On the dynamics of basic growth models: Ratio stability versus convergence and divergence in state space

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1 On the dynamics of basic growth models: Ratio stability versus convergence and divergence in state space Thorsten Pampel July 27 Discussion Paper No. 538 (revised Department of Business Administration and Economics, Bielefeld University, P. O. Box 1 131, D 3351 Bielefeld, Germany tpampel@wiwi.uni-bielefeld.de Abstract We show for a class of basic growth models that convergence in per capita terms does not imply the convergence to the corresponding balanced growth path. We derive conditions on parameters and on the elasticity of the savings function for convergence or divergence and apply our results to the Solow model, an augmented Solow model as well as to an optimal growth model. An implication for the convergence debate is, that two economies which differ only in the initial capital stock and converge in per capita terms might diverges to infinity in absolute terms. This paper is accepted for publication in the German Economic Review, a former version (25 was titled On the convergence of balanced growth in continuous time. This research is part of the project Stochastische Dynamische Konjunkturtheorie supported by the Deutsche Forschungsgemeinschaft under contract No. BO 635/9-1,3. I am indebted to V. Böhm, B. S. Jensen, L. Kaas, and J. Wenzelburger for important discussions and comments. 1

2 On the dynamics of basic growth models 2 1 Introduction Common perception in growth theory seems to suggest that trajectorial convergence to a balanced growth path in the state space (K, L is essentially guaranteed by the stability of the associated model in intensity form, but typically, the notion of convergence to a balanced growth path is not defined for the state space. For example, Romer (1996, p. 14 writes Thus the Solow model implies that, regardless of its starting point, the economy converges to a balanced growth path a situation where each variable of the model is growing at a constant rate, using convergence of rates, not convergence in the state space. An analysis of the dynamics in the original capital labor space and the question of convergence or divergence is widely missing. Indeed, for the Solow model without depreciation, Deardorff (197 shows, that the distance to a balanced growth path is always exploding in the capital labor space. We extend this result in the following directions: 1. We analyze growth models in a more general set up and derive conditions for both convergence and divergence in the state space. 2. With a positive depreciation rate > both convergence and divergence in the state space is possible, even for the Solow model. 3. In models with growth of labor productivity we provide conditions under which the values per effective labor unit converge, but the absolute values as well as the values per capita diverge. 4. In an optimal growth model we show that (even with = divergence and convergence to an optimal balanced growth path is both possible. The result in (Deardorff 197 seems widely unconsidered, notable exceptions 1 in this direction are (Jensen 1994 and actually (Jensen, Alsholm, Larsen & Jensen 25, where the influence of scale and substitution effects on the dynamics in capital labor 1 In the unpublished paper (Koch 1995, this issue is observed and stability in the state space for two endogenous growth models is analyzed.

3 On the dynamics of basic growth models 3 space is analyzed. In this article we concentrate on the influence of the elasticity of the savings function which corresponds to the elasticity of the production function in the Solow model and obtain a divergence result as in (Deardorff 197 or (Jensen 1994 for the case without depreciation of capital. For the case with a positive depreciation rate, we provide conditions 2 for convergence to a balanced growth path or divergence, depending on parameters and on properties of the savings function, in particular. For systems with an exponentially growing state variable we introduce a technique to decide whether convergence of ratios in the intensity system coincides with convergence of differences in the state space. The results are illustrated for basic growth models as the Solow model (see (Solow 1956, (Swan 1956 in 4.1 and an optimal growth model (see (Cass 1965, (Koopmans 1965 in 4.3. However, the technique can also be applied to other models with exponentially growing variables, as in endogenous growth models, or those with physical and human capital accumulation as the augmented Solow model in (Mankiw, Romer & Weil 1992, analyzed in 4.2. The central issue of this paper is to demonstrate, that convergence in ratios need not imply convergence in differences 3. A typical situation, when these issues may arise, is in the literature of convergence of countries, where models are analyzed by comparing ratios. Our result shows, that two identical countries with convergence in ratio (either conditional or club convergence 4 but arbitrarily small differences in initial capital might induce time series where differences in absolute terms like GDP, total investment, or total consumption diverge to infinity. 2 Similar conditions are obtained in (Böhm & Wenzelburger 1999 for discrete time models and in (Böhm, Pampel & Wenzelburger 25 for stochastic models. 3 To illustrate this as simply as possible, consider to have a growth model with a solution for capital K(t = L(t+ L(t and labor L(t satisfying lim t L(t =. Obviously, this implies convergence of the capital labor ratio k(t = K(t L(t to k = 1, i. e. lim t k(t k =. On the other hand the distance K(t kl(t to the balanced growth path is L(t and hence lim t K(t kl(t =. 4 For an overview article on absolute convergence, conditional convergence and club convergence see (Galor 1996.

