高宏课件Chapter1SolowGrowthModel.ppt

Chapter 1 Solow Growth Model,Facts about economic growth; Assumption and model structure; Dynamics;Comparative static;Convergence; Data and growth accounting; Homework and paper,1 Facts and key questions,Growth rates differ: as China,US, Africa; Productivity growth slowdown; See Pictures Key question: what drives economic growth Key question 2:why it differ so much? Now we start with Solow model Question:Why starts with Solow?,2Assumptions and Model Structure,Production:Neoclassical production function: Y = F(K, L, A). A: non-rival, non-excludible technology. One-sector technology: Y goes for C or K (gross investment). K, capital, is cumulated net investment. Can interpret K to include human capital. K depreciates at rate 0 (exogenous).,Labor,L = labor supply (market clears with full employment). Labor supply = population (or proportional to population). No labor leisure choice and labor-force-participation rate is constant. Can extend model to make these endogenous. Population grows exogenously at rate n 0: (1/L)(dL/dt) = n L(t) = L(0)exp(nt) Can extend model to make fertility, mortality, immigration endogenousthen n is endogenous.,Properties of neoclassical production function,Constant returns to scale (CRS) in K, L: F(K, L, A) =F(K, L, A) for all 0 Positive and diminishing marginal products: FK , FL 0; FKK, FLL 0. Limiting (Inada) conditions: Lim (FK) = , Lim (FL) = ; K0 L0 Lim (FK) = 0, Lim (FL) = 0; K L,Conditions,These conditions imply each input is essential: F(0, L) = F(K, 0) = 0. An example of a neoclassical production function is the Cobb-Douglas one: Y = AKL1-, A 0; 0 1. This form implies unit elasticity of substitution between K and L. If factors paid marginal products, factor shares of income are constant at for K and 1-,Technological progress,Basic model assumes exogenous technological progressA depends only on time, t: Y = F(K, L, t). Common assumption to get nice steady-state results, when n = (1/L)(dL/dt) is constant, is that technical progress takes labor augmenting, Kaldor form. Y depends on K and effective labor, L = L(t) Y = F(K, L ).,About A,If technical change takes place at constant rate x 0, L= L exp(xt), L grows at rate n + x. Technology might instead augment K (Solow) or F(K, L) overall (Hicks). If production function is Cobb-Douglas, the three forms are indistinguishable.,Model Structure,Standard model has labor-augmenting technical change at constant rate, x. CRS property (with = 1/L ) implies Y/L = FK/L , 1. Define y Y/L , k K/L . Then y = f( k ).,Price of K,L,Output per unit of effective labor depends only on capital per unit of effective labor. Can show: Y/K = f ( k ), Y/L = f( k ) k f ( k )exp(xt).,3 Solow equation,Change in capital stock is dK/dt = I K, where I is gross investment. Assume closed economy, so that I = S (gross saving). Let s be gross saving rate, ratio of S to gross income or GDP, Y. Then dK/dt = sY K.,Behavior of k,Convenient to focus on k : (1/ k )(d k /dt) = (1/K)(dK/dt) (n+x). Substituting for dK/dt, (1/ k )(d k /dt) = s(Y/K) - (n+x). Y/K is average product of capital, which equals f( k )/ k . Therefore, (1/ k )(d k /dt) = s f( k )/ k - (n+x+).,k and s,In level form, after multiplying by k : d k /dt = sf( k ) (n+x+) k . Solow equation holds whether s is constant or variable. Basic model has s exogenous and constant (0 s 1). Ramsey model (Ch. 2) makes s endogenous household choice. Cannot usually write solution as closed form, s = s( k ).,Solow diagram,Steady state is at k* where d k */dt = 0. Note stability. k* satisfies sf( k *) = (x+n+) k *. An increase in s raises k *. An increase in f() raises k *. An increase in x+n+ (effective depreciation rate) lowers k *.,Diagram,4 Comparative analysis,For given k , an increase in s raises (1/ k )(d k /dt) on impact. Same with a rise in f(). A rise in n+ lowers (1/ k )(d k /dt). A rise in x lowers (1/ k )(d k /dt) but leaves unchanged the growth rate of k.,5 Quantitative implication,The growth rate of y is (1/ y )(d y /dt) = k f( k )/f( k )(1/ k )(d k /dt) = (1/ k )(d k /dt). If constant, effects of s, f(), n+ work by affecting (1/ k )(d k /dt). An increase in x lowers (1/ y )(d y /dt) by therefore, raises growth rate of y on impact.,Convergence,Log-linear approximation around k * is (1/ k )(d k /dt) -log( k / k *). Model implies (1/ k )(d k /dt)/ log( k ) = -(1-)sA k -1 For first-order Taylor expansion, - is the value of this derivative at the steady state: = (1-)sA( k *)-1 = (1-)(x+n+).,convergence,Same form works for log y (t). is the rate of convergence