gq)¸gI (29 )
From equation(2 2 ) weget:
gwgq= g
1 ¾¡1
n (30)
Finally, from equation (2 3) wehave:
(gw)1¡¾(gq)1¡¾gI = 1 (31)
In correspondence ofa balanced growth path, the various growth rates mustthereforesatisfytherestrictionsexpressedbyequations(2 5)¡(31), and usingthese restrictions itis possible todetermine the relations amongthe di¤erentgrowthrates.
Firstofall, from (2 7) wehave:
gw = (gq) ¸® 1¡® From (2 5) weget: gK = (gq) ¸ 1¡®
From (2 6) wethen have:
gC = gI = gn = (gq)
¸ ® 1¡®
T hesameresultcanbeobtainedfrom (2 9 ), thatisthereforeredundant. From (30 ) wehave: gn = (gq)(¾¡1) 1¡® + ¸® 1¡® and sincewealsohavegn = (gq) ¸ ® 1¡® itmustbe¾ = 1¡® + 2 ¸® 1¡® + ¸® (this restriction
is a consequence ofthe ”lab-equipment” speci…cation forR &D ). W e then havethat(31) is redundant, …nallyfrom (2 8) weget:
r = (gq) ¸® 1¡®
½ ¡1
In conclusion, the …ve unknowns ofthe problem (gw, gq, gY, gK, r) are
related byasystem offourequations (theequations (2 5) ¡(2 8), while(2 9) and(31) areredundantand(30 ) isusedtoobtaintheexpressionfor¾ derived above), andforgivengqalltheotherunknowns canbefound, and therefore
they are parameterized by gq. T he results are expressed in the following
P roposition:
P roposition2 Ifqtgrows ata factorgq > 1, then allthe othervariables
growatstrictlypositive rates with:
gY = gC = gI = gn = gw = (gq) ¸® 1¡® gK = (gq) ¸ 1¡®
H ence, alongabalanced growth path output, consumption, investment, numberofvarietiesandwagesgrowatthesamerate, whilethestockofcapital grows faster(sinceitincludes improvements in theembodied productivity). T hesystem is thereforeabletodisplaygrowthoftheeconomy.
3.3 T he stationarizeddynamicsystem andthe steady
state system
T he nextstep ofthe analysis is the study ofthe restrictions on the long-run levels, thatgive the additionalinformation necessary to determine gq.
Computingthese restrictions from the dynamicsystem expressed by equa-tions(17)¡(2 3) (i.e. rewritingtheequationssubstitutingeachvariablewith the correspondingexpression similarto(2 4 )), we end with 7equations for
8 unknowns (w , q, C , I , K , n, r and gq - since allthe othergrowth rates
can be expressed in terms ofgq -). T he system in terms oflevels is
there-fore undetermined (this is ausualproperty ofendogenous growth models), butitis possibletorewriteitin such awaythatwegetrid ofthis indeter-minacy. T his is done by ”stationarizing” the equations by means ofsome auxiliaryvariables, thatis byrewritingthesystem interms ofvariables that are stationary (i.e. constant) in the steady state. M ore precisely, the dy-namic system (17) ¡(2 3) can be rewritten as a function ofthe following sevenstationaryvariables: b qt= qt w 1¡® ¸ ® t b Ct= Cwtt Ibt= wItt Kbt= Kt n 1 ® t b nt= wntt gt= nnt¡1t rt
T hese variables are such thatthey are constantin the steady state, forin-stanceforbqtwehave: b qt= qt w1¸®¡® t = qg t q (w gt w) 1¡® ¸ ® = qg t q w1¡®¸ ® (gq) ¸® 1¡®t 1¡® ¸® = q w1¸®¡® = constant
and thereforetheyarestationaryvariables. A s aconsequence, itis possible toobtain the stationarized dynamicsystem correspondingtothe equations (17)¡(2 3). Inparticular, from equation(17) weobtain:
((1¡® )z t) 1 ® Kb tbn 1 ® t+ µ ¾¡1 ¾ ¶¾ Á¾¿¾t¡1bntqbt1¡¾Ibt= 1 (32) From (18) wehave: z ®1 tKbtnb 1 ® t(1¡® ) 1¡® ® = bC t+ bIt+ ´bnt µ 1¡1 gt ¶ (33) From (19 ) weget: ® z ®1 t(1¡¸)etbqt¸(1¡® ) 1¡® ® + 1¡± 1 + rt+ 1 µ et et+ 1 ¶ µ b qt b qt+ 1 ¶¸µ b nt+ 1 b ntgt+ 1 ¶1¡® ® = 1 (34) From (2 0 ) weobtain: b Ct+ 1 b Ct b nt b nt+ 1 gt+ 1= (1 + rt+ 1)½ (35)
From (2 1) wethen have: b Kt¡(1 ¡±) bKt¡1g¡ 1 ® t = etqbt¸Ibtbn¡ 1 ® t (36) From (2 2 ) weget: b qt ¿tÁ = nb 1 ¾¡1 t µ ¾ ¡1 ¾ ¶ (37 ) Finally, from (2 3) wehave:
¿1t¡¾(¾ ¡1)1¡¾¾¾
Á¾
´rt+ 1
1 + rt+ 1
= qbt1¡¾Ibt (38)
T heequations (32 ) ¡(38) representthereforethestationarized dynamic system correspondingtotheoriginalsystem formedbyequations(17)¡(2 3). A s forthe originalsystem, alsoin the stationarized one there are twopre-determined variables, bKtand gt, thereforethestationarization doesn’talter
thedynamicorderoftheoriginalsystem.