4 On the dynamics of basic growth models 4 In econometric models often logarithms of variables or growth rates 5 are analyzed. It follows from our result, that some care should be taken, when interpreting the results as deviations from an exponential trend, since the deviation (a difference of absolute values might diverge, even if logarithms or growth rates converge. 2 Balanced growth paths in continuous time The typical structure of growth models in continuous time with depreciation of capital is given by a system of differential equations K = Ls ( K L K, K( = K, L = nl, L( = L, (1 where s( is the function of gross capital accumulation per capita, [, 1] is the depreciation rate and n > 1 is the growth rate of labor. An orbit ( K(t, L(t, t of (1 describes the development of capital and labor in time. A solution k(t, t of the differential equation in intensive form k = h(k := s(k (n + k, k( = k := K L (2 with k := K L induces the solution ( K(t, L(t = ( k(te nt L, e nt L, t of (1. In the literature, the dynamics of a growth model as defined by (1 are mostly analyzed using the intensive form (2 only. In particular, if the initial value k = k is a fixed point of (2, then ( K(t, L(t = ( kl(t, L(t is a solution, where the growth rates of capital and labor are both n. Thus, a balanced growth path can be defined as an orbit of the original system as follows: Definition 1 An orbit of ( K(t, L(t, t of (1 is called a growth path. A growth path is called balanced, if K K = L L for all t. (3 5 Note, that convergence of differences in logarithms or convergence of growth rates is essentially the same as convergence of ratios.

5 On the dynamics of basic growth models 5 Geometrically, a balanced growth path is located on a ray in the state space (K, L. Since K = s(k and L K k L following three assertions: = n for all t we obtain from 3 the equivalence of the 1. ( K(t, L(t := ( kl(t, L(t is a balanced growth path. 2. k is a fixed point of (2. 3. n + = s( k k. Therefore, each fixed point in ratios is related to a ray in the state space. Obviously, a necessary condition for the existence of a positive fixed point k of (2 and hence of a balanced growth path of (1 is n + >. A sufficient condition for asymptotic (ratio stability of a fixed point k of (2 is h ( k < or equivalently s ( k < n +. However, in the following we will see that such a condition for asymptotic (ratio stability need not imply convergence to the balanced growth path 6 (i. e. to the related ray in an expanding system with n >. 3 Convergence and divergence in the state space Deardorff (197 was the first to observe that for =, the convergence in absolute terms to a balanced growth path of the system (1 does not automatically follow, whenever k is an asymptotically stable fixed point of the ratio system (2. It is a surprising fact, however, that conditions for convergence and divergence in continuous time models are essentially the same as derived in (Böhm & Wenzelburger 1999 for discrete time models. We obtain the following properties of the savings function for convergence and divergence. 6 The notion of Lyapunov stability does not apply to expanding systems. However, if the differences to a balanced growth path D K (t := K(t kl(t converges globally, than the ray is an attracting set in the sense of (Guckenheimer & Holmes The ray cannot be an attractor, since the ω limit set of solutions converging to the ray is empty and the ray is not compact.

