Discussion Paper No.

588

ROBUST CYCLICAL GROWTH

Anjan Mukherji

July 2003

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Robust Cyclical Growth
Anjan Mukherji ∗Centre for Economic Studies and Planning
Jawaharlal Nehru University, New Delhi
andInstitute of Social and Economic Research
Osaka University
This version: April 2003∗

The paper was prepared during a visit to the Institute of Social and Economic Research of Osaka

University. The facilities provided at the Institute are gratefully acknowledged. An earlier paper “Robustness
and Stability of Cyclical Behavior”, contained preliminary versions of the results reported here. The current
version owes a lot to the comments received from seminar participants at Keio University, particularly
Hiroshi Matano, Toru Maruyama, and Michihiro Ohyama. I am indebted to Arnab Mukherji for help in thesimulation exercises and in preparation of the bifurcation diagrams.

Abstract
The stability of cyclical growth within the context of a model in Matsuyama (1999) is examined. It is shown that but for an extreme situation, the two-cycles are unique and arange of parameter values which imply the stability of such cyclical growth is derived. The
growth enhancing property of 2-cycles are shown to be retained by any cycle; the resultsof simulation exercises carried out are reported to show that for very wide range of parameter values, such cyclical growth paths are stable and thus robustness of the conclusions
are established. Finally, the configuration of parameters for which the dynamics exhibits
complicated (chaotic) behavior is also identified.

KEYWORDS : Cycles, stability of cycles, robustness, growth, innovation, chaos.

Robust Cyclical Growth1

1

Introduction

Studies of growth processes usually consider two sources of growth: one is through processof accumulation and the other is through processes of technical change and innovation. Matsuyama (1999)(and (2001)) provides a neat model which combines the two sources. This
strategy allows the feasibility of sustaining growth indefinitely through technical progress,
since without such an avenue, accumulation can only take an economy some distance before diminishing returns destroys growth possibilities. With accumulation and innovation
alternating, the studies referred to being in discrete time, growth possibilities are examined.
That neoclassical models of growth, in discrete time, may be capable of non-convergenceand generating complicated behavior has been known for some time, now. Consequently, it
is worthwhile to investigate whether the possibility of innovation affects long term behavior.
In the studies mentioned above, it is claimed that there is a trade-off between growthand innovation; while innovation is necessary for sustaining growth, during the process ofinnovation, growth rate is shown to be low; on the other hand, when no innovation occurs,
growth rate picks up. When the steady state is unstable, it is shown there are two-cycles,
which alternate between these two phases and offers the possibility of growing at rates which
are more than the one associated with the steady state.
While the existence of such cyclical possibilities are of interest, it is only when thesecycles are stable that the conclusion has some relevance. We examine this issue closely andshow that parameters of the process are such that with similar looking parameter values,
two-cycles may either be stable or unstable. Consequently it is necessary to investigate
the range of parameter values for which the results are valid. A range of parameter values
which imply stability of the 2-period cycles is obtained. The range which imply stability
for 2-cycles is shown to increase with an increase in a crucial parameter. However, if one

Robust Cyclical Growth2

is interested only in enhanced growth possibilities along cyclical paths, we show that any
cyclical path will have this property and so we have to examine only if the cycle is stable.
We report the results of some exercises to conclude that such cycles are stable for very wide
range of parameter values. Consequently the conclusions are robust.
The paper proceeds as follows: we introduce the model, to keep the treatment selfcontained1 , and consider the existence of two-cycles and their stability. It is shown that but
for an extreme situation, the two-cycles are unique; we also derive a stability condition forsuch cycles. This allows us to obtain a range over which the 2-cycles are stable. We also
investigate how this range changes when parameters change. Next, we focus attention onwhat happens when two-cycles lose stability. The growth enhancing property of 2-cycles areshown to be retained by any cycle and consequently it is of some interest to investigate thestability of such cycles; the results of simulation exercises carried out are reported to showthat for very wide range of parameter values, we have stable cycles; finally, the configurationof parameter values for which there may be complicated (chaotic) behavior is also identified.
The concluding section briefly discusses the significance of these results.

2

The Model

There is one final good which is produced under competitive conditions with the help oflabor (L) supplied inelastically and several intermediate goods; the single final good is bothconsumed and invested; say that at the end of period t − 1, Kt−1 units of the final good isavailable for production in period t and is to play the role of capital.
Additionally in period t, there are several types of intermediate goods available denoted1

But we refrain from a discussion of related literature and refer the interested reader to Matsuyma (1999)

and (2001).