A tthis pointwe can considerthe steady state system correspondingto the stationarized system, to this end itis possible to de…ne the following stationaryvariables (correspondingtothoseintroducedabove): b q = q w1¸®¡® b C = Cw I =b wI K =b K n®1 b n = n w g = gn r
and toobservethatwecan write: lim t! + 1 z t= z t! + 1lim et= e t! + 1lim ¿t= ¿ T hestationarizedsteadystatesystem isnowgivenbythefollowingequations: ((1¡® )z )®1 Kbnb®1 + µ ¾ ¡1 ¾ ¶¾ Á¾¿¾¡1bnqb1¡¾I = 1b (39 ) z ®1Kbbn®1 (1¡® ) 1¡® ® = bC + bI + ´bn µ 1¡1 g ¶ (40)
® z ®1(1¡¸)ebq¸ (1¡® ) 1¡® ® + 1¡± 1 + rg ¡1¡®® = 1 (41) g = (1 + r)½ (42) b K h1¡(1 ¡±)g¡®1 i b n®1 = eqb¸Ib (43) ¿Á µ ¾ ¡1 ¾ ¶ b n¾¡11 = bq (44) ¿1¡¾(¾ ¡1)1¡¾¾¾ Á¾ ´r 1 + r = qb 1¡¾Ib (45)
W ethereforehaveasystem of7equations with7unknowns (bq, bC , bI , bK , b
n, g, r) thatcan be solved (atleastfrom a theoreticalpointofview). In reality, given the complexity ofthe long-run steady state, itis impossible toderive an analyticalsolution. N evertheless itis possible toobtain some interestingintermediateresults, inparticularitis possibletoexpress eachof theotherunknowns as afunction ofthegrowth factorg, because there are explicitfunctionsexpressingthelong-runlevels (bq, bC , bI , bK ,bn , r) exclusively in terms ofg.
Firstofall, from equation(4 2 ) wehave: r = ªr(g) =
g ½ ¡1 From (4 1) wethen have:
b q = ªbq(g) = à 1¡1+ ª1¡±r(g)g¡1¡®® ® z ®1(1¡¸)e (1 ¡® ) 1¡® ® ! 1 ¸ From (4 4 ) weobtain: b n = ªnb(g) = µ ¿Á ªbq(g) ¶1¡¾µ¾ ¡1 ¾ ¶1¡¾
From (4 4 ) wealsohave: b nbq1¡¾ = ¿1¡¾Á1¡¾ µ ¾¡1 ¾ ¶1¡¾ (¤) whilefrom (4 3) wehave: b Kbn®1 = e (ªqb(g)) ¸ b I 1¡(1 ¡±)g¡®1 (¤¤) and putting(¤) and(¤¤) into(39) weget:
b I = ªIb(g) = 1 ((1¡® )z )®1 e 1¡(1¡±)g¡ 1® (ªqb(g)) ¸ + ¾¡1 ¾ Á From (4 0 ) wenowhave: b C = ªCb(g) = z 1 ® e 1¡(1 ¡±)g¡®1 (ªqb(g))¸ªIb(g)(1¡® ) 1¡® ® ¡ª b I(g)¡´ µ 1¡1 g ¶ ªnb(g)
and …nallyfrom (¤¤) wehave: b K = ªKb(g) = e 1¡(1 ¡±)g¡®1 (ªbq(g))¸ ªIb(g) (ªbn(g)) 1 ® In conclusion, thefollowingresultholds:
P roposition3 A tanygrowthfactorg; thereexistexplicitfunctionsexpress-ingthe long-run levels bq, bC , bI , bK ,n , r exclusivelyin terms ofg:b
b
q = ªqb(g) C = ªb Cb(g) I = ªb Ib(g)
b
K = ªKb(g) bn = ªnb(g) r = ªr(g)
From these expressions, some otherinteresting results can be deduced. Inparticular, itcanbeobservedthatªIb(g) doesn’tdependneitheronz nor
on e noron ¿ (the variables thatmeasure the productivity, respectively, in the …nalgood sector, in the e¢cientcapitalsectorand in the intermediate good sector) - in fact, substituting in ªIb(g) the expression corresponding
-. Furthermore, the function ªbq(g) depends (negatively) on z and e and
the functions ªnb(g), ªKb(g) and ªCb(g) depend on z , e and ¿ (in particular
the …rst function is positively related to these variables, while the other two functions are negatively related to them). In addition, ifwe consider equation (4 5) and wesubstitute bI andqb1¡¾ bytheirrespectiveg-functional
expressions, we obtain an equation involving only g thatdepends on z , e and ¿, and this means thatthe longterm growth factoris a¤ected by the productivityvariables z , e and¿.
A lltheseresults canbesummarized in thefollowingP roposition:
P roposition4 A ssumingthatasolution forthesteadystatesystem exists, thelongrun values ofz ande a¤ ectthestationaryvalues bq,n, bbK and bC and thelongrunvalueof¿ a¤ ectsthestationaryvalues bn , bK and bC . Furthermore z , e and¿ have an impacton the longterm growth factorg.
A ccordingtotheseresults, permanentchanges inz t(theproductivityin
the…nalgood sector), in et(theproductivity in thee¢cientcapitalsector)
andin¿t(thelaborproductivityintheintermediategoodsector) willa¤ ect
thelongrungrowthrateoftheeconomy. T his is themaindi¤erenceofthis version ofthe model(based on the ”lab-equipment” speci…cation ofR &D ) with respecttothe version without”lab-equipment” speci…cation ofR &D , in which longterm growth turns outtobe insensitive tochanges in z tand
et(in thatcaseonlyiftheproductivityofR &D is boosted, stimulatingthe
creationofsoftwares, thereislongterm growthoftheeconomy). Considering forinstancethee¤ ects ofchanges inz t, thedi¤erencebetweentheversionof
themodelwithout”lab-equipment” andtheversionwith”lab-equipment” is basedonthefactthatinbothversions longterm growthrelies onhorizontal di¤erentiation ofR &D , butin the ”lab-equipment” version the production functionintheR &D sectoris implicitlythesameas inthe…nalgoodsector, whileintheotherversiontheproductionfunctionintheR &D sectorismore laborintensive. T heresultis thatashockonthetotalfactorproductivityof the…nalgoodsectorhasane¤ ectonlongterm growthinthe”lab-equipment” model(because itcorresponds toa shock on the productivity ofthe R &D sector, thatis the engine ofgrowth in this kind ofmodel), while itdoesn’t havethis e¤ ectintheothermodel.