6 On the dynamics of basic growth models 6 Theorem 1 Let s be differentiable, n + > and let k be an asymptotically stable fixed point of k = s(k (n + k. For any k k in the basin of attraction of k the distance D K (t := K(t kl(t to the balanced growth path ( kl(t, L(t satisfies lim D K(t = if h ( k + n >, (4 t lim D K(t = if h ( k + n <, (5 t and equivalently, using the elasticity of the savings function E s (k := s (kk s(k, Proof: lim D K(t = if E s ( k > t n +, (6 lim D K(t = if E s ( k < t n +. (7 Since h (k = s (k (n +, the asymptotically stable fixed point of k satisfies h( k = and h ( k and an orbit with initial value k k in the basin of attraction of k converges to k as t. The distance function satisfies D K (t = ( k(t k L(t and Ḋ K = K k L = s(kl K n kl. (8 Therefore, the growth rate of the distance to the balanced growth path is Ḋ K D K = s(k k n k k k = n + h(k k k = s(k (n + k n + k k = h(k h( k n + k k h ( k + n. k k (9 If h ( k + n >, then the growth rate of the distance is eventually positive with a positive lower bound, implying (4. If h ( k + n <, then the growth rate of the distance is eventually negative with a negative upper bound, implying (5. The equivalent representations (6 and (7 are obtained by observing h ( k + n s ( k s( k k E s ( k n +. }{{} = s ( k k (1 s( k }{{} = E s( k Using the asymptotic growth rate h ( k + n in (9, one can show, that the distance function behaves like e (h ( k+n(t t D K (t for large t and t t. We formalize this

7 On the dynamics of basic growth models 7 in Lemma 1 in the appendix. These results show, that different asymptotic behavior of ratios and differences occur, whenever the expanding factor (here n in the denominator of k = K is stronger than the contracting factor (here L h ( k of the ratio system. The technique can be transfered to systems with more state variables with essentially the same observations. We illustrate this in the following section for the model in (Mankiw et al with physical and human capital and exogenous technological progress. In this model with exogenous technological progress, convergence of capital per unit of effective labor need not imply convergence of per capita values. 4 Basic growth models 4.1 The Solow model Consider the original Solow (1956 model, where the savings function is s(k = sf(k, s (, 1 and f(k is the production function in intensive form. For a concave, strictly increasing function f we get E s (k = E f (k (, 1], which is also the capital share of output. For > and n > the result of Theorem 1 implies 7, that convergence to a balanced growth path or divergence from the balanced growth path depends on the elasticity of the production function. Proposition 1 Let the assumptions of Theorem 1 hold with s(k = sf(k, s (, 1, then the distance D K (t := K(t kl(t to the balanced growth path satisfies lim D K(t = if E f ( k > t n +, (11 lim D K(t = if E f ( k < t n +, (12 where E f (k := f (kk f(k is the elasticity of the production function. 7 If n + > (necessary for the existence of a fixed point k and n < (decreasing labor we get E f ( k 1 < and hence convergence to the balanced growth path (and finally to (, in the state space. For n > and =, we obtain divergence to infinity, since E f ( k > = was also observed by Deardorff (197., which

8 On the dynamics of basic growth models 8 Proof: Since s(k = sf(k implies E s (k = E f (k, the proof is an immediate application of Theorem 1. Indeed, we have shown that the Solow model exhibits convergence of per capita values, but for E f ( k > the difference of real capital diverges to infinity. Later on in section 5 we visualized by numerical simulations of the Solow model, that several economies i with the same characteristics (saving rate s, depreciation rate, production function f and development of labor L(t = e nt L, but different initial real capital K i converge to the same steady state in capital intensities lim t k i (t = lim t k j (t = k, whereas the difference of real capital diverges, i. e. lim t K i (t K j (t =, if E f ( k >. The Cobb Douglas function8 F (K, L = AK α L 1 α has an isoelastic intensive form f(k = Ak α with constant elasticity E f (k = α (, 1, such that convergence or divergence depends on α. 4.2 The augmented Solow model Consider the model Mankiw et al. (1992 with physical capital, human capital accumulation, and exogenous technological progress K = s K F (K, H, AL K, K( = K, (13 Ḣ = s H F (K, H, AL H, H( = H, (14 AL = (n + gal, L( = L, A( = A. (15 This simple form allows us to show the main results very briefly: 8 For the Cobb Douglas function D K = kl ( 1 + bl 1 n (1 α 1 α kl (b R depends on initial values is computed in (Jensen et al. 25. There, D K is claimed to converge to as L, if >. However, for n > and sufficiently large L (such that bl n (1 α < 1 the binomial series implies D K = kl ( 1 + i=1 ( 1 1 α b i i L n (1 α i kl = k i=1 ( 1 1 α b i L i 1 i n (1 α and hence the sign of the largest exponent 1 n (1 α of L describes the convergence or divergence behavior. Since 1 n (1 α α are more obvious in case of α = 1 2. we get the result of Theorem 1. These arguments

