On the Stability of Balanced Growth

Volker Bohm Thorsten Pampel

Department of Business Administration and Economics

Bielefeld University

D-33501 Bielefeld, Germany

Jan Wenzelburger

Centre for Economic Research

Keele University

Keele, ST5 5BG, UK

Discussion Paper No. 548

Revised August 2007

Department of Business Administration and Economics

Bielefeld University

P. O. Box 100 131

33501 Bielefeld

Germany

Abstract

Common folklore in growth theory suggests that the stability of balanced growth paths in thecapital-labor space is essentially guaranteed by conditions which imply stability of the corre-sponding steady states of the models in intensity form. We show by means of simple examplesthat, in general, these well-known conditions are only necessary. For a class of deterministicgrowth models with discrete time we provide new structural insights into the nature of this phe-nomenon by stating additional requirements that ensure stability of balanced growth paths inthe original capital-labor space. We introduce a notion of path-wise convergence for stochasticgrowth models and generalize our sufficient conditions to the stochastic case.

1 Introduction

Dynamic properties of growth models are usually analyzed in per-capita terms or in growth rates.The reason for doing so is the fact that growth models which take the original capital-labor spaceas state space are expanding and thus inherently unstable when the labor force grows. A conceptof central importance for the analysis of the long-run development in growing economies is thenotion of balanced growth paths along which the growth rate of capital is equal to the growth rateof the labor force. Reformulating such a model in per-capita terms, balanced growth paths arecharacterized by steady states of a model in intensity form. It is tacitly presumed that the stabilityof balanced growth paths, i.e., the convergence of nearby unbalanced growth paths to a balancedgrowth path is implied by the stability of the corresponding steady states of the model in intensityform. This assumption of ‘path-wise convergence’ underlies almost all models of economic growth,including overlapping-generations models, optimal growth models, and endogenous growth models.A dynamic analysis of the growth model in the original state space is usually omitted. Indeed,the standard textbook literature, (e.g. Romer (1996), p. 14, Aghion & Howitt (1998), Chap. 1, orBarro & Sala-I-Martin 1995) seem to presume that the convergence in either per-capita terms or ingrowth rates implies the convergence in state space as well. However, the analysis typically lacks arigorous mathematical notion of path-wise convergence in the capital-labor space.

Economically the issue of path-wise convergence is at the center of the debate of comparative growthas for example in Galor (1996), yet a rigorous analysis of the dynamics in the capital-labor space ismissing. Notable exceptions are Deardorff (1970), Jensen (1994), Jensen, Alsholm, Larsen & Jensen(2005), and Pampel (2007). Deardorff (1970) shows that without depreciation the distance betweenunbalanced and balanced growth paths in the standard Solow model is always exploding. Jensen(1994) points out that, in general, path-wise convergence in continuous-time growth models cannotbe obtained from convergence in per-capita quantities. Jensen, Alsholm, Larsen & Jensen (2005)investigates convergence in the state space for models with arbitrary elasticity to scale. There it isconjectured that models with depreciation will converge to the balanced growth path. The literatureseems to have overlooked these findings. A rigorous analysis of continuous-time growth models isconducted by Pampel (2007) who demonstrates that two identical economies which converge inper-capita ratios but with arbitrarily small differences in initial capital may induce time seriesalong which absolute differences in GDP, aggregate investment, or aggregate consumption divergeto infinity. These findings indicate that convergence in per-capita quantities is only a necessarycondition for path-wise convergence and that one should be careful when interpreting convergenceresults on the level of per-capita quantities, growth rates, or logarithms as deviations from somegeneral trend.

This paper provides a more structural insight into this phenomenon by providing necessary andsufficient conditions for path-wise convergence for a class of growth models in discrete time and withpositive depreciation, both deterministic and stochastic. Based on the theory of random dynamicalsystems (Arnold 1998), the notion of path-wise convergence to a balanced growth path is generalizedto stochastic models and sufficient conditions for convergence and divergence are provided, evenfor expanding systems. These results have implications for many dynamic models which exhibitconvergence in intensity form only. Among these are models with overlapping generations, modelsof endogenous and of optimal growth, two sector models as initiated by Galor (1992) , and modelsin international trade, e.g., Mountford (1998, 1999).

The paper is organized as follows. Basic results on existence and uniqueness of balanced growth forthe deterministic Solow model are collected in Section 2. Section 3 provides necessary and sufficientconditions for path-wise convergence in the state space for growth models with more general savings

1

functions which admit asymptotically stable fixed points in per-capita terms. Stochastic modelswith discrete time are analyzed in Section 4.

2 Balanced Growth

2.1 The basic model with exogenous savings

The basic issues of the relationship of the stability of balanced growth in the two alternativeperspectives can best be exhibited and analyzed in the simplest of all growth models, the oneoriginally discussed by Solow (1956). Let aggregate real income be generated by a continuousconcave production function F : R2

+ → R+ which is homogeneous of degree one. For given laborsupply L ≥ 0 and given capital stock K ≥ 0, aggregate savings under full employment and fullcapital usage are given by a savings function

S = S(L,K) := sF (L,K), (L,K) ∈ R2+.

where 0 ≤ s ≤ 1 is a given constant savings propensity. Applying the fundamental principle ofcapital accumulation with a given rate of depreciation on capital 0 ≤ δ ≤ 1, the law of capitalaccumulation is a mapping Gs : R2

+ −→ R+ given by

Gs(L,K) := (1 − δ)K + S(L,K) (2.1)

which determines the capital stock of the subsequent period Kt+1 = Gs(Kt, Lt) for any t = 0, 1, . . ..If the labor force grows at a constant rate n > −1 so that Lt+1 = (1+n)Lt, one obtains a dynamicalsystem for L and K with state space R2

+ governed by the two maps

(L,Gs) : R2+ −→ R2

+, (L,K) 7→(L(L,K),Gs(L,K)

), (2.2)

where L(L,K) := (1 + n)L. Let (L,Gs)t denote the t-th iterate of the map (L,Gs), i.e.,

(Lt,Kt) = (L,Gs)t(L0,K0) := (L,Gs) · · · (L,Gs)

︸ ︷︷ ︸

t − times

(L0,K0).

Thus, an orbit γ(L0,K0) of the time-one map (2.2) is given by

γ(L0,K0) := (L,Gs)t(L0,K0)t∈N,

where the sequence (Lt,Kt)∞t=0 is also called a growth path.

The following properties are usually not stated explicitly, since they are simple and immediateconsequences of the underlying properties. The assumption of full–employment in each periodimplies that the change of employment is equal to the change of the population. Hence, employmentin each period grows independently of the capital stock and of income. Since F is homogeneous ofdegree one, the map (2.2) is homogeneous of degree one and the origin (L,K) = (0, 0) is a fixedpoint.

If n 6= 0, then the origin is the only fixed point. If (L, K) is a fixed point for n = 0, then (λL, λK)is also a fixed point for any λ > 0. Such fixed points do not describe interesting long run situationsof economic growth. However, since the system (2.2) is homogeneous of degree one, the notionof balanced growth for the long run development of a growing economy replaces the concept ofstationarity, being defined by orbits along which capital and labor grow at the same constant rate.

