Title PageIntroductionIntroductionSolow Model: IntroductionIThe production function is useful in describing the state of theeconomy, or compare economies at di↵erent time periodsIIt is also a useful starting place for accounting for the di↵erentfactors that contribute to growthIHowever, it does not provide an explanation for a number ofimportant issues:IWhat is the process of growth? is it a↵ected by someprimitive decisions, parameters, or policies?IWhat happens beyond GDP? How about consumption?The Solow Growth Model provides answers to some of these questions46 / 91Solow Model - Main Questions1. What is the relationship between long run standards of li vi n gand f u nd ame ntal fact ors such as saving rates, population growth,technological progress?2. How does growth evolve over time? Will it stabilize? Accelerate?Stop?3. Are there economic forces that generate “catch-up”? In levels orgrowth rates?47 / 91Afewthingstonote...From a “learning p e rs pect i ve”, a few remarks:IThis m odel is our first encounter wi t h dynamics, or the evolutionof the economy over timeIExogenous vs. Endogenous variablesISome assumptions may seem unreasonable, but we’ll see how ithelps simplifying the analysis48 / 91Assumptions and NotationINtis the number of workers (population) at time tIn =: population’ growth rate ) Nt+1=(1+n)NtIYt= Ct+ ItINo Government expenditure G =0IClosed economy: NX =0ICapital depreciates at a constant rate 0    1IWe focus on per-worker terms:Iyt⌘YtNtct⌘CtNtkt⌘KtNtit⌘ItNtIA standard production functionIIndividuals save (and invest) a const ant fraction s of theirincome: It= sYt49 / 91ProductionIWith constant returns, we know that:yt=YtNt=AtF (Kt,Nt)Nt= AtF✓KtNt,NtNt◆= Atf(kt)Ie.g. Cobb -D ougl as an d 2/3 labor share of out pu t : yt= Atk1/3tIThe rest of the assumptions imply that the shape of thisper-worker production function is concave50 / 91Investment and Capital AccumulationICapital (K) evolves over time:Kt+1=(1 )Kt+ ItISome of the existing stock depreciates (at rate )INew capital is accumulated through investment I.IWhat does this say about the conversion of consumption toinvestment goods?IDerive the evolution of kt= Kt/Ntkt+1=(1  )kt+ it1+n51 / 91Investment and Capital AccumulationICapital (K) evolves over time:Kt+1=(1 )Kt+ ItISome of the existing stock depreciates (at rate )INew capital is accumulated through investment I.IWhat does this say about the conversion of consumption toinvestment goods?IDerive the evolution of kt= Kt/Ntkt+1=(1  )kt+ it1+n52 / 91Investment and Capital AccumulationICapital (K) evolves over time:Kt+1=(1 )Kt+ ItISome of the existing stock depreciates (at rate )INew capital is accumulated through investment I.IWhat does this say about the conversion of consumption toinvestment goods?IDerive the evolution of kt= Kt/Ntkt+1=(1  )kt+ it1+n53 / 91Steady StateDefinition: A steady state is when kt, yt, ct, and itare all constants(i.e. don’t change over time).ILet the constants be kss= ktfor all t, css= ctfor all t,etc.IFor a steady state to exist, K, Y, C, and I all grow at a constantrate pausekt+1= kt= kss(1)Kt+1Nt+1=KtNt=) (2)Kt+1Kt=Nt+1Nt=1+n (3)IIntuition: steady s t at e provides average levels - adjusted forpopulation growth - of the economy in the (very) long run54 / 91Steady StateDefinition: A steady state is when kt, yt, ct, and itare all constants(i.e. don’t change over time).ILet the constants be kss= ktfor all t, css= ctfor all t,etc.IFor a steady state to exist, K, Y, C, and I all grow at a constantrate pausekt+1= kt= kss(1)Kt+1Nt+1=KtNt=) (2)Kt+1Kt=Nt+1Nt=1+n (3)IIntuition: steady s t at e provides average levels - adjusted forpopulation growth - of the economy in the (very) long run55 / 91Steady States a re useful because...IWe can calcu l at e t he le vels of c, i, k, and yIc is important. We’ll use it to argue that increasing y is notnecessarily the best thing to do...why?IWe can ask i f and how we expect the economy to actually reacha steady stateIIt is a useful departure point for analyzing the e↵ects of certainchanges (shocks) that an economy may experien ce56 / 91Steady States a re useful because...IWe can calcu l at e t he le vels of c, i, k, and yIc is important. We’ll use it to argue that increasing y is notnecessarily the best thing to do...why?IWe can ask i f and how we expect the economy to actually reacha steady stateIIt is a useful departure point for analyzing the e↵ects of certainchanges (shocks) that an economy may experien ce57 / 91Steady State Inves tm e ntIWe know that k =KNshould be constant, and that,Kt+1=(1+n)KtIDerive the equation for steady state investmentkt+1=(1  )kt+ it1+niss=(n + )kssIIn order to keep the capital-labor ratio constant, we must investin order to exactly compensate for depreciation and populationgrowth rates58 / 91Steady State Consum p tio nIHow to calculate? start with t he res our c e con st r ai ntYt= Ct+ It) yt= ct+ it) ct= yt it) css= yss iss= Af (kss)  (n + )kssIThe easiest way to understand this is to plot thisIConsumption is the di↵erence between the concave produc t ionfunction and the linear investment line59 / 9160 / 91Finding a steady stateThe importance of kIRecall, the fixed savings rate 0  s  1 is given exogenou sl yIIf we know kss, can characterize the state of the economy (i.e.the rest of the variables)yss= Af(kss)iss= sAf(kss)css=(1 s)Af (kss)IIn order to find the steady state, all we need is to find a solutionfor k that is consistent with a steady stateIRequired investment to compensate for depreciation andpopulation growth is iss=(n + )kssIActual investment comes from s, iss= syss= sAf(kss)IIn Steady state, actual=required:sAf(k)=(n + )k61 / 91Finding a steady state62 / 91Finding a steady state: exampleIAssume a Cobb-Douglas production function: y = Af(k)=Ak↵IIn a steady state, Derive kss,iss,yss, and css.IRecall the system of equations:sA (kss)↵=(n + )kssyss= A (kss)↵iss= sysscss=(1 s)yss63 / 91Convergence to steady state?A key question is whether the economy actually reaches the steadystate.IAssume an economy that is not in steady state? will it get there?if so, what’s the process?IAssume an economy in a steady state, and a “shock” happens.Will it reach a new steady state? what is the process of gettingthere?Let’s go back to the graph...64 / 91Convergence to steady stateIstart from k1: the ec onomy invests more that