9 On the dynamics of basic growth models 9 Define k := K AL and h := H AL and the dynamics in ratios is by k = s K f(k, h (n + g + k, (16 ḣ = s H f(k, h (n + g + h. (17 Under standard assumptions on f (strictly increasing, strictly concave there is a unique positive steady state ( k, h which is asymptotically stable, since the Jacobian at this steady state has eigenvalues λ 1 = (n+g+ < λ 2 = (E f ( k, h 1(n+g+, where E f,k ( k, h is the elasticity to scale satisfying E f ( k, h < 1 for strictly concave functions. Therefore both eigenvalues are negative 9. An eigenvector of λ 1 is v 1 = and an eigenvector of λ 2 is v 2 = ( f( k, h/ h f( k, h/ k ( sk sh. Thus, the dynamics eventually converge in the direction 1 of v 2, i. e. lim t h(t h k(t k = s H s K. This is important to anticipate the two dimensional dynamics, when computing the asymptotic growth rates of the differences D K = K kal and D H = H hal. Proposition 2 In the augmented Solow model the distance functions given by D K (t = K(t ka(tl(t and D H (t = H(t ha(tl(t satisfy: If > λ 2 > (n + g or equivalently E f ( k, h > If λ 2 < (n + g or equivalently E f ( k, h < n+g+, then lim D K(t = and lim D H (t = (18 t t, then n+g+ lim D K(t = and lim D H (t =. (19 t t 9 The eigenvalues are λ 1 = (n + g + < λ 2 = s K f( k, h k λ 2 = (E f ( k, h 1(n + g + results from + s H f( k, h h (n + g + and s K f( k, h k + s H f( k, h h f( k, = s h K }{{ k } = n+g+ f( k, h k k + s H f( k, h h f( k, }{{ h } =: E f,k ( k, h } {{ = n+g+ } = (n + g + ( E f,k ( k, h + E f,h ( k, h f( k, h h h f( k, }{{ h } =: E f,h ( k, h where E f,k ( k, h, E f,h ( k, h are the partial elasticities of f and E f ( k, h := E f,k ( k, h + E f,h ( k, h is the elasticity to scale satisfying E f ( k, h < 1 for strictly concave functions. 1 Except for initial values on the strongly stable manifold (which has Lebesgue measure.

10 On the dynamics of basic growth models 1 Proof: With ḊK = ( s K f(k, h k (n + g k AL we get a growth rate Ḋ K D K = n + g + s Kf(k, h s K f(k, h + s K f(k, h (n + g + k k k = n + g + s K f(k, h f(k, h h h t s H f( k, h h and analogously lim t } {{ } f( k, h s K h Ḋ H D H + s K f( k, h k = s K f( k, h k h h k k }{{} sh s K + s Kf(k, h (n + g + k k k }{{} f( k, h s K (n+g+ k = λ 2 + n + g (2 + s H f( k, h h = λ 2 + n + g. The equivalent condition is obtained since λ 2 + n + g = (n + g + E f ( k, h and hence λ 2 (n + g E f ( k, h n+g+ Again, we obtain convergence of ratios and divergence of differences, if the growth factor n+g of A(tL(t is stronger than the contracting factor (E f ( k, h 1(n+g+ of the ratio system. For a Cobb Douglas function F (K, H, AL = K α H β (AL 1 α β one has E f (k, h = α + β < 1, so that α + β > balanced growth path and α + β < n+g+ n+g+ implies convergence. implies divergence from the Exemplary for models with technological progress we observe, that per capita values might diverge, even though the values per effective labor unit converge. Corollary 1 In the augmented Solow model with exogenous technological progress g > and convergence of real and human capital per effective labor unit to ( k, h, all per capita values diverge from the balanced growth path, if E f ( k, h > converge to it, if E f ( k, h <. n+g+ and n+g+ Proof: Let κ(t := K(t be the capital labor ratio, then D L(t κ(t = κ(t ka(t = D K(t L(t and hence its growth rate is Ḋ κ D κ = ḊK D K L L = (λ 2 + n + g n = (n + g + E f ( k, h (n +. (21 This implies the result of the corollary. The consequences of this result for the convergence debate is striking. For example,