3

Robust Cyclical Growthby z, z ∈ [0, Nt ]; prior to period t, types of intermediate goods up to Nt−1 have been
introduced (N0 > 0); in period t, production of type z up to Nt−1 (the old intermediate
goods) require just a units of capital per each unit of any of the old intermediate goods. New
intermediates in the range [Nt−1 , Nt ] may also be introduced and sold exclusively by those
who choose to innovate; for each of the new intermediate good, in addition to the a units ofcapital per unit, a fixed cost of F units of capital has to be incurred before production canbe carried out. The old intermediate goods are sold competitively and hence, their pricereflect marginal costs i.e.,pt (z) = art , z ∈ [0, Nt−1 ]where rt is the price of a unit of capital in period t; the new intermediates are sold underconditions of monopoly by the investor who innovates and to compute sales we need to findout about the demand for the new intermediates.
Production of the final good in period t takes place according to the production function:

Z Nt

Yt = AL1/σ {[xt (z)]1−1/σ dz}(1)0

where xt (z), z ∈ [0, Nt ] is the intermediate good of type z in period t; σ ∈ [1, ∞] is theconstant direct partial elasticity of substitution between any two pairs of intermediates; thefollowing features should be noted:•given Nt , production in period t satisfies constant returns to scale.
•the price elasticity of demand for each intermediate is σ.
•the share of labor is 1/σ.
Given the above demand condition, the price of the new intermediates must satisfypmt (z) =σart
, z ∈ (Nt−1 , Nt ]σ−1

4

Robust Cyclical Growth
Since all intermediate goods enter the production function symmetrically, we may takext (z) = xct ∀z ∈ [0, Nt−1 ]; xt (z) = xmt ∀z ∈ (Nt−1 , Nt ]and given the demand conditions specified above,xct
pct −σ1=[] = [1 − ]−σ
m
m
xt
pt
σ

(2)

Note that the above has been obtained on the assumption that both the new and theold intermediate goods are produced. The new intermediates are produced by monopolistsmwhose one period profit πtm is the sole incentive for production. Note that π m (t) = pm
t .xt −m 1
rt (axmt + F ) = rt (axt σ−1 − F ). Free entry guarantees that profits are non-positive so that

we have axmt ≤ (σ − 1)F with the proviso that: Nt ≥ Nt−1 and(Nt − Nt−1 )[a.xmt − (σ − 1)F ] = 0(3)Thus, when Nt > Nt−1 , innovation occurs and new products are introduced; the innovator earns no extra profits and operates at its break even point; while if this is not possible,
potential sales do not break even, then Nt = Nt−1 and no innovation takes place. In thissituation, of course there is a constraint which needs to be looked into. Recall that available
capital is Kt−1 and the production of both types of intermediates requires the use of thisresource; hence we must haveKt−1 = Nt−1 a.xct + (Nt − Nt−1 )(a.xmt + F)
Recall that if production takes place for both the sets of intermediate products (the newσ − 1 −σ and from (3),and the old), we have, Nt > Nt−1 and from (2), a.xct = a.xmt { σ }
σ − 1 1−σ . It should be pointed out thatcaxm
t = (σ − 1)F , so that axt = θF σ where θ = { σ }Kas σ ∈ [1, ∞], θ ∈ [1, e]. If on the other hand, Nt = Nt−1 , axct = Nt−1 . Thus, we have:t−1

5Robust Cyclical Growthaxct = min[Kt−1
, θσF ]
Nt−1

(4)

And further,

Nt = Nt−1 + max[0,Kt−1− θNt−1 ]σF(5)Substituting the above into (1), we have:

Y (t) =







1
1
K
AL σ Nt−1 ( aNt−1 )1− σ

for σF θNt−1 ≥ Kt−1

t−1


1

(σ − 1)F 1− σ1
Kt−1
1− 1

 AL σ [Nt−1 ( θσF− θNt−1 )()]
a ) σ +(aσFotherwise(6)

And finally to close the model, we have that a constant fraction of the output is left unused
so that it may be used as capital in the next period,2 i.e.,

Kt = µYt(7)

To simplify matters, we introduce the following notation:1α=A(aL) σa(θσF )

1σ

, kt =Kt
Nt σF θThe dynamics of the entire system is captured through the equations (5), (6) and (7). And
we may combine them into a single equation, using the variable kt introduced above:

kt =







µαkt−1σif kt−1 ≤ 1






µαkt−1
1 + θ(kt−1 − 1)otherwise

1− 1

(8)

In short, we shall write
2

It is this which provides the point of departure for the contribution of Matsuyama (2001), where capital

accumulation is derived from intertemporal optimization of the infinitely lived agent. This does lead to acomplication in the dynamics.

