CHAPTER3Solutio
stotheProblemsi
theTextbookCo
ceptualProblems1Theproductio
fu
ctio
providesaqua
titativeli
kbetwee
i
putsa
doutputForexampletheCobbDouglasproductio
fu
ctio
me
tio
edi
thetextisoftheformYFNKAN1KwhereYreprese
tsthelevelofoutput1a
dareweightsequaltothesharesoflaborNa
dcapitalKi
productio
whileAisofte
usedasameasurefortheleveloftech
ologyItca
beeasilyshow
thatlabora
dcapitaleachco
tributetoeco
omicgrowthbya
amou
tthatisequaltotheiri
dividualgrowthratesmultipliedbytheirrespectivesharei
i
come2TheSolowmodelpredictsco
verge
cethatiscou
trieswiththesameproductio
fu
ctio
savi
gsratea
dpopulatio
growthwilleve
tuallyreachthesamelevelofi
comepercapitaI
otherwordsapoorcou
trymayeve
tuallycatchuptoarichero
ebysavi
gatthesameratea
dmaki
gtech
ologicali
ovatio
sHoweverifthesecou
trieshavediffere
tsavi
gsratestheywillreachdiffere
tlevelsofi
comepercapitaeve
thoughtheirlo
gtermgrowthrateswillbethesame3Aproductio
fu
ctio
thatomitsthestockof
aturalresourcesca
otadequatelypredicttheimpactofasig
ifica
tcha
gei
theexisti
gstockof
aturalresourceso
theeco
omicperforma
ceofacou
tryForexamplethediscoveryof
ewoilreservesora
e
tirely
ewresourcewouldhaveasig
ifica
teffecto
thelevelofoutputthatcould
otbepredictedbysuchaproductio
fu
ctio
4I
terpreti
gtheSolowresidualpurelyastech
ologicalprogresswouldig
oreforexampletheimpactthathuma
capitalhaso
thelevelofoutputI
otherwordsthisresidual
oto
lycapturestheeffectoftech
ologicalprogressbutalsotheeffectofcha
gesi
huma
capitalHo
thegrowthrateofoutputToelimi
atethisproblemweca
explicitlyi
cludehuma
capitali
theproductio
fu
ctio
suchthatYFKNHANaKbHcwithabc1The
thegrowthrateofoutputca
becalculatedasYYAAaNNbKKcHH5Thesavi
gsfu
ctio
sysfkassumesthataco
sta
tfractio
ofoutputissavedThei
vestme
trequireme
tthatisthe
dkli
ereprese
tstheamou
tofi
vestme
t
eededtomai
tai
aco
sta
tcapitallaborratiokAsteadystateequilibriumisreachedwhe
savi
gisequaltothei
vestme
trequireme
tthatiswhe
sy
dkAtthispoi
tthecapitallaborratiokKNis
otcha
gi
gsocapitalKlaborNa
doutputYallmustbegrowi
gatthesameratethatistherateofpopulatio
growth
NN6I
thelo
gru
therateofpopulatio
growth
NNdetermi
esthegrowthrateofthesteadystateoutputpercapitaI
theshortru
howeverthesavi
gsratetech
ologicalprogressa
dtherateofdepreciatio
ca
allaffectthegrowthrate7Laborproductivityisdefi
edasYNthatistheratioofr