T hese are the results thatcan be obtained analytically;in ordertoget furtherinsightsitis necessarytoresorttonumericalsimulation, infactfrom P roposition3 wecanderivethefollowingCorollary:
Corollary5 T here exists an explicitfunction ª (g) such thatthe longrun equilibrium growthfactorsolves the equation ª (g) = 0 .
T his means thatby using the g-functionalexpressions ofthe long run levels (thoseofP roposition 3) in anyequation ofthesteadystatesystem it is possible to obtain an explicitequation involving only g, in this way the system can be reduced to an explicitscalarequation involvingthe growth factorg, and oncethis equation is solved theremaininglongrun levels can bedeterminedusingtheexplicitg-functions. T heproblem is thattheequa-tion ª (g) = 0 (thatgives the eventualsteady state growth factor) is very complicated (given the complexity ofthe long run steady state), and itis impossibletoderiveananalyticalsolution. Forthisreason, toobtainfurther results itis necessarytoresorttothenumericalsimulationofthemodel.
4 Simulationofthe model
T he modeldescribed abovecan besimulated numericallyin ordertoverify theanalytical…ndingsandtogetsomeotherinsights, concerninginparticular the behaviorofthe economy as a consequence ofshocks thatcan hitthe system and therobustness ofthemodel. T his requires acalibration, thatis chosen in such a way thatitreproduces the essentialfeatures ofthe ”new economy” onthebasis oftheempiricalevidenceavailable.
4.1 Calibration
T he calibration ofthe model is tailored on the data ofthe U S economy, thereforethedi¤erentparameters are…xedtovalues thatcanbeconsidered reasonable on the basis ofthe empiricaldata, and they are also chosen in orderto match a series ofmoments ofthe steady state ofthe model. In particular, in the benchmark case the laborshare in the …nalgood sector 1¡® is equalto0 :65 (hence ® = 0 :35), while the share ofsoftware in the production ofe¢cientcapital¸ is equalto 0 :85 (this parameter, together with the laborproductivity in the intermediate good sector¿, is used to calibrate the size ofthe neweconomy in terms ofthe laborforce employed in the intermediate good and in the R &D sector). A s a consequence, the elasticity ofsubstitution between varieties ofsoftwares ¾ is equalto1:311.
1T hese values imply a mark-up rate ofabout4, which can be considered very high
T he rateofdepreciation ofphysicalcapital± is 10 % , and the psychological discountfactor½ is 97% . T heproductivityinthe…nalgoodsectorz is then equalto3, while the productivity in the equipmentsectore is …xed to12 (in this way we have a ratio capitalto outputof2 ) and the productivity in the intermediate good sector¿ is equalto0 :2 5 (this, togetherwith the value chosen forthe parameter¸, implies thatabout8% ofthe laborforce is employed in the intermediate good and in the R &D sector). Finally, the costofanewvarietyofsoftwareexpressedinunits ofoutput´ (thatderives from the”lab-equipment” speci…cationforR &D ) isusedtocalibratethesize ofthe R &D sectorand is equalto2 0 (in this way the R &D expenditure is approximatelyequalto3:5% ofG D P ). W ehavetherefore:
Param eter Sym bol V alue
L aborsharein the…nalgoodsector 1¡® 0 :65 Shareofsoftwarein theproductionofe¢cientcapital ¸ 0 :85 Elasticityofsubstitutionbetweenvarieties ofsofwares ¾ 1:31 R ateofdepreciationofphysicalcapital ± 0 :10 P sychologicaldiscountfactor ½ 0 :9 7 T otalfactorproductivityin the…nalgoodsector z 3 T otalfactorproductivityin theequipmentsector e 12 L aborproductivityintheintermediategood sector ¿ 0 :2 5
Costofanewvarietyofsoftwarein units ofoutput ´ 2 0
Table1: V alues oftheparameters, benchmarkcase
and the corresponding relevant moments of the steady state that are reproducedare:
L aborshareinintermediateandresearchsector 8%
Capital/outputratio 2
R &D expenditureinterms ofG D P 3:5%
Interestrate 3:9%
T able2: R elevantmoments ofthesteadystate, benchmarkcase assumptionintroducedinthis model, theelasticityofsubstitution¾ (andhencethemark-up, equalto ¾
¾¡1) is strictlylinked totheparameters ® and ¸. W hen wechooseforthese
parameters values thatallowtomatch themoments ofthesteadystateweareinterested in, this high mark-up rateappears.
W ith these values, the modelleads toagrowth rate ofoutputequalto 0 :8% peryear, thatcanbeinterpretedasthepartofoutputgrowthgenerated by embodied technicalprogress, and is in line with the available data (see forinstance G reenwood, H ercowitz and Krusell(19 9 7 )). Furthermore, for this parameterization themodelhas auniquesteadystateequilibrium with positivegrowth, andthisequilibrium islocallyasaddlepoint(forsimulations andstabilityassesmentsthe”D ynare” package- seeJuillard(19 9 6)- hasbeen used).
A tthis point, the benchmark case can be used …rstofalltoverify the e¤ectsonthecalibratedversionofthemodelofdi¤erenttypesofshocksthat can interesttheeconomy. M oreprecisely, afterthatthesteadystateforthe initialcalibration ofthe modelhas been computed, ashockon aparticular variableisconsidered(herealltheshockshaveanintensityequalto1% ), then thenewsteadystateis obtainedandthedynamicsoftransitionfrom theold steadystatetothenewoneis determined. In this wayitis alsopossibleto determinethemagnitudeofthee¤ectsoftheshocksontherelevantvariables in the shortand in the longrun, and this is an interestingcontribution of theanalysis developed.
