1. Basic Neoclassical Model (Solow Model) (April 14, 2014)

Transcription

1 Prof. Dr. Thomas Steger Advanced Macroeconomics I Lecture SS Basic Neoclassical Model (Solow Model) (April 14, 2014) Introduction Model setup Intensive production function Capital accumulation The steady state Technological progress Speed of convergence Summary

2 Introduction: ti Why thinking about economic growth? (1) Why are there poor and rich countries? The richest countries are about 30 times richer than the poorest countries (in 1988). (Parente and Prescott, 2000) Why do some economies experience rapid growth, while some other stagnate at the same time? Niger (NER) versus South Korea (KOR); see PWT! Manipulating an economy s (long run) growth rate seems to have massive welfare consequences. If per capita income grows at a growth rate of 1 percent it takes about 70 years until per capita income is doubled. If per capita income grows at a growth rate of 2 percent it takes about 35 years until per capita income is doubled. Lucas (1987, 2003) has argued that the complete removal of consumption volatility ( business cycle phenomenon) would imply a welfare gain which is equivalent to a permanent increase in consumption of about 0.1 percent to 1 percent. Taking heterogeneity into account should further increase the welfare gain. Follow up studies found substantially larger welfare gains of about 10 percent to 30 percent (see Lucas, 2003, p. 7). The potential welfare gain resulting from the implementation of optimal growth policies appears to be substantially higher compared to results mentioned above. An appropriate policy reform could achieve a welfare gain which is equivalent to a permanent doubling of per capita consumption (Grossmann, Steger, Trimborn, 2010). 2

3 Introduction: ti Why thinking about economic growth? (2) South Korea Niger The consequences for human welfare involved in questions like these are simply staggering: Once one starts thinking about them, it's hard to think about anything else. Robert Lucas (1988) (Constant Price es: Chain series s) Real GDP per capita

4 Introduction: ti Why thinking about economic growth? (3) xt () = xe g=0.1 0 gt g=0.02 g=0.01 How long does it take until x(t) is twice its initial value? Solve x 0 e gt =2x 0 for t to get t=ln(2)/g. 4

5 Reminder (1): The Kaldor Facts In 1961 Nicolas Kaldor listed 6 stylized facts that describe economic growth in advanced economies 1. Y/L (output per worker) exhibits continual growth. 2. K/L (capital per worker) exhibits continual growth. 3. r (real interest rate) is roughly constant. 4. K/Y (capital-output ratio) is roughly constant. Fact #4 is implied by #1&#2. Fact #5 is implied by #3&#4. 5. rk/y, wl/y (factor shares) are roughly constant. 6. There are wide differences in the rate of growth of productivity across countries. N. Kaldor (1961), Capital Accumulation and Economic Growth, in F. A. Lutz and D. C. Hague, editors, The Theory of Capital. New York: St. Martin's Press. 5

6 Reminder (2): Basic concepts and dtools - differential equations (1) First-order, linear, differential equations y + ut () y = wt () t R denotes the independent variable (often the time index) and y:=dy/dt. first-order DE: only the first derivative of y wrt w.r.t. time occurs linear DE: both y and y appear only in first degree and there is no such term y The homogenous case, w(t)=0 t, with constant coefficient, u(t)=a, reads y + ay =0 A solution is easily found as follows 1 dy 1 dy = a dt = adt ydt ydt LHS RHS 1 dy d ln y 1 dy d ln y LHS : = dt = dt = ln y + c y dt dt y dt dt 1 RHS: adt = at + c2 ln y = at+ c with c: = c c 2 1 at yt ( ) = Ae with c A: = e (general solution) at yt ( ) = y( 0) e (definite solution) 6

7 Reminder (3): Basic concepts and dtools - differential equations (2) The non-homogenous case, w(t)=b t, with constant coefficient, u(t)=a, reads y + ay = b Solution comprises two terms: the complementary function y c and the particular integral y p Complementary function y c is the general solution of the homogenous DE Particular integral y p is simply any particular solution of the non-homogenous equation Complementary function y c ( discussion of the homogenous DE) yc = Particular integral y p : try the simplest possible type of solution, i.e. y=constant, which gives Ae at yp = b/ a y=constant implies y=0y 0 and hence y+ay=by b yields y p =b/a (assuming a 0) The general solution of the non-homogenous DE is b y = yc + y = + Ae p a at (general solution) b at b y = y( ) e + (definite solution; from t = ) 0 a a 0 7

8 Reminder (4): Basic concepts and dtools - continuous-time growth rates The growth rate of a time-dependent variable x(t) with t R is defined as follows xt () dxt ()/ dt xt ˆ( ): = = xt () xt () Consider the following dependent variable (time index t is suppressed) y: = x x β 1 2 Question: What is the growth rate of y in terms of x 1 and x 2? Answer: The growth rate reads as follows (reasoning: next slide) yˆ = xˆ + β xˆ 1 2 8