11 On the dynamics of basic growth models 11 for constant population n =, exogenous technological progress g >, and a Cobb Douglas function with α + β > g+, two identical economies with different initial K or H will induce convergence of values per effective labor unit, but divergence of per capita values. 4.3 Optimal growth Now we apply the results to an optimal growth model as introduced in (Cass 1965, (Koopmans Consider the textbook version of an optimal growth model as in Barro & Sala-I-Martin ( , where discounted utility c(ν 1 θ e (ρ nν dν with 1 θ ρ > n, θ > 12 is to be maximized subject to k = f(k c (n + k and some transversality conditions. It follows that an optimal solution ( k(t, c(t of the system of ordinary differential equations in intensive form k = f(k c (n + k, k( = k (22 ċ = 1 θ ( f (k ρc (23 converges monotonically for t to the unique positive steady state (k, c given by k = (f 1 ( + ρ and c = f(k (n + k. We obtain the following convergence and divergence result for this optimal growth model. Proposition 3 In the optimal growth model the distance D K (t := K(t k L(t to the optimal balanced growth path satisfies lim D K(t = if θρn > f (k c, (24 t lim D K(t = if θρn < f (k c. (25 t 11 We apply the model with x =, i. e. without technological progress. With technological progress x > we have to normalize all variables by A(tL(t = e (x+nt A L. Further analysis is essentially the same if n is replaced by n + x, except for the fact that k = (f 1 ( + ρ + θx depends on θ. 12 As usual, for θ = 1 we define log(c(ν = lim θ 1 c(ν 1 θ 1 1 θ 1 1 θ has no influence on the decisions. for all ν >, where the constant

12 On the dynamics of basic growth models 12 Proof: To apply Theorem 1, we have to find an optimal consumption function c (k, which is indeed a parameterization of the stable manifold of the saddle (k, c. Defining an optimal savings function s (k = f(k c (k, convergence to the optimal balanced growth path or divergence in the (K, L space depends by Theorem 1 on the sign of s (k = f (k c (k = ρ c (k. To compute c (k, we observe that any parameterization c(k of a solution of (22, (23 satisfies by chain rule applied to c(k(t that ċ = c (k k. Hence, with (22, (23 we obtain 1( f (k ρc(k = ċ = c (k θ k = c (k ( f(k c(k (n + k. (26 As c (k is a parameterization of a solution converging to (k, c we get c (k = lim k k 1 θ( f 1 (k ρc(k f(k c(k (n + k = f (k θ c ρ n c (k by l Hôspital, since f (k ρ = and f(k c (n + k =. Solving for c (k we get 13 for c (k ρ n (27 c (k = ρ n 2 + (ρ n θ f (k c > ρ n. (28 Summarizing, with the definition (27 the optimal consumption function c (k is differentiable, at least in a neighborhood of the asymptotically stable steady state (k, c. Therefore, we can apply Theorem 1 to s (k = f(k c (k and by E s (k n + we get the results in (24 and (25. s (k ρ c (k ρ + n (ρ n θ f (k c (29 (ρ + n2 (ρ n2 ρn = θ f (k c Since f (k c >, one obtains convergence or divergence depending on parameters. In particular, f (k c is independent of θ (, if x =. Thus, for n > 13 Analyzing the dynamics of (22, (23 one observes that 1 θ (f (k ρc(k f(k c(k (k must be positive along an optimal solution. Thus, the negative solution of the quadratic term can be excluded here. Indeed, the negative solution belongs to the unstable manifold.