T he benchmark case, then, can be used alsotoverify the robustness of the model. W ith reference to this aspect, starting from the benchmark it is possibletoconsidersigni…cantchanges in therelevantparameters and to simulateagainthemodel(in ordertoverifytheresponseoftheeconomyto the di¤ erentshocks when the newvalues ofthe parameters are taken into account). T he results obtained showthatthe modelis su¢ciently robust, andrepresentthereforeanotherimportantcontributionofthepresentstudy.
4.2 T he benchmarkcase: productivityshocks
T he…rstresultthatcanbeobtainedwiththecalibratedversionofthemodel described above is represented by the analysis ofthe e¤ects ofshocks that can hitthe economy. T he shocks considered are productivity shocks, more preciselyitis possibletoconsiderashockonz (theproductivityinthe…nal good sector), a shock on e (the productivity in the equipment - e¢cient capital- sector) andashockon¿ (theproductivityintheintermediategood - software- sector);itis alsopossibletoanalysehowtheeconomyreacts to areduction in ´ (the costofanewvariety ofsoftware in units ofoutput). A lltheshocks considered arepermanent(from t= 0 ) andhavean intensity
equalto1% .
T he…rstsimulationconcernsashockonthe…nalgoodsectorproductivity parameter z , that is increased permanently by 1% . From the analytical results (P roposition4 ) weknowthatthis shouldhaveanimpactonthelong term growthrate(thisisacentraldi¤erenceoftheversionofthemodelwith ”lab-equipment” with respecttotheversionwithout”lab-equipment”), and thesimulationcon…rmsthisresult. Ine¤ect, boththegrowthrateofe¢cient capitalandthegrowthrateofproduction(that, inthisversionofthemodel, is alsoequaltothe growth rate ofthenumberofpatents, i.e. ofsoftwares) increasein thelongrun (Figures 1:1 and 1:2 - thevalues reported arethose ofthe growth factors, from which those ofthe growth rates can be easily deduced -). T his is due tothe factthatthe ”lab-equipment” speci…cation implies thattheproductionfunctionintheR &D sectoris thesameas inthe …nalgood sector. A s aconsequence, an increase in the productivity ofthe …nalgoodsectoris equivalenttoanincreaseintheproductivityoftheR &D sector, and since this sectoris the engine ofgrowth in the model(through theexpansioninthevarieties ofsoftwares) this determines ane¤ ectonlong term growth.
In particular, the increase in the productivity ofthe …nalgood sector reduces the cost ofproduction of the …nal good and initially determines a reallocation ofthe laborforce favorable to the …nalgood sectorand at the expenses of the intermediate good sector. A s a consequence, in the …rstperiod immediately afterthe shock there is a very strong increase in the growth rates (almost8% ), due tothe contemporaneous increase in the productivity and in the laborforce ofthe …nalgood sector. A tthe same time, the laborforce employed in the intermediate good sector(i.e. in the production ofsoftware) is characterised by an importantreduction (about 1:3% ). A s time passes, then, the labor force employed in the …nal good sectorreduces (becauseproductivecapacity has reached its maximum) and there is again areallocation ofthis laborforce in favorofthe intermediate good sector. A s a consequence, the growth rates partially reduce, and the longrune¤ectis anincreasein both thegrowth rateofe¢cientcapitaland thegrowthrateofproduction(ofmorethan4 :5% ) withrespecttotheinitial steady state values. Forthe same reason the initialreduction in the labor forceemployedintheintermediategoodsectorisalmostcompletelyrecovered in thefollowingtwoperiods, and from t= 4 thereis asmallincrease(with respecttothe initialsteady state value). T he longrun resultis that, as a consequenceofanincreaseintheproductivityofthe…nalgoodsector, there
is asmallreallocation(about0 :1% ) ofworkers from the…nalgoodsector tothe intermediate good sector(Figure 1:3). T he lastresultconcerns the interest rate (Figure 1:4 ), that decreases immediately afterthe shock (of about0 :5% ), thenrecovers andinthelongrunhas anincreaseofabout1% withrespecttotheinitialvalue.
T hesecondsimulationconsideredconcernsashockontheequipmentsec-torproductivityparametere, thatis increased permanentlyby1% . A lsoin this case we knowfrom the analyticalresults thatthis should have an im-pacton thelongrun growth rate(contrary towhathappens in the version ofthe modelwithout”lab-equipment”), and the simulation again con…rms this result. A s before, this longrun e¤ectis due tothe factthatthe ”lab-equipment” speci…cation implies the same production function in the …nal good sectorand in theR &D sector. In this casean increasein theproduc-tivity ofthe equipmentsectorincreases the production ofe¢cientcapital andthereforeofthe…nalgood, thiscorresponds toanexpansionintheR &D sectorandsincethis sectoris theengineofgrowthinthemodeltheresultis an e¤ecton longterm growth. N evertheless, the shortrun behaviorofthe variables is very di¤ erentwith respecttothe situation thatwe have in the caseofashockon theproductivityofthe…nalgoodsector(in factitis the opposite).