2

Define the growth rate of capital by the usual formula

nK := (Kt+1 − Kt)/Kt.

The capital accumulation law (2.2) implies

1 + nK =Gs(Lt,Kt)

Kt= (1 − δ) + sF (Lt,Kt).

Rewriting Gs(L,K) using the homogeneity property, one has

Gs(L,K) = K[(1 − δ) + sf(k)k

] = K[1 + nK(k)],

so that the growth rate of capital becomes nK(k) := sf(k)k

− δ, where k = KL

is the capital-laborratio or the so called capital intensity. Thus, the growth rate of capital at any time is homogeneousof degree zero in (L,K) and determined completely by the capital-labor ratio K/L.

The requirement that the capital stock grows at the same rate as the population, i. e.

Kt+1 − Kt

Kt= nK

(Kt

Lt

)

= n =Lt+1 − Lt

Lt

implies that the capital labor ratio is constant over time. This follows from

kt+1 =Kt+1

Lt+1=

(1 + nK(kt)

)Kt

(1 + n)Lt= kt, for all t.

Definition 2.1 A growth path γ(L0,K0) of (2.2) with (L0,K0) 0 is called balanced, if

Kt+1 − Kt

Kt= nK

(Kt

Lt

)

= n =Lt+1 − Lt

Lt

for all points (Lt,Kt), t ∈ N of the orbit γ(L0,K0).

Let k > 0 be the capital labor ratio associated with a balanced growth path γ(L0,K0) of the system(2.2). Then, according to Definition 2.1, the initial capital stock satisfies K0 = kL0. Moreover,capital and labor evolve over time with a constant capital labor ratio, that is, for all t one hasKt = kLt. In other words, in state space R2

+ capital and labor evolve along the ray K = kL, asillustrated in Fig. 1 for n > 0.

2.2 Existence of balanced growth paths

Let f : R+ → R+, f(KL

) := F (1, KL

) denote the so called intensive form1 of the production functionF and consider an economy of the above type with population growth rate n > −1 and depreciationrate 0 ≤ δ ≤ 1.

Lemma 2.1 A growth path γ(L, K) is balanced, if and only if the capital labor ratio k = K/L isstrictly positive and satisfies sf(k) = (n + δ)k.

1As is customary in this literature, the capital labor ratio K

L≡ k is also called the capital intensity, where both

terms are used interchangeably and equivalently.

3

K0

L(1 + n)2L0(1 + n)L0L0

kL

K

Figure 1: Balanced growth path.

Proof:By Definition 2.1, γ(L, K) is a balanced growth path, if the growth rate of capital is constant overtime, that is

nK

(Kt

Lt

)

:=sf

(Kt

Lt

)

Kt

Lt

− δ = n, for all (Lt,Kt) ∈ γ(L, K)

implying that k = k0 = KL

satisfies sf(k) = (n + δ)k. On the other hand k0 = k and kt = k ⇒

kt+1 = k implies by induction kt = k for all t and hence nK

(Kt

Lt

)

= nK(k) = n for all t ∈ N. QED.

Lemma 2.1 states that balanced growth paths exist, if and only if there exists a positive capitalintensity k > 0 such that the average capital productivity satisfies f(k)/k = (n + δ)/s. If ν(k) :=

kf(k) denotes the so called capital coefficient, the growth rate of the capital stock nK(k) can be

rewritten as nK(k) = sν(k) − δ. Lemma 2.1 then implies that balanced growth paths must satisfy

sν(k)

= n + δ or, equivalently,

savings propensity

capital coefficient= rate of population growth + rate of depreciation.

Since (n, δ, s) are given parameters of the economy, these considerations imply that the capital

productivity f(k)k

(or the capital out ratio ν(k)) must be sufficiently flexible as a function of k toguarantee existence of steady states for all economies. To obtain a general existence result thefollowing generalization of the so called Inada2 conditions becomes useful.

Definition 2.2 A continuous function f : R+ −→ R+ is said to satisfy the weak Inada conditions,if

(a) limk→0

f(k)

k= ∞ and (b) lim

k→∞

f(k)

k= 0.

2A differentiable function f : R+ −→ R+ is said to satisfy the Inada conditions, if (a) limk→0 f ′(k) = ∞ and(b) limk→∞ f ′(k) = 0. They were introduced in Inada (1963) for the first time. Clearly, the Inada conditions implythe weak Inada conditions for any differentiable function f : R+ → R+.

4

The weak Inada conditions have a straightforward economic interpretation. Since k/f(k) measuresthe capital input requirement for a unit of output, they imply that under efficient substitutionbetween labor and capital, the capital productivity/the capital coefficient may attain any positivevalue (assuming continuity of f). In economic terms this means that capital can be substituted bylabor without restrictions. Thus, to produce a unit of output efficiently any positive level of capitalK > 0 requires a positive input level L > 0 of labor. In other words, labor is an essential inputfactor implying F (0,K) = 0 for all K ≥ 0 (see Panel (a) of Figure 2), while capital does not haveto be an essential input factor.

K

L1/f(0)

PSfrag replacements

xη

(a) Both weak Inada conditions satisfied

K

1/M

L

1/M

PSfrag replacements

xη

(b) Both weak Inada conditions violated

Figure 2: Isoquants and weak Inada conditions

To describe the implication for the unit isoquant geometrically, suppose that both limits of f(k)/kare finite and positive, i. e.

(a) limk→0

f(k)

k= M < ∞ and (b) lim

k→∞

f(k)

k= M > 0.

Then the capital input requirement function κ(k) := k/f(k) has the limiting properties

(a) limk→0

κ(k) = limk→0

k

f(k)=

1

Mand (b) lim

k→∞κ(k) = lim

k→∞

k

f(k)=

1

M.

with 0 < 1/M < 1/M < ∞, implying a capital input requirement function κ : R+ → R+ whichis not surjective. In this case, the associated unit isoquant has the form as drawn in panel (b) ofFigure 2, starting at the point (0, 1/M ) becoming tangent or equal to the horizontal line at 1/M .

The following Proposition 2.1 provides the weakest existence result.

Proposition 2.1 Let f : R+ −→ R+ be continuous, and the map k 7→ f(k)/k be decreasing. Theequation (n + δ)k = sf(k) has a unique positive solution k = k(n, δ, s) > 0 if and only if

0 ≤ limk→∞

f(k)

k<

n + δ

s< lim

k→0

f(k)

k≤ ∞. (2.3)

5

Proof:Existence follows from (2.3) and the intermediate value theorem. Uniqueness is consequence of themonotonicity of k 7→ f(k)/k. QED.

Obviously, the weak Inada conditions imply (2.3). On the other hand, for all (n, δ, s) with n+δ > 0and 0 < s ≤ 1 the range n+δ

sis R+, such that the Inada conditions are necessary to get steady

states for all parameters (n, δ, s) with n + δ > 0 and 0 < s ≤ 1.