9 Reminder (5): Basic concepts and dtools - continuous-time growth rates (a) Reasoning (1): Differentiate both side of y=x 1 x 2β w.r.t. time β dy dx1 β dx2 = x2 + x1 (product rule) dt dt dt 1 β β 1 y = x 1 x 1 x2 + x1 βx 2 x 2 outer derivative inner derivative outer derivative inner derivative (chain rule) Subsequently, divide both sides by y=x 1 x 2 β to get y x x2 x βx x2 x x x = + = + β yˆ = xˆ + β xˆ y 1 β β β β x1 x2 x1 x2 x1 x2 Reasoning (2): Form the natural logarithm of y=x 1 x 2β to get ln( y ) = ln( x ) + β ln( x ) Subsequently, take the derivative w.r.t. time on both sides Noting Gives dln( y) dln( x1) dln( x2) = + β dt dt dt dln( xt ) 1 dx( t) x = = = xˆ (chain rule) dt x() t dt x yˆ = xˆ + β xˆ 1 2 9

10 Reminder (6): Basic concepts and dtools - growth accounting Growth accounting is due to Solow (1957). This procedure decomposes the growth rate of GDP (over longer period of time) into several supply side components. As a by-product, growth accounting allows an assessment of the technological change component. The point of departure is a usual Cobb-Douglas output technology Yt () = AtKt () () Lt () 1 Expressed in terms of growth rates we have Yˆ = Aˆ + Kˆ + (1 ) Lˆ Y, K, L and are observable or can be determined empirically. Growth accounting allows to decompose the growth rate of GDP Y into the contribution of capital accumulation (K), the contribution of a change in labor input (1-)L and the contribution of technical change A. 10

11 Reminder (7): Basic concepts and dtools - growth regressions Growth regressions are due to Barro (1991) and Mankiw, Romer, Weil (1992). The last mentioned authors have estimated the following cross-sectional growth regression yˆ = ln( y ) ln( in ) 0.505ln( n + g + δ ) SCHOOL + u i i,1960 i,1960 i i i (3.66) (4.66) (6.02) (1.75) (3.88) Remarks : Sample size: 98 countries; t-values in brackets; estimation procedure: OLS Limitations Endogeneity of right hand side variables yˆ i : average annual growth rate of real GDP per capita in country i over the period d yi,1960 : real GDP per capita in country iin 1960 ini,1960 : investmentratein country i (average over ) n i : population growth ratein country i SCHOOL i : average school enrollment rate in country i (secundary education, average over ) g: rate of technical progress δ: capital depreciation rate u :error term (i.i.d.) The right hand side variables are often determined together with the dependent variable. As a consequence, estimation results are likely to be biased. Way out: instrumental variables approach. Model uncertainty Results are often highly sensitive w.r.t. to the underlying empirical model (i.e. number and type of explanatory variables). The true empirical model is not known. Way out: robustness checks and model averaging procedure. Reference: Mankiw, N. Gregory, David Romer, and David N. Weil, A Contribution to the Empirics of Economic Growth, The Quarterly Journal of Economics, Vol. 107, No. 2. (May, 1992), pp i 11

12 Model setup (1) al = K a (capital market equilibrium) i rk s a L households Y = AK L 1 final-output sector (origin) wl (distribution) 1 s C households (use) There are L households, every HH is endowed with one unit of time which is supplied inelastically to the labor market. The numeraire good is Y. Notice that financial wealth a and K are both measured in units of Y. s: exogenous saving rate 12

13 Model setup (2) The stock of physical capital per capita seems to matter for the standard of living! This is, in a nutshell, the story of the neoclassical model. The Solow model lhas some more insights: i role of technological progress transitional dynamics 13

14 Model setup (3): two versions The real economy (Robinson Crusoe) A consumer-producer household with technology Y Decentralized financial market economy Firms have access to the following technology 1 = K L Competitive factor prices read Y = K L The resource constraint reads Y Y r = δ, w= (1 ) K L Y = K + δ K + C There are L identical households. Every household is endowed with one unit of time per period, which is supplied inelastically to the labor Define the (gross) saving rate market. Households have also access to the capital market. Individual (gross) household income X is given by Y C s : = X = ( r+ δ ) a+ w Y Capital market equilibrium requires The K-equation is then given by K = al Some terminology K = sy δ K Y: gross output (=gross factor income) Y-δK: net output (=net factor income) K+δK: gross investment K: net investment capital demand 1 capital supply Total household income (assuming K=aL) reads Y Y XL = ( r + δ) al+ wl = K+ (1 ) L = Y K L Individual stock of assets evolves according to a = X δ a C net factor income Defining the (gross) saving rate s:=(x-c)/x,, we have a = sx δ a From K=aL and LX=Y we get K = al = sxl δal = sy δk 14

