14.06 Lecture N otesInterm ediate M acroeconom icsGeorge-Marios AngeletosMIT Depar tment of EconomicsSpring 2004Chapter 6Endogenous Grow th I: AK, H, and G6.1 The Simple AK Model6.1.1 Pareto Allocations• Total output in the economy is given b yYt= F (Kt,Lt)=AKt,where A>0 is an exogenous parameter. In intensive form ,yt= f(kt)=Akt.• The social planner’s problem is the same as in the Ramsey model, except for the factthat output is linear in capital:max∞Xt=0u(ct)s.t. ct+ kt+1≤ f(kt)+(1− δ)kt113George-Marios Angeletos• The Euler condition givesu0(ct)u0(ct+1)= β (1 + A − δ)Assuming CEIS, this reduces toct+1ct=[β (1 + A − δ)]θorln ct+1− ln ct≈ θ(R − ρ)where R = A − δ is the net social return on capital. That is, consumption gro wthis proportional to the difference between the real return on capital and the discoun trate. Note that now the real return is a constant, rather than diminishing with capitalaccumulation.• Note that the resource constraint can be rewritten asct+ kt+1=(1+A − δ)kt.Since total resources (the RHS) are linear in k, an educated guess is that optimalconsumption and in vestment are also linear in k. We th us proposect=(1− s)(1 + A − δ)ktkt+1= s(1 + A − δ)ktwhere the coefficien t s is to be determin ed and m u st satisfy s ∈ (0, 1) for the solutionto exist.• It follows thatct+1ct=kt+1kt=yt+1yt114Lecture Notesso that consumption, capital and income all grow at the same rate. To ensure perpetualgro w th, we thus need to imposeβ (1 + A − δ) > 1,or equivalen tly A − δ>ρ.If that condition were not satisfied, and instead A − δ<ρ,then the economy w o uld shrink at a constant rate towards zero.• Fromtheresourceconstraintwethenhavectkt+kt+1kt=(1+A − δ),implying that the consumption-capital ratio is giv en byctkt=(1+A − δ) − [β (1 + A − δ)]θUsing ct=(1− s)(1 + A − δ)ktand solving for s w e conclude that the optimal savingrate iss = βθ(1 + A − δ)θ−1.Equivalen tly, s = βθ(1 + R)θ−1, where R = A − δ is the net social return on capital.Note that the sa v ing rate is increasing (decreasing) in the real return if and only if theEIS is higher (lower) than unit, and s = β for θ =1. Finally, to ensure s ∈ (0, 1), weimposeβθ(1 + A − δ)θ−1< 1.This is automatically ensured when θ ≤ 1 and β (1 + A − δ) > 1, as then s =βθ(1 + A − δ)θ−1≤ β<1. But when θ>1, this puts an upper bound on A.IfA exceeded that upper bound, then the social planner could attain infinite utility, andthe problem is not w ell- de fined.115George-Marios Angeletos• We conclude thatProposition 24 Consider the social planne r’s problem with linear technology f(k)=Akand CEIS preferences. Suppose (β, θ,A, δ) satisfy β (1 + A − δ) > 1 >βθ(1 + A − δ)θ−1.Then, the economy ex hibits a balanced growth path. Capital, output, and consumption allgrow at a constant rate given bykt+1kt=yt+1yt=ct+1ct=[β (1 + A − δ)]θ> 1.while the investment rate out of total resources is given bys = βθ(1 + A − δ)θ−1.The growth rate is increasing in productivity A, increasing in the elasticity of intertemporalsubstitution θ, and decreasing in the discount rate ρ (where β =11+ρ).• Differences in productivities and preferences may thus help explain differences, notonly in the lev el of output and the rate of inv estm ent, but also in growth rates.6.1.2 The Frictionless Competitiv e Economy• Con side r now how the social planne r’s allocation is decen traliz ed in a competitivemarket economy.• Suppose that the same tec hnology that is a vailable to the social planner is available toeac h single firm in the economy. Then, the equilibrium rental rate of capital and theequilibrium wage rate will be given simplyr = A and w =0.116Lecture Notes• The arbitrage condition between bonds and capital will imply that the in terest rate isR = r − δ = A − δ.• Finally, the Euler condition for the household will givect+1ct=[β (1 + R)]θ.• We conclude that the competitive mark et allocations coincide with the Pareto optimalplan. Note that this is true only because the private and the social return to capitalcoincide.6.1.3 What is next• The analysis here has assumed a single type of capital and a single sector of produc-tion. We next consider multiple t ypes of capital and m u ltiple sectors. In essence, we“endogenize” the capital K and the productivit y A — for example, in terms of ph ysicalversus hu man capital, intentional capital accumulation v e rsus unintensional spillovers,inno vation and know ledge creation, etc. The level of productivity and the growth ratewill then depend how the economy allocates resources across differen t types of capitaland differen t sectors of production. What is important to keep in mind from the sim pleAK model is the importance of linear returns for delivering perpetual gro wth.6.2 A Sim ple Model of H um an Capital• We no w consider a variant of the AK m odel, where there are two t ypes of capital,ph ys ical (or tangible) and human (or intangible). We start by assuming that both117George-Marios Angeletost ypes of capital are produced with the same technology, that is, the absorb resourcesin the same intensities. We later consider the case that the production of hum an capitalis more in ten sive in time/effort/s kills than in machines/fac tories .6.2.1 Pareto Allocations• Total output in the economy is giv en byYt= F (Kt,Ht)=F (Kt,htLt),where F is a neoclassical production function, Ktis aggregate capital in period t, htis human capital per worker, and Ht= htLtis effectiv e labor.• Note that, due to CRS, we can rewrite output per capita asyt= F (kt,ht)=Fµktht, 1¶htkt+ ht[kt+ ht]=or equivalen tlyyt= F (kt,ht)=A (κt)[kt+ ht],where κt= kt/ht= Kt/Htis the ratio of ph ys ical to human capital, kt+ htmeasurestotal capital, andA (κ) ≡F (κ, 1)1+κ≡f(κ)1+κrepresents the return to total capital.• Total output can be used for consumption or inv estment in either t ype of capital, sothat the resource constrain t of the econom y is giv en byct+ ikt+ iht≤ yt.118Lecture NotesThe laws of motion for two types of capital arekt+1=(1− δk)kt+ iktht+1=(1− δh)ht+ ihtAs long as neither iktnor ihtare constrained to be positive , the resource constraint andthe t wo laws of motion are equivalent to a single constraint, namelyct+ kt+1+ ht+1≤ F (kt,ht)+(1− δk)kt+(1− δh)ht• The social planner’s problem thus