Ine¤ect, theincreaseofproductivityintheequipmentsectorincreasesthe pro…tabilityofproducinge¢cientcapitalandincreasesthemarginalreturnto bothsoftwares andhardware, stimulatingthedemandfortheseinputs. T his in turn stimulates the creation and production ofsoftwares and determines an initialstrongincrease (about2 % ) in the laborfraction employed in the intermediate good sector(while in the case ofa shock on the productivity ofthe…nalgood sectorintheshortrunthereis areallocation ofworkers in favorofthe …nalgood sector), launchingthe growth ofthe economy. Since this growth is based on the expansion ofthe intermediategood sector(and notdirectlyofthe…nalgoodsector), nevertheless, theincreaseofthegrowth rates is less strong than in the …rstsimulation considered, and itrequires moretime. In fact, initially there is areduction ofboth the growth rate of e¢cientcapitaland the growth rate ofproduction (due tothe reduction of the laborforce in the …nalgood sector, thata¤ects growth directly, while in the…rstsimulation considered initiallythereis astrongincreaseofthese growthrates) - thedecreasewithrespecttotheinitialsteadystatevalues is oftheorderof3% -, then from t= 2 thesegrowthrates recoverandinthe
longruntheyincrease(ofabout1:5% ) withrespecttotheirinitialsteady statevalues(Figures2 :1 and2 :2 ). Furthermore, theinitialstrongreallocation ofworkersinfavoroftheintermediategoodsectoris notlonglasting, infact in a fewperiods the laborfraction in the software sectorreturns toalevel thatis onlyslightlyhigher(about0 :0 5% ) thantheinitialsteadystatevalue (sincelaborforcereallocates infavorofthe…nalgoodsector, contributingto theincreaseinthegrowthrates from t= 2 ), andremains tothis levelinthe longrun (Figure 2 :3). A similarbehavioris thatofthe interestrate, that immediately afterthe shock increases of2 :5% , then returns toalevelonly slightlyhigher(about0 :3% ) thantheinitialone(Figure2 :4 ).
T hesamekindofresultsobtainedconsideringashockontheproductivity oftheequipmentsectorhold when weconsiderashockon theintermediate good sectorproductivity parameter¿, increased permanently by 1% . A lso in this case, in fact, there is a permanente¤ecton growth (always due to thefactthatthe”lab-equipment” speci…cation implies thesameproduction functioninthe…nalgoodsectorandintheR &D sector). Inthissituationthe increaseintheproductivityoftheintermediategoodsectorreduces thecost ofproductionofsoftwaresandhencedeterminesaninitialstrongreallocation ofthe laborforce favorable tothis sector. In fact, in the …rstperiod after theshockthereis anincrease(ofmorethan1:5% ) inthelaborfractionthat is employed in the intermediate good sector, atthe expenses ofthe labor forceemployedinthe…nalgoodsector. A s before, sinceinthis situationthe growth ofthe economy is based on the expansion ofthe intermediate good sector(andnotdirectlyofthe…nalgoodsector), itis less strongthaninthe …rstsimulationconsideredand itrequires moretime. In fact, initiallythere is a reduction ofboth the growth rate ofe¢cientcapitaland the growth rateofproduction (duetothereduction ofthelaborforcein the…nalgood sector, thata¤ ects growthdirectly) - thedecreasewithrespecttotheinitial steady state values is ofthe orderof2 :5% -, then from t= 2 these growth rates starttorise again and in the longrun they increase (ofabout1:5% ) with respectto theirinitialsteady state values (Figures 3:1 and 3:2 ). W e alsohave thatthe increase in the laborforce employed in the intermediate goodsectoris notlonglasting, in factafterafewperiods thelaborfraction in the software sectorreturns toa levelonly slightly higher(about0 :0 5% ) than theinitialvalue(sincelaborforcereallocates infavorofthe…nalgood sector, determiningtheincreaseinthegrowthratesfrom t= 2 ) andremains tothis levelinthelongrun(Figure3:3). T hesamekindofbehavior, …nally,
can be observed forthe interestrate, thatincreases (ofabout2 % ) im-mediatelyaftertheshock, thendecreases andinthelongrunreaches alevel slightlyhigher(about0 :3% ) thantheinitialone(Figure3:4 ).
A ftertheshocks ontheproductivityvariables describedabove, itis also possible to analyse how the economy reacts to a decrease in ´, the cost to create a new variety ofsoftware in units of output (as a consequence ofthe ”lab-equipment” speci…cation adopted forR &D ), equalto1% . T he results turn outto be very close tothose obtained in the …rstsimulation, whereanincreaseintheproductivityofthe…nalgoodsectorwasconsidered. In particular, the e¤ect on the long run growth rate of a reduction in ´ is permanent(due to the factthat, in this model, the R &D sectoris the engine ofgrowth), and itis an increase in both the growth rate ofe¢cient capitalandthegrowthrateofproduction(Figures4 :1 and4 :2 ). Inparticular, the reduction in ´ determines a temporary reallocation ofthe laborforce favorable to the …nalgood sectorand atthe expenses ofthe intermediate good sector. A s a consequence, in the …rstperiod afterthe decrease of´ there is a very strongincrease in the growth rates (about8% ), due tothe increase in the laborforce ofthe …nalgood sector. A tthe same time, the labor force employed in the intermediate good sector is characterised by an importantreduction (ofthe orderof1:5% ). A s time passes, then, the laborforce employed in the …nalgood sectorreduces and there is again a reallocation ofthis laborforce in favorofthe intermediate good sector. A s aconsequence, the growth rates partiallyreduce, and the longrun e¤ ectis an increase in both thegrowth rateofe¢cientcapitaland the growth rate ofproduction ofabout4 :5% with respecttothe initialsteady state values. Forthesamereason, theinitialreductioninthelaborforceemployedinthe intermediategoodsectoris almostcompletelyrecoveredinthefollowingtwo periods, andthelongrune¤ ectisasmallincrease(about0 :1% ) withrespect tothe initialsteady state value (Figure 4 :3). T he lastresultconcerns the interestrate(Figure4 :4 ), thatdecreases immediatelyafterthedecreasein´ (ofabout1% ), then recovers and in the longrun has an increase (ofabout 1% ) with respecttotheinitialsteadystatelevel.
