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1、14.452 Economic Growth: Lecture 8, Neoclassical Endogenous GrowthDaron AcemogluMITNovember 19, 2009.Daron Acemoglu (MIT)Economic Growth Lecture 8November 19, 2009.1 / 47First-Generation Models of Endogenous GrowthModels so far: no sustained long-run growth; relatively little to say about sources of
2、technology dierences. Models in which technology evolves as a result of rmsand workers decisions are most attractive in this regard. But sustained economic growth is possible in the neoclassical model as well: AK model before: relaxed Assumption 2 and prevented diminishing returns to capital. Capita
3、l accumulation could act as the engine of sustained economic growth. Neoclassical version of the AK model: Very tractable and applications in many areas. Shortcoming: capital is essentially the only factor of production, asymptotically share of income accruing to it tends to 1. Two-sector endogenous
4、 growth models behave very similarly to the baseline AK model, but avoid this.Daron Acemoglu (MIT)Economic Growth Lecture 8November 19, 2009.2 / 47Demographics, Preferences and Technology IFocus on balanced economic growth, i.e. consistent with the Kaldor facts.Thus CRRA preferences as in the canoni
5、cal neoclassical growth model.Economy admits an innitely-lived representative household, household size growing at the exponential rate n.PreferencesU=Z0exp(?(?n)t)“ c(t)1?1 1?#dt.(1)Labor is supplied inelastically.Flow budget constraint, a(t) = (r(t) ?n)a(t) +w(t) ?c(t),(2)Daron Acemoglu (MIT)Econo
6、mic Growth Lecture 8November 19, 2009.3 / 47Demographics, Preferences and Technology IINo-Ponzi game constraint:limt!? a(t)exp? ?Zt0r(s) ?nds? ?0.(3)Euler equation: c(t) c(t)=1 (r(t) ?).(4)Transversality condition,limt!? a(t)exp? ?Zt0r(s) ?nds? =0.(5)Problem is concave, solution to these necessary c
7、onditions is in fact an optimal plan. Final good sector similar to before, but Assumptions 1 and 2 are not satised.Daron Acemoglu (MIT)Economic Growth Lecture 8November 19, 2009.4 / 47Demographics, Preferences and Technology IIIMore specically, Y(t) =AK(t),with A0.Does not depend on labor, thus w(t)
8、in (2) will be equal to zero.Dening k(t) ?K(t)/L(t)as the capital-labor ratio,y(t)?Y(t) L(t) =Ak(t).(6)Notice output is only a function of capital, and there are no diminishing returnsBut introducing diminishing returns to capital does not aect the main results in this section.Daron Acemoglu (MIT)Ec
9、onomic Growth Lecture 8November 19, 2009.5 / 47Demographics, Preferences and Technology IVMore important assumption is that the Inada conditions embedded in Assumption 2 are no longer satised,lim k!f0(k) =A0.Conditions for prot-maximization are similar to before, and require R(t) =r(t) +.From (6) th
10、e marginal product of capital is A, thus R(t) =A for all t, r(t) =r=A?, for all t.(7)Daron Acemoglu (MIT)Economic Growth Lecture 8November 19, 2009.6 / 47Equilibrium IA competitive equilibrium of this economy consists of paths c(t),k(t),w(t),R(t)t=0, such that the representative household maximizes
11、(1) subject to (2) and (3) given initial capital-labor ratio k(0)andw(t),r(t)t=0such that w(t) =0 for all t, and r(t)is given by (7).Note that a(t) =k(t).Using the fact that r=A? and w=0, equations (2), (4), and (5) imply k(t) = (A?n)k(t) ?c(t)(8) c(t) c(t)=1 (A?),(9)limt!k(t)exp(?(A?n)t) =0.(10)Dar
12、on Acemoglu (MIT)Economic Growth Lecture 8November 19, 2009.7 / 47Equilibrium IIThe important result immediately follows from (9).Since the right-hand side is constant, there must be a constant rate of consumption growth (as long as A?0). Growth of consumption is independent of the level of capital
13、stock per person, k(t). No transitional dynamics in this model.To develop, integrate (9) starting from some c(0), to be determined from the lifetime budget constraint,c(t) =c(0)exp?1 (A?)t? .(11)Need to ensure that the transversality condition is satised and ensure positive growth (A?0). Impose:A+ (
14、1?)(A?) +n+.(12)Daron Acemoglu (MIT)Economic Growth Lecture 8November 19, 2009.8 / 47Equilibrium CharacterizationEquilibrium Characterization INo transitional dynamics: growth rates of consumption, capital and output are constant and given in (9).Substitute for c(t)from equation (11) into equation (
15、8),k(t) = (A?n)k(t) ?c(0)exp?1 (A?)t? ,(13)First-order, non-autonomous linear dierential equation in k(t). Recall that if z(t) =az(t) +b(t),then, the solution isz(t) =z0exp(at) +exp(at)Zt0exp(?as)b(s)ds,for some constant z0chosen to satisfy the boundary conditions.Daron Acemoglu (MIT)Economic Growth
16、 Lecture 8November 19, 2009.9 / 47Equilibrium CharacterizationEquilibrium Characterization IITherefore, equation (13) solves for:k(t) =( exp(A?n)t) +?(A?)(?1)?1+?1?n?1 ?c(0)exp?1(A?)t?),(14) where is a constant to be determined. Assumption (12) ensures that(A?)(?1)?1+?1?n0.Substitute from (14) into the transversality condition, (10),0=limt!+?(A?)(?1)?1+?1?n?1?c(0)exp?A?)(?1)