Proposition 2.2 Let f : R+ −→ R+ be continuous, and the map k 7→ f(k)/k be decreasing. Thenthe following statements are equivalent:

(i) f satisfies the weak Inada conditions,

(ii) for every (n, δ, s) with n + δ > 0 and 0 < s ≤ 1, there exists a unique k = k(n, δ, s) > 0 withs f(k) = (n + δ)k.

Proof:If f satisfies the weak Inada conditions, then the map k 7→ f(k)/k is surjective. Since this map isdecreasing by assumption, it is also injective. This implies (ii). On the other hand, if f satisfies(ii), the map k 7→ f(k)/k has to be one-to-one on R+. Since it is decreasing, it follows that f mustsatisfy the weak Inada conditions. QED.

Observe that any differentiable and strictly concave function f : R+ → R+ is strictly monotonicallyincreasing with f ′(k) < f(k)/k for all k. Hence

(f(k)

k

)′

=f ′(k)k − f(k)

k2< 0 for all k > 0,

implying that the capital product f(k)/k is strictly monotonically decreasing in k. It then followsfrom Proposition 2.2 that balanced growth paths γ(L, K) with k = K/L exist for all possiblechoices of n + δ > 0 and s.

If either Conditions (a) or (b) of the weak Inada conditions is not satisfied, then the existence of

k may fail, see Fig. 3. However, as long as the weak Inada condition limk→0f(k)

k= ∞ holds, the

existence of k > 0 can be guaranteed for sufficiently small s.

2.3 Stability of steady states in intensity form

In order to analyze the stability of balanced growth paths, let us first examine the dynamics ofthe corresponding capital intensities (the capital labor ratios) k = K/L. Throughout this sectionwe assume the production function f : R+ → R+ to be differentiable and concave (and thereforenon-decreasing).

Definition 2.3 The dynamical system in intensity form associated with (L,Gs) is defined by thetime-one map

Gs : R+ −→ R+, k 7−→ Gs(k) = 11+n

[(1 − δ)k + sf(k)].

6

n+δs

f(k)k

k

(a)

f(k)k

k

n+δs

(b)

Figure 3: Examples of nonexistence of balanced growth.

Since f(k) and (1− δ)k are both concave and non-decreasing functions, the time-one map Gs againis concave and non-decreasing. Moreover, Gs(0) = s

1+nf(0) and

G′s(k) = 1

1+n[1 − δ + sf ′(k)] ≥ 0 for all k ∈ R+.

For δ < 1, Gs is strictly monotonically increasing. Observe that k = Gs(k) is a steady state ofGs, if and only if (n + δ)k = sf(k). It follows from Lemma 2.1, that balanced growth paths of thesystem (L,Gs) correspond exactly to steady states of the associated system Gs. The next lemmafollows from Proposition 2.2 and our assumptions on the production function f .

Lemma 2.2 Let n + δ > 0 and 0 < s ≤ 1 be arbitrary. Then Gs has a unique positive steadystate k = k(n, δ, s) > 0, if and only if the concave production function f satisfies the weak Inadaconditions.

Fig. 4 indicates geometrically that the dynamics of the capital intensities always converges mono-tonically to the unique positive steady state k. The following proposition makes this graphicalobservation precise.

Proposition 2.3 Let n + δ > 0 and 0 < s ≤ 1 be arbitrary and assume, in addition that theconcave production function f : R+ −→ R+ satisfies the weak Inada conditions. Then Gs hasa unique positive steady state k = k(n, δ, s) > 0 which is globally asymptotically stable on R++.Moreover, all orbits γ(k0), k0 ∈ R++ are monotonic sequences.

Proof:The existence of a unique positive steady state k > 0 follows from Lemma 2.2. Let k > k bearbitrary. Since Gs is non-decreasing and concave, k ≤ Gs(k) < k and G2

s(k) ≤ Gs(k). It followsby induction that k ≤ Gt

s(k) ≤ Gt−1s (k) for all t ∈ N. Hence, the sequence Gt

s(k)t∈N is monoton-ically decreasing. Since it is bounded from below by k, there exists a unique limit point k0 withlimt→∞ Gt

s(k) = k0. The limit point must be a steady state of Gs. Since k was the unique steadystate, k = k0. For k < k the argument is analogous. QED.

7

It was seen in the previous section that there are two reasons why existence of a positive steadystate k > 0 of Gs may fail. These are violations of either part of the weak Inada conditions. First,let f(0) = 0 and suppose that limk→0 f(k)/k ≤ M < ∞, such that condition (a) of the weakInada conditions is violated. Since f is concave, f ′(k)k ≤ f(k) for all k yields (taking right limits)f ′(0) ≤ M . This implies

G′s(0) = 1

1+n[1 − δ + sf ′(0)] ≤ 1 ⇐⇒ f ′(0) ≤ n+δ

s.

Since Gs is concave with Gs(0) = 0, 0 is the only steady state if (n + δ)/s ≥ M . Consequently,if the savings propensity s is too small, 0 is globally asymptotically stable, meaning that theeconomy will impoverish in the long run (see Fig. 4 (b)). Second, suppose that f(0) ≥ 0 and

k

450

Gs(k)

k

(a) Positive steady state

k

450

Gs(k)

(b) No positive steady state

k

450

Gs(k)

(c) No positive steady state

Figure 4: Dynamics of Gs.

limk→∞ f(k)/k ≥ n + δ > 0 such that f violates condition (b) of the weak Inada conditions. Thenthere exists s > 0 with f(k)/k > n+δ

sfor all k > 0. The monotonicity and the concavity of Gs

imply that Gs(k) > k for all k > 0, see Fig. 4. Thus, the capital intensities kt = Gts(k0) diverge to

infinity for all initial values k0 > 0 as t tends to infinity. Therefore, output per capita yt = f(kt)as well as consumption per capita cs(kt) = (1 − s)f(kt) tend to infinity as well. Hence, no positivesteady state k exists where all variables grow at the same constant rate (see Fig. 4 (c)).

3 Stable Balanced Growth

Given the results on existence and uniqueness of balanced growth paths, it was a tempting conclu-sion that the stability of the balanced growth path would follow also from the stability in intensivefrom. Unfortunately, this is not the case. As the results of the next section show, the stabil-ity of balanced growth requires additional conditions which cannot be captured by the intensityconditions alone. The instability arises in a structural way in all models in intensity form as aconsequence of the reduction of the dimension.

Let us return to the dynamical system (2.2) which describes the evolution of the capital stockK and of the population L, and consider a more general continuous aggregate savings functionS(K,L) which is homogeneous of degree one. Define s(k) := S(k, 1) to be the associated per capita

8

function, where k = KL

as before3 . Then, the dynamics in state space is described by the differenceequations

Kt+1 = (1 − δ)Kt + s

(Kt

Lt

)

Lt, (3.1)

Lt+1 = (1 + n)Lt. (3.2)

An orbit γ(L0,K0) is said to converge to the balanced growth path defined by k, if the difference

4t := Kt − kLt = Gs(Lt−1,Kt−1) − kLt

between the actual capital stock Kt = Gs(Lt−1,Kt−1) = (1 − δ)Kt−1 + S(Kt−1, Lt−1) and thebalanced capital stock kLt converges to zero for t → ∞.