T heresults ofthesimulations illustratedabovecanbeusedalsotocom-pare the di¤ erent shocks. First of all, it is possible to observe that the various shocks considered a¤ ectin adi¤ erentmeasurethegrowth rates and theallocationofthelaborforcebetween…nalgood sectorand intermediate
good sector(italsoturns outthatthe shockon the productivity ofthe …nalgoodsectorandthatonthecostoftheR &D sectorproduceverysimilar consequences, and the same is true for the shock on the productivity of the equipmentsectorand forthaton the productivity ofthe intermediate goodsector). Inparticular, theincreaseintheproductivityofthe…nalgood sectorz and the decrease in the costto create a new variety ofsoftware ´ both determine a strong increase in the growth rate of e¢cientcapital and in thegrowthrateofproduction(thatthenpartiallyreduce), whilethe increase in the productivity ofthe equipmentsectore and the increase in the productivity ofthe intermediate good sector¿ have aless stronge¤ ect onthesegrowthrates (thatinitiallydecrease, thenincrease). W ithreference to the allocation ofthe laborforce, the increase in z and the decrease in ´ initially determine a reallocation ofthe laborforce favorable tothe …nal good sectorand atthe expenses ofthe intermediate good sector(then the situation reverses), while the increase in e and the increase in ¿ have the opposite e¤ ect(initially the laborforce employed in the intermediate good sectorincreases, thendecreases). Finally, withreferencetotheinterestrate, theincreasein z and thedecreasein ´ initiallydetermineadecreasein the interestrate(thatthenincreases), whiletheincreaseine andtheincreasein ¿ havetheoppositee¤ ect(initallytheinterestrateincreases, thendecreases).
A lltheseresults canbesummarized in thefollowingT able:
increaseinz increaseine
decreasein´ increasein¿
growthrateofe¢cientcapital …rst" …rst#
then # (partially) then "
growthrateofproduction …rst" …rst#
then # (partially) then "
laborforcein intermediatesector …rst# …rst"
then " then #
interestrate …rst# …rst"
then " then #
T helastaspectconcernsthequantitativee¤ectsdeterminedbythedi¤ er-enttypes ofshocks ontherelevantvariables inthelongrun. T heconclusion is thatthe increase in z and the decrease in ´ have the strongeste¤ecton growth(since, ontheonehand, the…nalgoodsectora¤ects growthdirectly, and on the otherhand the R &D sectorrepresents the engine ofgrowth in this model) - thelongrune¤ ectofthetwotypes ofshocks is an increasein the growth rate ofe¢cientcapitaland in the growth rate ofproduction of about4 :5% withrespecttotheinitialsteadystatevalues -. O nthecontrary, the increase in e and the increase in ¿ have a less stronge¤ ecton growth (since these sectors a¤ectgrowth indirectly) - the long run e¤ectofthese twoothertypes ofshocks is an increase in the growth rates ofabout1:5% withrespecttotheinitialsteadystatevalues -. Furthermore, theshocks on z and on ´ have a strongere¤ ectalso on the laborforce employed in the intermediategoodsector(that, inthelongrun, increasesofabout0 :1% with respecttotheinitialsteadystatevalue, whiletheincreaseis oftheorderof 0 :0 5% whentheshocksone andon¿ areconsidered) andontheinterestrate (the increase in the longrun is 1% with respecttothe initialsteady state value, whileitis only0 :3% whentheshocks one andon ¿ areanalysed).
A lltheseresults arereportedinthefollowingT able:
shockonz shockon e shockon´ shockon ¿ growthrateofe¢cientcapital + 4 :5% + 1:5% growthrateofproduction + 4 :5% + 1:5% laborforceinintermediatesector + 0 :1% + 0 :0 5% interestrate + 1% + 0 :3%
Table4: L ongrunquantitativee¤ ects (with respecttotheinitialsteady statelevels) ofdi¤ erenttypes ofshocks, benchmarkcase
4.3 T he benchmarkcase: robustness analysis
T hebenchmarkcasepresentedabovecanbeusedalsotoverifytherobustness ofthe model. T he idea in this case is tostartfrom the benchmarkand to modifysigni…cantlysomeparameter. W ith this newvalueoftheparameter thathas beenchanged themodelis then simulatedagain, andthee¤ects of the di¤ erentshocks are studied (exactly as in the benchmark case). Ifthe resultsaresu¢cientlyclosetothoseobtainedfortheinitialparameterization,
itispossibletoconcludethatthemodelisrobust, andthisrepresentsagood propertyofthemodelitself.
In particular, three types ofchanges in the parameters are considered. T he …rstis a change in ¸ (the share ofsoftware in the production ofe¢-cientcapital), thatis used tostudy the e¤ectofavariation in the levelof competitionintheeconomy(sincethis parameterin‡uences theelasticityof substitution ¾ and hence the mark-up rate). T he second is a change in z (theproductivityin the…nalgood sector), thatis usedtoanalysethee¤ ect ofachangein thetechnology. T hethird is achangein ´ (thecostofanew varietyofsoftwareinunitsofoutput), thatisusedtostudytheconsequences ofavariation inthecostofR &D .
T he mostinteresting aspectforthis kind ofanalysis is represented by the e¤ects ofthe di¤ erentshocks on the relevantvariables in the longrun. T hese e¤ects have been discussed, forthe benchmarkcase, in the previous subsection (and are summarized in T able 4 ); they can nowbe determined forthedi¤ erentcasesthatdepartfrom thebenchmark, inordertoverifythe robustness ofthemodel.