15 Model setup (4) The typical final output firm has access to the following Cobb-Douglas technology Yt () = FKt [ (), Lt ()] = Kt () Lt () This technology exhibits constant returns to scale (CRS), i.e. F(λK,λL)=λY for all λ R. The problem of the typical firm reads (time index t is omitted in what follows) 1 max pfkl Y (, ) ( r+ δ ) K wl KL, user cost of capital The resulting inverse demand functions are given by FKL (, ) Y w= = (1 ) K L = (1 ) L L Notice that we have set p Y =1. FKL (, ) r = δ = K L = K Y K 1 1 δ δ The labor income share: (wl)/y=1-; the capital income share: (rk)/y= (assuming δ=0). Output is exactly exhausted, i.e. Y=rK+wL ( Euler's Theorem). 15

16 Intensive production function Output per worker (=labor productivity) reads Y L K = K L = L y = k with k:=k/l and y:=y/l The 1 st derivative is positive ( y/ k=k -1 >0) and the 2 nd derivative is negative ( ²y/ k²=(-1)k -2 <0), i.e. there are diminishing returns to capital per worker. Moreover, the following relations hold true Y K y = K L = = k = K L k In words: The marginal product of capital equals the marginal product of capital per worker, evaluated at corresponding levels of capital and capital per worker. 16

17 Capital accumulation The capital accumulation equation is given by K = sy δ K The equation of motion for k:=k/l is obtained as follows k K L k Y L = = s δ k K L k K L YK L The last equation uses k= s δ + k k = sy ( δ + n ) k KL L L L/L n implying L L 0 e L=L/L=n implying L=L 0 e nt Two basic questions How does long-run output per worker compare in two economies that have different investment rates and population growth rates? How does output per worker evolve over time? 17

18 Capital accumulation (a) Possibility 1 Differentiating k(t)=k(t)/l(t) w.r.t. time by applying the quotient rule, noting d[k(t)/l(t)]/dt=k, gives Kt () d dk( t) dl( t) Lt () Kt () Lt () dt dt K() tlt () KtLt () () Kt () Kt () Lt () = = kt () = 2 2 dt Lt () Lt () Lt () Lt () Lt () Next, we divide both sides of this equation by k(t)=k(t)/l(t) to get k K L L K L = = k L K L K L Possibility 2 Take the natural logarithm on both sides of k(t)=k(t)/l(t) to get Next differentiate w.r.t. time ln[ kt ( )] = ln[ Kt ( )] ln[ Lt ( )] Recall dln[ Xt ( )] dln[ ( )] dx( ) 1 dt = Xt t X dx( t) dt = X( t) dln[ kt ( )] dln[ Kt ( )] dln[ Lt ( )] k K L = = dt dt dt k K L 18

19 The steady state t We need a dynamic equilibrium concept that describes the long-run state of a dynamic economy. The usual equilibrium concept is that of steady state equilibrium. A steady state is state of the economy such that the growth rates of the endogenous variables are constant. From the k-equation the growth rate of capital per worker is ˆ 1 = ( δ + ) k sk n Since 0<<1, sk -1 declines as k increases, lim k 0 sk -1 = and lim k sk -1 =0 (see next slide). This indicates that there is a level of k such that k=0 such that k=const. The steady state level of k is as follows (by setting k=0 we ignore the trivial steady state k*=0) ˆ 1 ( ) 0 k = sk δ + n = k 1 1 s 1 s = y = n+ δ n+ δ s c = (1 s) n + δ 1 19

20 Graphical representations ti of the steady state t y=k per capita consumption (n+δ)k sy k(0) k* k (0) sk kˆk 0.05 n+ δ k 20

21 Technological l progress (1) The production function now reads Y = F( K, AL) = K ( AL) Technological l progress (A>0) is labor-augmenting and exogenous A gt g A Ae 0 A = ó = 1 The production function in intensive form is then given by y = k A 1 y and k are still defined by: k:=k/l and y:=y/l. Expressed in growth rates the preceding equation reads as follows y = k + (1 ) A (*) y k A 21

22 Technological l progress (2) To determine the steady state growth rate of y, we first consider the growth rate of k ˆ y k = s ( δ + n) k Hence, k = const. requires y =k. Using (*) we find A yˆ = yˆ + (1 ) A A yˆ = = g A The growth rate of output per worker equals the rate of technical progress. Moreover, since y =kk the following relation holds ŷ = kˆ = g A growth path along which all variables grow at the same constant rate is called a balanced growth path (BGP). 22