6Robust Cyclical Growthkt = φ(kt−1 )(9)and given any initial k 0 , we shall study the iterates φt (k 0 ), for t = 1, 2, ...
11 )Gk − σ
> 0 if k < 1
It should be noted φ(1) = G = µα; further that φ0 (k) = (1 − σand φ0 (k) =G(1 − θ)< 0 if k > 1. Thus the map φ(.) is of the standard uni-modal(1 + θ(k − 1))2

variety with a maximum value at 1.

FIGURES 1a and 1b HERE

It may be easily seen by referring to (8) that a crucial parameter for the system is
G = φ(1). Note also that as k → ∞, φ(k) → G
θ , which is finite, so that for all k large
enough, φ(k) < k; also since φ(k) > k for all k small enough (since φ(0) = 0 and φ0 (k) > 1
for all k small enough), it follows that there exists k > 0 such that k = φ(k). Further sucha positive k is unique too.
There are two possibilities depending on the magnitude of G; first of all notice that if
G < 1, k ∗ is an equilibrium ⇒ k ∗ < 1; consequently k ∗ ≤ 1 ⇒ k ∗ = Gσ ; and if G > 1 thenk ∗ > 1 ⇒ k ∗ = G θ− 1 + 1. Consider then these two cases one by one.
Consider first, G ≤ 1; as indicated above, in this case, k ∗ = Gσ . In this equilibrium,
notice that N remains fixed and hence no innovation occurs; since N is constant and given
the definition of the variable k, K too remains fixed and hence Y , so that no growth takes
place. Beginning from any arbitrary k 0 , the iterates settle to the regime k < 1 (Matsuyama
1 )| < 1; thus k ∗ iscalls this the Solow regime) and converges to k ∗ since, |φ0 (k ∗ )| = |(1 − σlocally asymptotically stable.
− 1 |; thus it is seen that
Consider next G > 1; k ∗ = G θ− 1 + 1 and |φ0 (k ∗ )| = | θ G
local asymptotic stability requires that θ − 1 < G. If on the other hand, θ − 1 > G, the7

Robust Cyclical Growthequilibrium is unstable. At this equilibrium notice that N is increasing continuously andkeeping pace with it is K and as we may see Y too is increasing all at the same rate: thecase of balanced growth. Along the balanced growth path, where kt = k ∗ > 1, it follows
N
Y
K
from (5) that Nt+1 = G and consequently, Yt+1 = Kt+1 = G as well; thus G − 1 is thettt
growth rate per period along the balanced growth path.

32-cycles and Their Stability

When G > 1 and θ − 1 > G, the equilibrium is unstable as we have shown above; but there
is a 2-cycle, i.e., ∃k1 > 1 > k2 such that k1 = φ(k2 ) and k2 = φ(k1 ).
Thus it is asserted that there exist k1 > 1 > k2 satisfying the following:

k2 =

Gk1
1 + θ(k1 − 1)

and
1− σ1

k1 = Gk2

Using the second relation to eliminate k1 , we have the following claim: there is 0 < k2 < 1such that
1− σ1

k2

1

[(1 − θ)k2σ + Gθk2 − G2 ] = 0
1

Note that writing f (k2 ) = [(1 − θ)k2σ + Gθk2 − G2 ], f (0) = −G2 < 0 while f (1) =(G − 1)(θ − 1 − G) > 0 and hence the claim follows. Also note that f 0 (0) < 0 and f 0 (k) =σ− 1 } σ−10 ⇒ k = k = { θσθG
and f (.) is decreasing for k < k and increasing for k > k;consequently, there is a unique k2 satisfying the above claim.
It is easy to see that there cannot be any two period cycle (k1 , k2 ), 0 < k1 < k2 < 1. For1− σ1then k1 = Gk2

1− σ1

and k2 = Gk1; this implies that k1 = Gσ > 1 which is a contradiction.