T he…rstvariation thatis consideredwith respecttothebenchmarkis a changeintheparameter¸ (thata¤ ects thecompetitionintheeconomy). In particular, the value ofthis parameterchanges from 0 :85 (as in the bench-mark) to0 :95 (hencethechangeis largerthan 10 % ), whilethevalues ofall theotherparameters remain unchanged with respecttothebenchmark. A t this point, thenewsteadystateofthemodelis computed, then thesimula-tions are performed (exactly as in the initialversion ofthe model) and the di¤erenttypes ofshocks considered in the benchmark case are introduced. Firstofall, withthenewvalueof¸ thegrowthrateinthesteadystateisequal to0 :96% , wethen havethatthebehavioroftheeconomy, as aconsequence oftheshocks, isveryclosetotheoneobtainedintheinitialcalibrationofthe model.Inparticular, boththeshocksonz andon´ determineaninitialvery strongincreaseinthegrowthrates(thanthenpartiallyreduce) andaninitial reductioninthelaborfractionoftheintermediatesectorandintheinterest rate(thatthenrecoverandinthelongrunincreasewithrespecttotheinitial steadystatevalues). O nthecontrary, theshocks one and on ¿ producean initialreduction in the growth rates (thatthen recoverand in thelongrun increasewithrespecttotheinitialvalues) andanincreaseinthelaborforce ofthe intermediate sectorand in the interestrate (thatthen decrease and reachalevel, inthelongrun, onlyslightlyhigherthantheinitialone). Itis thenimportanttoevaluatethemagnitudeofthee¤ ects oftheshocks onthe
di¤erentvariables in the longrun, in ordertomake acomparison with the benchmarkcase;thevalues obtainedarethefollowing:
shock shock shock shock
on z on e on¿ on´
growthrateofe¢cientcapital + 4 % + 1:4 % + 1:4 % + 4 %
growthrateofproduction + 4 % + 1:4 % + 1:3% + 4 %
laborforceinintermediatesector + 0 :2 % + 0 :1% + 0 :1% + 0 :2 %
interestrate + 1% + 0 :5% + 0 :5% + 1%
Table5: L ong-runquantitativee¤ ects (with respecttotheinitialsteady statelevels) ofdi¤ erenttypes ofshocks, case1 (higher¸ withrespecttothe
benchmark)
From theiranalysisitturnsoutthattheyareveryclosetothoseobtained in thebenchmarkcase, hencethemodelproves toberobustwithrespectto thechangeintroduced in the parameter¸. W ith referencetothis …rstcase considered in the robustness study ofthemodel, …rstofallitis possibleto observethatthe increase in ¸ determines an increase in ¾ (the elasticity of substitution between varieties ofsoftwares) and a decrease in the mark-up rate, thatcorresponds toanincreaseinthelevelofcompetitionintheecon-omy. Itis known thatin thevarietyexpandinggrowth models (liketheone consideredhere) therelationship betweenmark-up andgrowth is notneces-sarily positive (as a consequence ofthe presence ofthe so-called ”resource competition e¤ect”), andin factthis is whathappens in thepresentmodel, where adecrease in the mark-up is accompanied byan increase(and nota decrease) inthegrowthrate.
W hen, inthiscontext, weconsiderthedi¤ erenttypesofshocksintroduced in the previous subsection, then, we getthat the long run e¤ects on the growth rates are lowerthan in the benchmark (this is true especially for theshocks on z and on ´, whosee¤ects on thegrowth rates are0 :5% lower than in the benchmark). T his is due to the factthata highervalue of¸ determinesnotonlyanincreaseintheelasticityofsubstitution¾ , butalsoin thedemandfortheintermediategood(software).A s aconsequence, ashock on z oron ´ determines (as in the benchmark case) an initialreallocation of workers favorable to the …nal good sector and at the expenses of the intermediate good sector, nevertheless this reallocation is less strong than
in thebenchmarkcase(becausehigherlaborforcein theintermediategood sector is now necessary to face the increased demand determined by the increasein¸). T hedynamicsisthenofthesametypeanalysedintheprevious subsection, and the …nalresultis an increase in the growth rates, butthis increaseis less strongthaninthebenchmarkcase. Forthesamereason(i.e. the increased demand ofthe intermediate good) the shocks on z and on ´ have ahigherlongrun e¤ect(with respecttothe benchmark) on thelabor forceemployed in theintermediategood sector. W henashockon e oron ¿ is considered thesituation is similar(even if, as in thebenchmarkcase, the shortrun dynamics aredi¤erentwith respecttothecaseofashockon z or on ´) and thelongrun e¤ects areless strongthan thoseobtained when the shocks on z oron ´ are considered (since the shocks on e and on ¿ a¤ ect growth onlyindirectly). A lsoin this casethefactthathigherlaborforcein theintermediatesectorisneededtosatisfytheincreaseddemanddetermines, inthelongrun, anincreaseinthegrowthrates lowerthaninthebenchmark case, andanincreaseinthelaborforceoftheintermediatesectorhigherthan in thebenchmark.
T he second variation considered concerns the parameter z (hence the technology ofthe economic system), thatchanges from 3 to 3:5 (i. e. of more than 15% ), while again the values ofallthe otherparameters remain unchangedwithrespecttothebenchmark. A lsoin this casethenewsteady state ofthe model is computed and the simulations are performed. T he newgrowth rate in the steady state turns outto be 1:4 5% , and again the economyreactsqualitativelytothedi¤erentshocksasinthebenchmarkcase; in particular, then, the quantitative e¤ects ofthe shocks on the di¤ erent variables arethefollowing:
shock shock shock shock
on z on e on¿ on´
growthrateofe¢cientcapital + 3:4 % + 1:2 % + 1% + 3%
growthrateofproduction + 3:4 % + 1:4 % + 1:4 % + 3%
laborforceinintermediatesector + 0 :1% + 0 :1% + 0 :1% + 0 :1%
interestrate + 1% + 0 :4 % + 0 :4 % + 1%
Table6:L ong-runquantitativee¤ ects (with respecttotheinitialsteady statelevels) ofdi¤ erenttypes ofshocks, case2 (higherz withrespecttothe
A gainitispossibletoobservethattheyaresu¢cientlyclosetothevalues obtainedinthebenchmarkcase, hencealsowhenachangeinz isintroduced themodelprovestoberobust. Consideringthedi¤ erentshocks, then, inthis situationwehavethatthelongrune¤ectsonthegrowthratesarelowerthan inthebenchmarkcase(andalsolowerthaninthesituationexaminedbefore, wherean increase in ¸ was assumed). T his is trueespecially fortheshocks onz andon´ (whoselongrune¤ectsonthegrowthratesareabout1% lower thaninthebenchmarkcase), andcanbeexplainedobservingthat, withthe newvalueofz , theinitialgrowthratesinthesteadystatearequitelarge. A s aconsequence, shocksonthedi¤erentparametersstilldetermineincreasesin thesegrowth rates (followingthesamedynamics ofthebenchmarkmodel), buttheseincreasesareproportionallylessstrongthaninthebenchmarkcase, wheretheinitiallevelofthegrowthrates was lower.