23 Technological l progress (3) We now solve the model for the steady state. Since there is permanent growth in per capita variables (y, k), we define new variables according to y:= Y AL k:= K AL (output per efficiency units of labor) (capital per efficiency units of labor) It s all about detrending! lim Xt ˆ () = g t Xt () xt (): = gt e lim x( t) = const. t To derive the equation of motion of k, we differentiate k:=k/(al) w.r.t. time to get ˆ K L A ˆ sy K k δ = k = n g K L A K ˆ sy ˆ sy /( AL) k = δ n g k = δ n g K K /( AL) ˆ y k = s δ n g k = sy ( δ + n+ g) k k 23

24 Technological l progress (4) Noting y=k the steady state in terms of k and y reads as follows k 1 1 s 1 s = y = n δ g n δ g (**) To give a clear economic interpretation, let us rewrite (**) as follows s y ( t) = A( t) n + δ + g 1 Notice: y=ay (from y:=y/l and y:=y/(al) Along the BGP output per worker y grows at the constant rate g The level of the BGP is determined by economic fundamentals (s, n etc.) An increase in s does not affect the long-run growth rate. It does, however, affect the level l of the BGP and hence the growth rate along the transition to the (new) BGP. 24

25 Speed of convergence (1) The rate of convergence We now turn to the speed at which the economy converges to its steady state. This is measured by the rate of convergence (ROC). The ROC of any variable x(t) is defined by Important tool: linearization xt () ψ ():= t x xt () x Consider the following general, possibly non-linear, differential equation (DE) xt () = Fxt [ ()] Ψ x >0:convergence Ψ x Ψ x <0: divergence This DE is assumed to posses a stationary equilibrium defined by F(x*)=0. Linearization of F(x) around x* by means of a first-order Taylor approximation gives ( ) ( ) x () t F x + F x x() t x Hence, noting that F(x*)=0, x(t) converges at the following rate against x* ():= x ψ x t = F x x x ( ) 25

26 Speed of convergence (2) To determine the ROC for the Solow model recall that s k = s k ( δ + n+ g) k and k = n+ δ + g Linearizing the above DE around k* gives 1 1 ( ) 1 k = s k ( δ + n+ g ) k k Recall ( ) x () t F x x() t x 1 1 s k = s ( δ + n+ g) k k n+ δ + g ( k ) k = ( 1)( δ + n+ g) k (*) ψ k ( ) Recall ψ := k k k k Hence, the (local) rate of convergence reads Ψ k =(1-)(δ+n+g)>0. 26

27 Speed of convergence (3) k(t) Ψ k =0.05 (t 0.5 =13.8) Ψ k =0.03 (t 0.5 =23.1) Ψ k =0.01 (t 0.5 =69.3) LHS is the solution to d(k-k*)/dt=-ψk*)/d Ψ k (k-k*); k*) see (*) on previous slide. To see the quantitative implication, we assign empirically plausible parameter values: =0.35; δ=0.01; n=0.015; g=0.02. Ψ k =(1-)(δ+n+g)= More intuitive is the implied half life (t 0.5 ), which results from: [k*-k(0)]exp( t 0.5 )=0.5[k*-k(0)] t 0.5 =-ln(0.5)/0.0293=

28 Summary The Solow model is compatible with the Kaldor facts output per worker and capital per worker grow over time the ratio of capital over output is largely constant therealwageisgrowingovertime growing over time the rate of interest is largely constant the labor and capital income shares are largely constant The long run level of GDP per worker is determined by economic fundamentals, i.e. the saving rate and the population growth rate. Long run growth in GDP per worker is primarily driven by technological progress accompanied by capital accumulation. The growth rate of GDP per worker may temporarily exceeds its long run level. This is due to transitional dynamics in response to, say, a destruction in the stock of capital or an increase in the in the saving rate. 28

29 Notation ti and dabbreviations Notation a financial i wealth 0<<1 technology parameter A>0 technology parameter (either TFP or labor efficiency) i C consumption g growth rate of technology K physical capital k* steady state value of k k* steady state value of k K:=dK/dt k:=k/al derivative of K w.r.t. time capital per efficiency units of labor K:=K/K K/K growth rate of capital k:=k/l capital per worker L labor input n growth rate of population p Y price of Y (we use p Y =1) r interest rate 0<s<1 saving rate (=investment rate) t R time index w wage rate Y final output y:=y/al output per efficiency units of labor y:=y/l output per worker δ>0 capital depreciation rate Ψ x x rate of convergence ce of variable ab x Abbreviations BGP balanced growth path DE differential equation CRS constant returns to scale GDP gross domestic product OLS ordinary least squares PWT Penn World Tables ROC rate of convergence TFP total factor productivity i.i.d. independent identically distributed w.r.t. with respect to 29