Let k > 0 denote an asymptotically stable fixed point of the dynamical system Gs associated withGs, i. e. of Gs(k) = 1

1+n

((1 − δ)k + s(k)

).

Using the definition of Gs, one has

4t = Lt[Gs(kt−1) − k] for all t.

If an orbit of Gs is monotonically increasing, the differences 4t are either positive for all t ornegative for all t, depending on whether the initial 40 is positive or negative. As seen above, thespecial case 40 = K0 − kL0 = 0 corresponds to balanced growth paths of (2.2), implying that4t = 0 for all times t, see Fig. 1.

The next Theorem links the stability of balanced growth paths to the elasticity of the productionfunction f at the steady state k of Gs. Recall that the elasticity of a function f at some point k isdefined by Ef (k) = f ′(k)k/f(k).

Theorem 3.1 Let s be differentiable and let k be an asymptotically stable fixed point of Gs. Let4

B(k) ∩ (Gs)−1(k) = k, where B(k) is the basin of attraction of k and (Gs)

−1(k) is thepreimage of k. Consider the time-one map (L,Gs) as given in (2.2). Let γ(L0,K0) be an arbitraryorbit of (L,Gs) with K0/L0 ∈ B(k), K0/L0 6= k, implying 40 = K0 − kL0 6= 0. Then the followingholds:

If Es(k) <δ

n + δ, then lim

t→∞4t = 0; (3.3)

If Es(k) >δ

n + δ, then lim

t→∞|4t| = ∞, (3.4)

where Es(k) = ks′(k)s(k)

denotes the elasticity of the function s at the steady state k.

Proof:Let k0 > 0 and k0 6= k arbitrary but fixed. Under the hypotheses of this Theorem, one has

4t+1 = Kt+1 − kLt+1 = Lt+1[Gs(kt) − k], t ∈ N,

where k denotes the steady state of Gs. Then,

4t+1

4t= (1 + n)

Gs(kt) − k

kt − k.

3We continue to use the notation Gs and Gs for the case where s is now a per capita savings function s(k). Noconfusion with the Solow case where s(k) = sf(k) should arise.

4This assumption is always satisfied, if s is strictly increasing.

9

kt converges to k since k0 ∈ B(k). Therefore,

limt→∞

4t+1

4t= (1 + n)G′

s(k).

This implies that |4t+1

4t− (1 + n)G′

s(k)| < ε for all t larger than some t0. It follows that

[(1 + n)G′s(k) − ε]|4t| < |4t+1| < [(1 + n)G′

s(k) + ε]|4t|, t ≥ t0,

and by induction

[(1 + n)G′s(k) − ε]τ |4τ+t0 | < |4t+t0 | < [(1 + n)G′

s(k) + ε]τ |4t0 |, τ > 0.

Now, if (1+n)G′s(k) < 1, then (1+n)G′

s(k)+ε < 1 for sufficiently small ε such that limt→∞4t = 0.On the other hand, if (1+n)G′

s(k) > 1, then (1+n)G′s(k)− ε > 1 for sufficiently small ε and hence

limt→∞ |4t| = ∞. The rest of the statement follows from

(1 + n)G′s(k) < 1 ⇐⇒ Es(k) < δ

n+δ

and the fact that k0 ∈ B(k), k0 6= k was arbitrary. QED.

Theorem 3.1 applies to functions Gs which are strictly increasing or strictly decreasing in a neigh-borhood of a locally asymptotically stable fixed point. As an immediate implication of Theorem3.1 one obtains the following corollary for the Solow model.

Corollary 3.1 A balanced growth path of the Solow model is stable if

Ef (k) <δ

n + δ, (3.5)

and unstable if

Ef (k) >δ

n + δ. (3.6)

For the Solow model, the elasticity of the savings function always coincides with the elasticity ofthe production function. Since the production function f is strictly concave, its elasticity Ef (k)is always less than one. Therefore, for the Solow model, balanced growth paths are always stablefor non negative growth rates n ≤ 05. Notice, however, that for positive rates n the stability ofthe balanced growth path is not always guaranteed and that the distance from the balanced pathdiverges to infinity in case of (3.4) in spite of the fact that the intensity converges.

In order to understand the implications of the above result for the unstable case, consider twoeconomies (with a balanced path k > 0) which are identical in all of their characteristics includingthe initial level of labor supply L0, but which differ only in their initial capital with K 1

0/L0 < k <K2

0/L0. Then, in the long run the capital stock Kt of both economies will be infinitely far away(below resp. above) from the balanced path implying that their difference will also become infinite.In any period t the rate of capital accumulation of the economy with the lower initial capital willalways be larger than 1+n while the rate of capital accumulation of the economy with higher initialcapital will always be less than 1+n. However, the level of capital of the economy with initial undercapitalization will never catch up with that of an economy with initial over capitalization. In other

5This was recognized already by Deardorff (1970); for positive population growth see also Bohm & Wenzelburger(1999).

10

words, the capitalization of two identical economies will be diverging. Therefore, the property ofconvergence of capital intensities alone is a weak concept which does not guarantee the convergenceto the balanced path. This feature is a common property of all one dimensional models in intensityform derived from two factor models of growth which has largely been ignored or overlooked in theliterature6.

While the proof of Theorem 3.1 is somewhat technical and not too revealing, the following Lemmaprovides a useful check of the local asymptotic stability of balanced growth paths for arbitrarymodels in intensity form. It also exhibits the role of the elasticity of the mapping G and of the rateof population growth for the stability of the balanced path.

Lemma 3.1 Consider a twice differentiable time one map G : R+ −→ R+ of a growth model inintensity form with a positive fixed point k = G(k) where the labor force grows at the rate n > −1.Then, the positive balanced growth path associated with k is asymptotically stable if

max(|(1 + n)G′(k)|, |G′(k)|) < 1 (3.7)

Proof:Define an associated two dimensional system

kt+1 = G(kt) (3.8)

4t+1 = (1 + n)G(kt) − k

kt − k4t, (3.9)

where 4 := (k − k)L measures the deviation from the ray kL associated with the balanced path.Then:

i) A positive fixed point k of Gs defines a positive balanced path if and only (k, 0) is a fixedpoint of (3.8), (3.9)

ii) (3.8) - (3.9) has the two eigenvalues λ1 = G′(k) and λ2 = (1 + n)G′(k) .

Therefore, for any L0 > 0 and sufficiently small ε such that

|k0 − k| =|40|

L0≤

ε

L0

one has limt→∞4t = 0. QED.

Observe that Lemma 3.1 applies to very general one dimensional growth models in intensity form,including OLG models and optimal growth models. In addition, it provides clear cut informationabout the asymptotic stability of any number of hyperbolic fixed points of the augmented system(3.8) - (3.9). In other words, if the elasticity of G at any steady state is greater than 1/(1+n), thenthe basin of attraction of the balanced path is empty. Thus, even in the case of multiple balancedgrowth paths the stability of each one depends on the elasticity of G at each fixed point7. It mayvery well be the case that none of them is stable!