T helastvariationconsideredisthatoftheparameter´ (thatisameasure ofthecostofR &D ), thatchangesfrom 2 0 to17(henceexactlyof15% ), while theotherparametersremainatthevaluesofthebenchmark. T henewgrowth rate in the steady state in this case is 1:4 5% ;the modelis then simulated and the di¤erenttypes ofshocks are introduced, and alsoin this situation the qualitative behaviorofthe economy in response tothese shocks is the sameofthebenchmarkcase, whilethequantitativee¤ ects arethefollowing:
shock shock shock shock
on z on e on¿ on´
growthrateofe¢cientcapital + 3% + 1% + 1% + 3%
growthrateofproduction + 3:5% + 1% + 1:4 % + 3%
laborforceinintermediatesector + 0 :1% + 0 :1% + 0 :1% + 0 :1%
interestrate + 1% + 0 :4 % + 0 :2 % + 1%
Table7 : L ong-runquantitativee¤ ects (with respecttotheinitialsteady statelevels) ofdi¤ erenttypes ofshocks, case3 (lower´ withrespecttothe
benchmark)
A lsointhissituationtheyareclosetothevaluesobtainedfortheoriginal parameterizationofthemodel, thatthereforeis su¢cienltyrobustalsowith respecttochanges in´. Inparticular, itis possibletoobservethat, as inthe previous situation, thelongrune¤ects ofthedi¤ erentshocks on thegrowth rates arelowerthaninthebenchmarkcase(andalsolowerthan in thecase
in which ahigher¸ was assumed). T his is truealsofortheshocks one and on ¿ (whoselongrun e¤ects on thegrowth rates are0 :5% lowerthaninthe benchmark, whilein theothertwocases considered in therobustness study thedi¤erencewith respecttothebenchmarkwas smaller). T heexplanation thatcan be given is that, …rstofall, with the new value of´ the initial growth rates in the steady state are quite large, hence the shocks on the di¤erent parameters determine increases in these growth rates, but these increasesareproportionallylowerthaninthebenchmarkcase. Furthermore, in this situation theimportanceoftheR &D sectorin stimulatinggrowth is even largerthan in thebenchmark, becausethecostofR &D is lower. A s a consequence, whentheshocksone andon¿ areconsidered, sincetheseshocks determinegrowth onlyindirectlythroughtheexpansionoftheintermediate goodsector, theire¤ectsonthelongrungrowthrates areless strongthanin thebenchmarkcase(wherethecostofR &D is higher, hencetheimportance oftheR &D sectorin promotinggrowthis lower).
Inconclusion, from thediscussionpresentedwecandeducethatthemodel considered is quite robust, since the results obtained forthe initialcalibra-tion remain valid when some parameteris signi…cantly altered. T his is a goodpropertyofthemodel, andrepresentsanimportantachievementofthe analysis developed.
5 Conclusion
T his paperhas been devoted to the presentation ofa modelthattries to explain some of the characteristics ofthe recent ”ICT R evolution” using the framework ofendogenous growth theory. M ore precisely, itis a multi-sectoralgrowth modelwith embodied technologicalprogress, horizontaldif-ferentiation and ”lab-equipment” speci…cation ofR &D . A s a consequence ofthis latterassumption, in particular, itis possible to showanalytically thatan increase in theproductivity ofthe di¤erentsectors (…nalgood sec-tor, equipment secsec-tor, intermediate good sector) has an everlasting e¤ ect on growth (contrary towhathappens in the version ofthe modelwithout ”lab-equipment”).
T his resultis then con…rmed resortingtonumericalsimulation, thatal-lowsalsotogetsomefurtherinsights. Inordertodothis, acalibratedversion ofthemodelhasbeenconsidered, wherethevaluesofthedi¤erentparameters havebeenchosensoas toreproducetheempiricalevidencethatis available
concerningin particularthe U S economy. In this simulation, then, the dif-ferenttypes ofproductivity shocks have been analysed, and alsothe shock represented by a decrease in the costofR &D in terms ofoutput(strictly linkedtothepeculiarityofthe”lab-equipment” speci…cationadoptedinthis sector) has beenstudied.
T hemainconclusionobtainedfrom thesimulationisthatalltheseshocks have permanente¤ ects on longterm growth (and this is the centraldi¤ er-ence with respecttothe version ofthemodelwithoutthe ”lab-equipment” assumption, whereonlyashockontheproductivityoftheR &D sectorin‡u-encesthegrowthoftheeconomyinthelongrun). Inaddition, theshockson theproductivityofthe…nalgoodsectorandonthecostofR &D ontheone hand, andtheshocks ontheproductivityoftheequipmentsectorandofthe intermediate good sectoron the otherhand, a¤ectdi¤erently, in the short run, theeconomy, andin‡uencethegrowthwithdi¤erentintensity. Finally, the modelturns outtobe su¢ciently robust;when some parameteris sig-ni…cantlymodi…edwithrespecttothebenchmarkcase, boththequalitative and thequantitativeimplications ofthemodelremainvalid.
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