Transcription of Chapter 7: Solow Model I
1 Chapter7:SolowMo delI11 TheSolow-SwanMo ductionfunction:Y=F(K,L) =K L1 Outputp erworker:y YL;capitalp erworker:k KL;consumptionp erworker:c CL;investmentp erworker:i L1 YL= (KL) y=k C= (1 s)Y CL= (1 s)YL c= (1 s)yI=sY IL=sYL i= :A= opulation/lab orforcegrowth:L= ,constantsavingsrate:s= ,constantdepreciationrate: = :G=T= :X=M= 0G=NX= 0 Y=C+ ck Capitalgain :investment,i=sf(k) =sk Capitalloss :depreciatedcapital, kLawofmotion(discretetime): k=i k=sk ki < k k <0i= k k= 0i > k k >01 Econ302,Week9,10/30/2009; (Equilibrium)Steady-stateintheSolowmodel :inlong-runequilibrium,capitalp erworker(thecapital-lab orratio) :thefollowingequationde nesasteady-stateintheSolowmo :sf(kss) = kss kssf(kss)=s (1)Cobb-Douglascase:sk ss= kss kss= (s )11 (2)Ifthissteady-stateconditionholds,the owsinto(investment)andoutof(depreciation ) < kss sf(k)> k k >0k=kss sf(k) = k k= 0k > kss sf(k)< k k <0 Steady-statequantitiesasso ciatedwithkss:yss,css,iss(k :y ,c ,i ).
2 Delpredictsthatcountrieswithhigherrateso fsavingsandinvestmentwillhavehigherlevel sofcapitalandoutput/incomep erworkerinthelong-run, ,andthereforeyss? :s kss yss : kss yss Goldenrule capital-laborratio:Thelevelofcapitalp erworkerkgrthatmaximizescss=f(k) ecttok: y k= f (kgr) = k 1gr= k1 gr= kgr= ( )11 22 Exercise:SolowMo delConsidertheSolowgrowthmo delwithoutp delaregivenbys= (savingsrate)and = (depreciationrate).Letkdenotecapitalp erworker;youtputp erworker;cconsumptionp erworker;iinvestmentp )Rewritepro ductionfunctionY=K13L23inp )Findthesteady-statelevelofthecapitalsto ck, )Whatisthe goldenrule levelofkforthiseconomy?Recallthatthegold enrulelevelofthecapitalsto ckkgrmaximizesconsumptionp )Let'ssaythatab enevolentso cialplannerwishestoobtaink= ciatedsavingsratesgrthatmustb eimp osedbytheso cialplannertosupp ortkgr?
3 E) ,docitizensneedtosavemoreorless?f )Plotthefollowingonasinglegraph:y=f(k), k,sf(k),andsgrf(k).Do esthesavingscurvepivotupordown,relativet oitsinitialp osition,whentheplanner'ssgrisimplemented ?g)Discusstwotothreeeconomicp oliciesthatcouldhelptheso