6Deardorff (1970) and Jensen (1994) are two notable exceptions.7Compare Galor (1996).

11

Stable growth paths in the Solow model with different technologies

Example 3.1 Consider linear production functions of the form

f(k) = a + bk, a, b > 0.

Since

limk→0

f(k)

k= ∞ and lim

k→∞

f(k)

k= b,

f does not satisfy property (b) of the weak Inada conditions. However, for the associated time onemap Gs

kt+1 =1

1 + n((1 − δ)kt + s(a + bkt))

one finds a unique steady state

k =sa

n + δ − bs

which is greater than zero for n + δ > sb. Under this condition a unique positive balanced growthpath exists. The elasticity of f on the balanced path is Ef (k) = sb

n+δ. Therefore, the balanced

growth paths arestable if sb < δ and unstable if sb > δ.

Thus, stable permanent balanced growth occurs for all parameter values whenever8

n + δ > sb and sb < δ.

The balanced path exists but is unstable for

n + δ > sb > δ

with capital diverging from the balanced path for any K0/L0 6= sa/(n + δ − bs) since

∆t+1 := Kt+1 − kLt+1

= ∆t + k(n + δ − sb)Lt + (sb − δ)Kt − knLt

= ∆t + (sb − δ)(Kt − kLt) = (1 + sb − δ)∆t.

Example 3.2 Consider isoelastic production functions of the form

f(k) = Akα, A > 0, 0 < α < 1.

Then f(k)k

= Akα−1 and

limk→0

f(k)

k= ∞ and lim

k→∞

f(k)

k= 0.

Hence, isoelastic production functions satisfy the weak Inada conditions. In this case the uniquepositive steady state of the associated system Gs is given by

k =

(

As

n + δ

) 11−α

, n + δ > 0.

8Compare for example Barro & Sala-I-Martin (1995) and Aghion & Howitt (1998)

12

Since Ef (k) ≡ α for all k, balanced paths are stable for all 0 < s ≤ 1, if

α <δ

n + δ.

Therefore, they are stable for sufficiently small growth rates n of the population, in particular forall n ≤ 0. However, for n > 0,

α >δ

n + δ

implies an unstable balanced growth path for any s, with capital diverging whenever

K0/L0 6= k =

(

As

n + δ

) 11−α

.

∆t+1 := Kt+1 − kLt+1

= ∆t + (s(f(kt) − f(k)(n + δ)kLt + δkt − nk)Lt

= ∆t +

(

sf(kt) − f(k)

kt − k− δ

)

(kt − k)Lt

=

(

sf(kt) − f(k)

kt − k− δ

)

∆t

Since

f ′(k) = αn + δ

s,

one has

limt→∞

sf(kt) − f(k)

kt − k− δ = α(n + δ) − δ > 1.

Therefore, for all t ≥ t0|∆t+1| > |∆t|.

Figure 5 summarizes the qualitative results of Examples 3.1 and 3.2, by displaying the ranges ofparameters (shaded region) for which unstable positive balanced growth paths exist.

Example 3.3 Consider the case of Leontief production functions of the form

f(k) = mina, bk, a > 0, b > 0,

withf(k)

k= min

a

k, b

.

As can be seen from Fig. 6, the weak Inada conditions are violated. Therefore a fixed point kexists, if and only if

0 <n + δ

s≤ b.

Since the savings propensity s is bounded by 1, let us assume that (n + δ)/b ≤ 1. Otherwise nosavings propensity can lead to a positive steady state.

The evolution of the capital intensity is driven by

Gs(k) =

1−δ+sb1+n

k, k ≤ ab,

1−δ1+n

k + sa1+n

, k > ab.

(3.10)

There are three distinct cases (a) – (c) depending on the savings propensity s (see Figure 7).

13

1

stable δ

δ

balancedgrowth

balancedgrowth

no

growth

sb

n+

balancedunstable

(a) linear production function

1

stable

1

δ

δδ

E (k)=f

balanced

balancedgrowth

no

growth

α

n+

unstable

balancedgrowth

(b) isoelastic production function

Figure 5: Regions of parameters with unstable balanced growth; n > 0

ab

k

f(k)k

Figure 6: Capital productivity of the Leontief production function

Case (a): s > n+δb

.

Then, 1−δ+sb1+n

> 1 and there exists a unique fixed point k(n, δ, s) > ab

of Gs, see Fig. 7(a). The

case when k is strictly larger than ab

is usually referred to as a situation of over-accumulation,where labor is the binding input factor.

Case (b): s = n+δb

.

Then, 1−δ+sb1+n

= 1 and Gs has a continuum of fixed points, namely for each k ∈ [0, ab],

see Fig. 7(b). Notice that ab

is a particular fixed point of Gs. All steady states with 0 <k < a

bhave excess usage (input) of labor, while capital is fully employed representing the

binding input factor. Such situations are often referred to as cases of under-accumulationwith unemployment (see Figure 6).

Case (c): s < n+δb

.

The slope of the first function 1−δ+sb1+n

in (3.10) is less than 1, so that the graph Gs lies below

14

a

b

450

kk

(a) Unique steady state

k

450

a

b

(b) Continuum of steady states

k

450

a

b

(c) No positive steady states

Figure 7: Steady states for Leontief production functions.

the 45-degree line, see Fig. 7(c). This implies that the only fixed point of Gs is k = 0.

As is known from production theory, Leontief production functions do not allow efficient substitu-tion between labor and capital, the capital intensity k = a

bbeing the unique efficient factor input

ratio. The associated isoquants of the Leontief production function

F (L,K) = Lf(k) = minaL, bK, (L,K) ∈ R2+

are straight lines forming right angles along the ray abL. Therefore (see Fig. 8), any point (L,K)

with a capital-labor ratio k = KL

different from ab

is inefficient, i. e. the same aggregate output couldbe reached with either less capital stock or less labor input. Thus growth paths lying above the rayabL are paths with over-accumulation of capital, whereas paths lying below are paths with under-

accumulation (relative to efficient factor inputs). In order to examine the stability of balanced

K

L

kL

ab L

(a) over accumulation of capital

K

kL

kL

L

ab L

(b) under accumulation of capital

Figure 8: Positive balanced growth paths for Leontief production function.

15

growth paths in the Leontief case, the elasticity of the production function is given by

Ef (k) =

1, k < ab

0, k > ab

.

It follows from Theorem 3.1 that, for growing populations n > 0), the balanced growth pathswith over-accumulation k(n, δ, s) > a

bare always stable in state space. If balanced growth paths

with under-accumulation k(n, δ, s) < ab

exist, there must be a continuum of them corresponding tocase (b). However, any two of them are mutually diverging along different rays . Therefore, theHarrod–Domar growth model has a stable balanced growth path with over-accumulation whenevern+δ

s< b. In all other cases stable balanced growth does not exist.