13 On the dynamics of basic growth models 13 the optimal growth model with converging capital labor ratio might induce both, convergence to an optimal balanced growth path or divergence in the (K, L space, depending on θ, namely convergence if θ < f (k c ρn and divergence if θ > f (k c ρn. Therefore, two economies with the same characteristics except for the utility parameter θ converge to the same steady state (k, c. However, for θ 1 < f (k c ρn < θ 2 they have totally different limiting behavior in absolute values, i. e. economy 1 converges to the balanced growth path and economy 2 diverges from it. In particular, for initial K 1 > K2 > k L one observes an overtaking in the sense that K 1 (t < K 2 (t holds for sufficiently large t. 5 Numerical simulations We illustrate the previous results for the Solow model with Cobb Douglas production function s(k = sf(k = sak α, where E s (k = E f (k = α is constant. Moreover, we choose A =, such that the fixed point of k = h(k := sak α (k is normalized s to k = 1 for all parameter values α. As approximation we use the Euler method 14 which is here an appropriate method since the convergence and divergence behavior in (K, L space of the approximating solution and the continuous time solution are (even theoretically the same. the Euler method with step size > applied to (1 is to solve the difference equation K t+1 = (1 K t + s ( Kt L t Lt, Lt+1 = (1 + ñ L t with ñ = n, = and s( = s(. This is a discrete version of the Solow model, where the fixed points k of the intensive forms are the same, E s = E s and E s ( k ñ+ E s ( k =. Therefore ñ+ implies, that the convergence and divergence behavior is the same for the continuous time model and its Euler approximation. The result for discrete time models is obtained in (Böhm & Wenzelburger In Figures 1 and 2 we use a parameter set s = 1, n =.1, =.2, A = =.6 2 s 14 All simulations are carried out using ΛACRODYN, a software for discrete time dynamical systems (see (Böhm, Lohmann & Middelberg 1998, (Böhm 23 applied to the Euler approximation.

14 On the dynamics of basic growth models k Converging k for α < t D K (t Converging D K for α < k Converging k for α > t 35D K (t Diverging D K for α > ging k for α < k Converging k for α > (a α =.4 < 2 3 = t (b α =.8 > 2 3 = 6 8 t Figure 1: Time series of the capital labor ratio k and of the distance to balanced growth D K for k = sak α (n + k for different α =.4 and α =.8. and initial values L = 1 and different K =.1,.1, , In Figure 1 time series with of capital labor ratio k (above and of differences to balanced growth D K (t = K(t kl(t (below are plotted, in (a for a parameter α =.4 < 2 3 = α =.8 > 2 3 = with convergence to balanced growth and (b for a parameter with divergence. However, the capital labor ratio converges to k = 1 in both cases, whereas the difference converges in (a and diverges in (b. Figure 2 illustrates the behavior of the growth paths in the (L, K space, converging in (a and diverging in (b for different initial real capital K =.1,.1, , Since 1 + n = 1.1 in the upper plots we see ( L(t, K(t, t [, 31.1] and the lower plots we see ( L(t, K(t for a longer time interval t [, 53.6]. Finally, Figure 3 displays time series of the capital labor ratio k in (a and of the

15 Sfrag replacements PSfrag replacements On2the dynamics of basic growth models K K Convergence in (L, K [, 2] 2 K4 K Divergence in (L, K [, 2] ag replacements D K (t in (L, K [, 2] L k L K Convergence in (L, K [, 15] 2 K Divergence in (L, K [, 15] (a α =.4 < 2 3 = L PSfrag replacements K Divergence in (L, K [, 2] (b α =.8 > 2 3 = Figure 2: Phase portraits in (L, K space for different α =.4 and α =.8. k 1 D K (t L t t (a Converging k(t for all α (b Different asymptotics of D K (t Figure 3: Time series for different α =.1,.5,.6,.64,.66,.68,.72,.76,.8. difference D K (t = K(t kl(t in (b with the same initial K = 1, L = 1, but for different α =.1,.5,.6,.64,.66,.68,.72,.76,.8. This shows, that the convergence behavior of the difference changes at the critical value α = = 2. The 3 capital labor ratio k converges in both cases to 1. Note, that the parameterization

16 On the dynamics of basic growth models 16 A = implies, that the asymptotically stable fixed point of the intensive form s k = sak α (n + k is normalized to k = 1 for all these parameters α. 6 Summary We have shown for different models with growing economies, that convergence in ratios need not imply convergence in the state space itself, a phenomenon which might occur in most models of economic growth. Even if in general, this does not falsify theoretical results using ratios, it should be considered, when interpreting the results for the original state variables. As typical situation, our results should be taken into account, when comparing the development of countries in absolute values like GDP, total investment or total consumption, since these values might diverge, even if the system in ratios exhibits conditional convergence. Since Theorem 1 only assume k to be in the basin of attraction of a steady state k, the studies apply to economies with locally asymptotically stable fixed points and therefore also to models with club convergence.