4 Stable Balanced Growth with random perturbations

In this section we show that the issues of the stability of intensity form models and of their respectivestate space versions arise in the same way for stochastic growth models. In other words, convergence(in a stochastic sense) for models in intensity form does not guarantee convergence in state space.Moreover the structural source of the instability can be identified in the same way arising from aproperty of the per capita savings function. This translates into a condition of the elasticity of theproduction function for the Solow model9.

Consider the typical random growth model (in absolut terms) as described by two random differenceequation in the space of capital K and labor L given by

Kt+1 = (1 − δ)Kt + s

(

ξt,Kt

Lt

)

Lt, K0 given, (4.1)

Lt+1 = (1 + nt)Lt, L0 given, (4.2)

with an exogenous noise process ω := (ξi, ni)i∈Z,

• a per capita savings function s : R+×R+ → R+ : (ξ, k) 7→ s(ξ, k), where k ≥ 0 is the capitalintensity and ξ ∈ [ξmin, ξmax] ⊂ (0,∞) is a parameter, describing random productivity,random income, or random savings behavior,

• a random rate of population growth, n ∈ [nmin, nmax] ⊂ (−1,∞),

• and a rate of depreciation100 ≤ δ ≤ 1.

A path of random parameters is assumed to follow a stationary stochastic process described byω := ωii∈Z

= (ξi, ni)i∈Z∈ Ω := ZZ, Z := [ξmin, ξmax] × [nmin, nmax], such that (Ω,F , P) is

a probability space of random perturbations. By the assumption of an exogenous random growthrate nt, t ∈ Z, the labor force follows an exogenous time path Lt = L(t, ω, L0) driven by thelinear random difference equation Lt+1 = (1 + nt)Lt. A solution of (4.1), (4.2) is denoted by(K(t, ω,K0, L0), L(t, ω, L0)

).

9This section uses the description, techniques, and results of Schenk-Hoppe & Schmalfuss (2001), Bohm & Wen-zelburger (2002)

10To reduce notation we chose a deterministic rate of capital depreciation since a random one does not implyadditional results.

16

The induced random growth model in intensity11 form is given by

kt+1 = h(ξt, nt; kt) :=(1 − δ)kt + s

(ξt, kt

)

1 + ntwith k0 := K0

L0(4.3)

and a solution of (4.3) is denoted by k(t, ω, k0). Using a measurable invertible shift–operatorθ : Ω → Ω, θω = (ξi+1, ni+1)i∈Z

, the parameters in period t are (ξt, nt) = ωt = (θtω)0. With thisnotation (4.3) takes the form kt+1 = h(ωt, kt) = h

((θtω)0, kt

).

In a stochastic environment the analogue to a steady state in a deterministic system is eithera random fixed point, where time paths are invariant against the shift operator, or a stationarysolution, where the distribution is time invariant. Following (Schmalfuß 1996, 1998), a randomfixed point of h is a random variable k? : Ω −→ R+ on the probability space of the noise process(Ω,F , P) such that P–almost surely12

k?(θω) = h(ω0, k?(ω)

)for all ω ∈ Ω′, (4.4)

where Ω′ ⊂ Ω is a ϑ-invariant set of full measure, P(Ω′) = 1.

With this notion of a random fixed point k?t (ω) := k?(θ

tω), t ∈ Z defines a stationary solutionwith a time independent distribution ν(B) := P(k?

t (ω) ∈ B) =∫

1k−1? (B)P(dω) of k?

t . It cannotbe expected to observe solutions, where K and L have the same growth rates since kt is random.Instead, for a given noise process ω ∈ Ω we use k?

t (ω) to define a random balanced growth path tobe the following random variable:

Definition 4.1 Let ω ∈ Ω, K0 ∈ R+, L0 ∈ R+ and k?t be a stationary solution of (4.3). An orbit

of(Kt, Lt

)=

(K(t, ω,K0, L0), L(t, ω, L0)

)of (4.1), (4.2) is called a random balanced growth

path, ifKt = k?

t (ω)Lt for all t ≥ 0.

This definition implies the equivalence of the following three assertions:

1)(k?

t (ω)L(t, ω, L0), L(t, ω, L0))

is a balanced growth path for almost all ω ∈ Ω.

2) k? is a random fixed point of (4.3) satisfying (4.4).

3) (1 + n0)k?1(ω)

k?0(ω)

− (1 − δ) =s(ξ0, k

?0(ω))

k?0(ω)

for almost all ω ∈ Ω.

In the following section we analyze conditions, where convergence in ratios to k?t , i. e.

limt→∞

||k(t, ω, k0) − k?t (ω)|| = 0

implies convergence in absolute terms to a random balanced growth path as asymptote, i. e.

limt→∞

||K(t, ω,K0, L0) − k?t (ω)L(t, ω, L0)|| = 0

or divergence in absolute terms from the asymptote to infinity, i. e.

limt→∞

||K(t, ω,K0, L0) − k?t (ω)L(t, ω, L0)|| = ∞.

Definition 4.2 A random balanced growth path(Kt, Lt

)=

(k?

t (ω)L(t, ω, L0), L(t, ω, L0))

is calledasymptotically stable, if for all initial values K0 in a neighborhood of K0 and L0 = L0 thedistance ∆t = ∆

(t, ω,K0, L0

):= K

(t, ω,K0, L0

)− k?

t (ω)L(t, ω, L0

)from

(Kt, Lt

)to the balanced

growth path(Kt, Lt

)satisfies limt→∞ |∆t| = 0.

11Further on we abbreviate h′ := ∂h

∂k12Here, the term almost surely (a.s.) is used in a non-standard sense: a property holds P–almost surely (P–a.s.) if

there exists a ϑ-invariant set Ω′⊂ Ω (i.e. ϑΩ′ = Ω′) with P(ω′) = 1, such that the property holds for all ω ∈ Ω′.

17

4.1 Stability of balanced growth paths

Using these notions and concepts one can now establish similar properties of the savings functionfor convergence and divergence as above.

Theorem 4.1 Let s be differentiable and increasing with respect to k and let k?t be a stationary

solution induced by an asymptotically stable random fixed point of

kt+1 = h(ωt, kt) =(1 − δ)kt + s

(ξt, kt

)

1 + nt

.

Let [k, k], 0 ≤ k < k ≤ ∞ be a positive invariant interval with k?0(ω) ∈ [k, k] for all ω ∈ Ω and let

infZ×[k,k]h′(ξ, n; k) > 0.

For almost all ω ∈ Ω and any k0 ∈ (k, k), k0 6= k?0(ω) with limt→∞ |k(t, ω, k0)− k?

t (ω)| = 0 (i. e. k0

is in the basin of attraction of k?0(ω)) the distance ∆t := Kt − k?

t (ω)Lt to the balanced growth path(Kt, Lt

)=

(k?

t (ω)L(t, ω, L0), L(t, ω, L0))

satisfies

limt→∞

|∆t| = 0 if E log(∣∣h′(ω0, k

?0(ω))

∣∣)

+ E log (1 + n0) < 0, (4.5)

limt→∞

|∆t| = ∞ if E log(∣∣h′(ω0, k

?0(ω))

∣∣)

+ E log (1 + n0) > 0. (4.6)

Proof:First notice that k0 6= k?

0(ω), positive invariance of [k, k] and infZ×[k,k]h′(ξ, n; k) > 0 induces

kt 6= k?t (ω) and hence ∆t 6= 0 for all t ≥ 0.

¿From the definition ∆t = Kt − k?t (ω)Lt and by (4.1), (4.2) we obtain

∆t+1 = (1 − δ)Kt + s(ξt, kt

)Lt − k?

t+1(ω)(1 + nt)Lt

and hence using the solution property k?t+1(ω) = h(ωt, k

?t (ω)) we get

∆t+1

∆t=

(1 − δ)kt + s(ξt, kt

)− k?

t+1(ω)(1 + nt)

kt − k?(θtω)

= (1 + nt)h(ωt, kt) − h(ωt, k

?t (ω))

kt − k?t (ω)

.

Since limt→∞ |kt−k?t (ω)| = 0 we obtain for an arbitrary sufficiently small ε > 0 and t0 = t0(ε, ω) > 0

sufficiently large that∣∣∣∣

h(ωt, kt) − h(ωt, k?(θtω))

kt − k?t (ω)

− h′(ωt, k?t (ω))

∣∣∣∣< ε for all t ≥ t0(ε, ω). (4.7)

Note that (ξ, n, k) 7→ (1−δ)k+s(ξ,k)1+n

and (ξ, n, k) 7→ 1−δ+s′(ξ,k)1+n

are uniformly continuous on the

compact set [ξmin, ξmax] × [nmin, nmax] × [k, k].

Thus, the distance |∆t| = |Kt − k?t (ω)Lt| to the random balanced growth path satisfies

(1 + nt)(h′(ωt, k

?t (ω)) − ε

)|∆t| < |∆t+1| < (1 + nt)

(h′(ωt, k

?t (ω)) + ε

)|∆t|,

for all t > t0(ε, ω), if ε < infZ×[k,k]h′(ξ, n, k)|. By induction we see ∆t ≤ |∆t| ≤ ∆t for all t ≥ t0,

where ∆t solves the linear random dynamical system

∆t+1 = (1 + nt)(∣∣h′(ωt, k

?t (ω))

∣∣ + ε

)∆t, ∆t0 = |∆t0 |

18

and ∆t solves the linear random dynamical system

∆t+1 = (1 + nt)(∣∣h′(ωt, k

?t (ω))

∣∣ − ε

)∆t, ∆t0

= |∆t0 |.

In the appendix we show for E log (h′(ω0, k?0(ω))) + E log (1 + n0) < 0 that the upper bound ∆t

eventually converges to 0 P -a. s. inducing the convergence result. On the other hand forE log (h′(ω0, k

?0(ω))) + E log (1 + n0) > 0 we show that the lower bound ∆t eventually goes to

infinity P -a. s. implying the divergence result. QED.

Finally, we provide rather restrictive sufficient conditions for the convergence and divergence asanalogous results to the deterministic case.

Lemma 4.1 Let the assumptions of Theorem 4.1 be satisfied. Then

limt→∞

|∆t| = 0 if Es

(ξ0, k

?0(ω)

)<

δ(

n0k?1(ω)

k?0(ω) +

k?1(ω)−k?

0 (ω)k?0(ω)

)

+ δfor all ω ∈ Ω,

limt→∞

|∆t| = ∞ if Es

(ξ0, k

?0(ω)

)>

δ(

n0k?1(ω)

k?0(ω) +

k?1(ω)−k?

0 (ω)k?0(ω)

)

+ δfor all ω ∈ Ω.

Proof:If Es

(ξ0, k

?0(ω)

)< δ

„

n0k?1(ω)

k?0(ω)

+k?1(ω)−k?

0(ω)

k?0(ω)

«

+δ

for all ω ∈ Ω, then

E log(h′(ω0, k

?0(ω))

)+ E log (1 + n0) = E log

(s′(ξ0, k

?0(ω)) + (1 − δ)

)

= E log

(s(ξ0, k

?0(ω))

k?0(ω)

Es(ξ0, k?0(ω)) + (1 − δ)

)

= E log

((

(1 + n0)k?1(ω)

k?0(ω)

− (1 − δ)

)

Es(ξ0, k?0(ω)) + (1 − δ)

)

< E log

(

(1 + n0)k?1(ω)

k?0(ω)

− (1 − δ)

)δ

(

n0k?1(ω)

k?0(ω) +

k?1(ω)−k?

0 (ω)k?0(ω)

)

+ δ+ (1 − δ)

= E log(1) = 0.

The same arguments can be used for divergence when the reverse inequality “>” holds. QED.

Since it is difficult to verify such a condition for all ω ∈ Ω we provide the two more restrictivesufficient conditions on the elasticity for convergence. The first one

limt→∞

|∆t| = 0 if sup[ξmin,ξmax]×[k,k]

Es(ξ, k) <δ

(

(1 + nmax) kk− 1

)

+ δ

describes an upper bound for the elasticity of the savings function which depends only on featuresof the support of the noise process. The second one

limt→∞

|∆t| = 0 if Es

(ξ0, k

?0(ω)

)<

δ

E(

n0k?1(ω)

k?0(ω) +

k?1(ω)−k?

0 (ω)k?0(ω)

)

+ δfor ω ∈ Ω.

19

is less restrictive but more difficult to verify, and it may depend on initial the conditions (n0, k0).Its validity can be seen from

E log

((

(1 + n0)k?1(ω)

k?0(ω)

− (1 − δ)

)

Es(ξ0, k?0(ω)) + (1 − δ)

)

≤ E

((

(1 + n0)k?1(ω)

k?0(ω)

− (1 − δ)

)

Es(ξ0, k?0(ω)) − δ

)

< E

(

(1 + n0)k?1(ω)

k?0(ω)

− (1 − δ)

)δ

E(

n0k?1(ω)

k?0(ω) +

k?1(ω)−k?

0(ω)k?0(ω)

)

+ δ− δ

= 0.

These properties show that the results from the deterministic case carry over to the stochastic casewhen the perturbations are sufficiently small.

4.2 Numerical simulations

2

1.5

1

0.5

0

0 250 500 750 1000

PSfrag replacements

kt

t

(a) β = 0.4 < 23

= δ

n+δ

2

1.5

1

0.5

0

0 250 500 750 1000

PSfrag replacements

kt

t

(b) β = 0.8 > 23

= δ

n+δ

20

14.5

9

3.5

−2

0 250 500 750 1000

PSfrag replacements

kt

t

∆t

t

(c) β = 0.4 < 23

= δ

n+δ

200

100

0

−100

−200

0 250 500 750 1000

PSfrag replacements

kt

t

∆t

t

(d) β = 0.8 > 23

= δ

n+δ

Figure 9: Convergence and Divergence of the capital intensity kt and of the distance ∆t

We illustrate the results for the stochastic case using the Solow–model with a constant savingspropensity, a Cobb–Douglas production function with constant production elasticity α, and a mul-tiplicative Hicks neutral technological shock.