17 On the dynamics of basic growth models 17 References Barro, R. J. & Sala-I-Martin, X. (1995, Economic Growth, McGraw-Hill Book Company, New York a.o. Böhm, V. (23, ΛACRODYN the handbook, Discussion Paper 498, Department of Economics, University of Bielefeld. Böhm, V. & Wenzelburger, J. (1999, Lectures on the theory of economic growth, Mimeo, Department of Economics, Bielefeld University, Bielefeld. Böhm, V., Lohmann, M. & Middelberg, U. (1998, ΛACRODYN1.-98 a dynamical systems toolkit, Software package, Department of Economics, University of Bielefeld. Böhm, V., Pampel, T. & Wenzelburger, J. (25, On the stability of balanced growth, Discussion paper no. 548, Department of Economics, Bielefeld University, Bielefeld. Cass, D. (1965, Optimal growth in an aggregative model of capital accumulation., Review of economic studies 32, Deardorff, A. V. (197, Growth paths in the Solow neoclassical growth model., Quarterly Journal of Economics 84, Galor, O. (1996, Convergence? inferences from theoretical models., The Economic Journal 16, Guckenheimer, J. & Holmes, P. (1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York. Jensen, B. S. (1994, The dynamic systems of basic economic growth models, Mathematics and its applications ; 32, Kluwer Acad. Publ., Dordrecht [u.a.]. Jensen, B. S., Alsholm, P. K., Larsen, M. E. & Jensen, J. M. (25, Dynamic structure, exogeneity, phase portraits, growth paths, and scale and substitution elasticities., Review of International Economics 13(1,

18 On the dynamics of basic growth models 18 Koch, K.-J. (1995, Stability issues of endogenous growth models, Diskussionsbeiträge, Serie II 27, Universität Konstanz, Sonderforschungsbereich 178, Konstanz. Koopmans, T. (1965, On the concept of optimal economic growth, in Scientific Papers of Tjalling C. Koopmans, Springer Verlag, Berlin, New York. Mankiw, N. G., Romer, D. & Weil, N. D. (1992, A contribution to the empirics of economic growth., Quarterly Journal of Economics 17, Romer, D. (1996, Advanced Macroeconomics, McGraw Hill, New York. Solow, R. M. (1956, A contribution to the theory of economic growth, Quarterly Journal of Economics 7(1, Swan, T. W. (1956, Economic growth and capital accumulation, Economic Record 32,

19 On the dynamics of basic growth models 19 A Appendix In the following lemma it is shown that a lower and an upper bound for the distance to the balanced growth path exists. Lemma 1 Let the assumption of Theorem 1 hold and n >. Moreover, let ɛ (, h ( k + n be arbitrary and t > sufficiently large, such that Ḋ K (t D K (t (h ( k + n < ɛ, for all t t. (3 Then the distance D K (t := K(t kl(t to ( kl(t, L(t satisfies e (h ( k+n ɛ(t t D K (t D K (t e (h ( k+n+ɛ(t t D K (t. (31 Proof: Let k( > k, implying k(t > k for all t and D K (t > for all t. Let t > be sufficiently large, such that h ( k + n ɛ < ḊK(t D K (t < h ( k + n + ɛ for all t t. Then we obtain by multiplying with D K (t and integration for all t t D K (t = D K (t + t t Ḋ K (sds D K (t + (h ( k + n + ɛ t t D K (sds. Applying the Gronwall Lemma saying g(t G + M t t g(sds = g(t e M(t t G with g := D K (, G := D K (t and M := h ( k + n + ɛ we get the right inequality of (31. Analogous arguments hold for the lower bound and for k( < k.