20

The random perturbations are modeled as an i.i.d. process of the form nt = n(1 + 2 · Xn) ∼[−0.01, 0.03], and ξt = ξ(1 + 0.5 · Xξ) ∼ [0.5, 1.5], where all Xi ∼ [−1, 1] are uniformly distributed.We assume s(ξ, k) = ξsf(k) = ξsAkα, where Es(ξ, k) = Ef (k) = α is constant. Moreover, wechoose A = n+δ

s, such that the fixed point of the deterministic model with n, ξ = 1 is normalized

to k = 1 for all parameter values α. All simulations are carried out using ΛACRODYN, a softwarepackage for discrete time dynamical systems (see Bohm 2003)).

Figure 9 displays the time series of kt and ∆t of the intensity form

kt+1 =(1 − δ)kt + ξtsAkα

t

1 + nt

with parameters s = 12 , n = 0.01, δ = 0.02, A = n+δ

s= 0.06, for the same noise path, and for

multiple initial values K0 = 0.001, 0.1, 0.5 1.0 1.5 2.0 10.0 20.0 and L0 = 1. Panels (a) and (c)correspond to the situation with α = 0.4 < 2

3 = δn+δ

which induces a stable balanced growth path in(K,L)–space. Thus, one observes the point wise convergence of both variables k and ∆ for arbitraryinitial conditions. In contrast, panels (b) and (d) display the results for α = 0.8 > 2

3 = δn+δ

, whichinduces unstable balanced growth paths in (K,L)–space. Notice that the speed of convergence ofthe capital intensities to the random fixed point is much slower than in panels (a), (c), while thedistance ∆ diverges.

0

5

10

15

20

0 5 10 15 20

PSfrag replacements

LK

(a) α = 0.4 < 23

= δ

n+δ, L ∈ [0, 20]

0

5

10

15

20

0 5 10 15 20

PSfrag replacements

LK

(b) α = 0.8 > 23

= δ

n+δ, L ∈ [0, 20]

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

PSfrag replacements

LK

(c) α = 0.4 < 23

= δ

n+δ, L ∈ [0, 150]

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

PSfrag replacements

LK

(d) α = 0.8 > 23

= δ

n+δ, L ∈ [0, 150]

Figure 10: Convergence and Divergence in (L,K)–space

21

10

7.5

5

2.5

0

0 250 500 750 1000

PSfrag replacements

kt − k?t (ω)

t

(a) Convergence of kt − k?t (ω) for all α

20

15

10

5

0

0 800 1600 2400 3200

PSfrag replacements

kt − k?t (ω)

t

∆t

(b) Convergence and Divergence of ∆t

Figure 11: The role of α = 0.1, 0.5, 0.6, 0.64, 0.66, 0.68, 0.72, 0.76, 0.8 .

Figure 10 illustrates the behavior of the growth paths(L(t),K(t)

)in the (L,K)–space, correspond-

ing to the numerical experiment of Figure 9, converging to each other in (a) and (c) and divergingin (b) and (d).

Finally, Figure 11 displays time series with the same initial K0 = 10, L0 = 1, the same noise pathω as before, but for different α = 0.1, 0.5, 0.6, 0.64, 0.66, 0.68, 0.72, 0.76, 0.8. Thus, as α increases,the growth paths become unstable.

Appendix

First, let M(ω) := (1 + n0) (h′(ω0, k?0(ω)) + ε), such that M(θtω) = (1 + nt) (h′(ωt, k

?t (ω)) + ε) and

∆t+1 = M(θtω) ∆t, ∆t0 = |∆t0 | implying convergence of |∆t| to 0 if E log(M(ω)

)< 0.

We show:for sufficiently small ε > 0, E log (h′(ω0, k

?0(ω))) + E log (1 + n0) < 0 implies E log

(M(ω)

)< 0.

Let r := E log (h′(ω0, k?0(ω))) + E log (1 + n0) < 0 and ε > 0 with ε < ε infZ×[k,k]h

′(ξ, n, k), where

ε := e−r − 1 > 0. Then, εh′(ω,k?

0(ω)) < ε and ε = e−r − 1 implies

E log(∣∣h′(ω0, k

?0(ω))

∣∣ + ε

)= E log

(∣∣h′(ω0, k

?0(ω))

∣∣

(

1 + ε|h′(ω0,k?

0(ω))|

))

< E log(∣∣h′(ω0, k

?0(ω))

∣∣ (1 + ε)

)

= E log(∣∣h′(ω0, k

?0(ω))

∣∣)

+ log (1 + ε)

= E log(∣∣h′(ω0, k

?0(ω))

∣∣)− r

= −E log (1 + n0) .

and E log(M(ω)

)= E log (h′(ω0, k

?0(ω)) + ε) + E log (1 + n0) < 0. Hence, the linear random dy-

namical system ∆t+1 = M(θtω) ∆t, ∆t0 = |∆t0 | has a unique random fixed point O(ω) ≡ 0, whichis asymptotically stable. Thus, |∆t| is bounded above by ∆t satisfying limt→∞ ∆t = 0 P -a. s.implying limt→∞ |∆t| = 0 P -a. s.

Second, for r := E log (h′(ω0, k?0(ω))) + E log (1 + n0) > 0 we show E log

(M(ω)

)> 0, M(ω) :=

(1 + n0) (h′(ω0, k?0(ω)) − ε) for sufficiently small ε > 0. Let ε > 0 with ε < ε infZ×[k,k]h

′(ξ, n, k),

22

where ε := 1 − e−r ∈ (0, 1). Then − ε|h′(ω0,k?

0(ω))| > −ε > −1 implies

E log(∣∣h′(ω0, k

?0(ω))

∣∣ − ε

)= E log

(∣∣h′(ω0, k

?0(ω))

∣∣

(

1 − ε|h′(ω0,k?

0(ω))|

))

> E log(∣∣h′(ω0, k

?0(ω))

∣∣ (1 − ε)

)

= E log(∣∣h′(ω0, k

?0(ω))

∣∣)

+ log (1 − ε)

= E log(∣∣h′(ω0, k

?0(ω))

∣∣)− r

= −E log (1 + n0)

and hence E log(M(ω)

)= E log (|h′(ω0, k

?0(ω))| − ε) + E log (1 + n0) > 0.

If, in addition, ε < infZ×[k,k]h′(ξ, n, k) we see M(ω) > 0 and hence ∆t0

6= 0 implies ∆t 6= 0 for

all t ≥ t0, such that Qt := 1∆

t

solves

Qt+1 =1

M(θtω)Qt, Qt0 =

1

∆t0

.

Therefore, O(ω) ≡ 0 is a unique random fixed point of Qt+1 = 1M(θtω)Qt which is asymptotically

stable since E log(

1M(ω)

)

= −E log(M(ω)

)< 0. Thus, limt→∞ Qt = 0 P -a. s. implies limt→∞ ∆t =

∞ P -a. s. and limt→∞ |∆t| = ∞ P -a. s. and we get the divergence result.

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