8

Robust Cyclical Growth

Nor can there be a cycle (k1 , k2 ), 1 < k2 < k1 ; for other wise k2 + k2 θ(k1 − 1) = Gk1
and k1 + k1 θ(k2 − 1) = Gk2 so that (k2 − k1 )(1 − θ + G) = 0 which too is a contradiction.
Thus:

1 There is a unique 2-period cycle (k1 , k2 ) with 0 < k2 < 1 < k1 when θ − 1 > G > 1.

The stability of this cycle depends on whether the following is greater or less than unity:|φ0 (k1 ).φ0 (k2 )| = |G2 (1 − σ1 )(θ − 1)
1|=[

G2 (θ − 1)k2σ {1 + θ(k1 − 1)}2
=

1

k2σ

].

(1 − σ1 ){1 + θ(k1 − 1)}2

(θ − 1)(1 − σ1 ).G2 .k12
1

{1 + θ(k1 − 1)}2 .k12 .k2σ1

θ − 1 k2σ (1 − σ1 )=.
G
G
= A.B sayNote that A > 1, while B < 1 so that the product could go either way. If the above
happens to be less than unity then, the 2-period cycle is attracting while if the quantity isgreater than unity the cycle is unstable. It should be clear that the smaller is the ratio A,
the greater is the chance of the cycle being stable. As we shall see, this point is of crucial
importance for locating a robust range of parameter values for which the cycles will bestable. There can be nothing more definite about the aspect of stability since as is clear
from the above expression, both possibilities exist.
For instance, consider the following situations for σ = 5 and hence, θ = 2.44; but foralternative similar looking values of G, we have:

Example 1 Let G = 1.075 < θ − 1 = 1.44. Then the two period cycle is given by k1 =1.06487 > 1 > k2 = 0.988226; |φ0 (k1 ).φ0 (k2 )| = 0.995503 hence the cycle is stable.

9

Robust Cyclical Growth

Example 2 Let G = 1.070 < θ − 1 = 1.44. Then the two period cycle is given by k1 =1.06041 > 1 > k2 = 0.988805; |φ0 (k1 ).φ0 (k2 )| = 1.00491 hence the cycle is unstable.

Since both kinds of cycles are possible, consider then the merit of the following proposition (Proposition 2, Matsuyama (1999)):
Let gx be the gross growth rate of the variable x. Along the period-2 cycles
1(a) gN = 1 < G < G(k2 )− σ = gK = gY in the Solow Regime.
(b) gN = 1 + θ(k1 − 1) > G = gk = gY in the Romer Regime.
11(c) gN = gK = gY = {1 + θ(k1 − 1} 2 = Gk22σ > G over the cycles.
The claims made above were meant to show that output (Y ) and investment (K) growfaster in the Solow Regime: i.e., (when k < 1 and no innovation takes place); in other words,
even though innovation is essential to sustain growth, during the process of innovation (theRomer regime when k > 1) the economy registers lower growth.
As should be obvious, this conclusion is of some interest only if the cycle is stable, sinceotherwise the fact, that along the cycle a higher growth rate is possible, is of little interest.
It is because of this that the calculations about the stability of the cycles provided above
assume significance.

3.1

Are Robust Stable 2-Cycles Plausible?

What kind of parameter values are sure to provide us with stable cycles ? The Examples(1) and (2) show that with roughly similar looking values of G one may get different results.
But there is some thing more that one may say in addition to the above.
Consider the situation when G = θ − 1; it is immediate, by referring back to (8), that inthis case, φ(1) = G and φ(G) = 1; further |φ0 (1).φ0 (G)| = (1 − σ1 )/G < 1 so that this cycle
is stable (here φ0 (1) is to be interpreted as the derivative on the left); this assertion follows

10

Robust Cyclical Growthby considering the expression A.B derived earlier; now A = 1 and hence the assertion
follows. But in this particular situation there is an embarrassment of cycles; since for any
k1 > 1, k1 < G there is k2 > 1 such that φ(k1 ) = k2 and φ(k2 ) = k1 . But these cycles
disappear as soon as G < θ − 1, as we have seen above; but, it should be clear, that for G
close to but less than θ − 1, the cycle close to (1, G) remains and that cycle retains stability.
Thus if G is close but less than θ−1 then the resultant 2-cycle is stable. Note that for thesituation described in Examples 1 and 2, θ − 1 is about 1.44; thus some where between thevalues for G given by 1.07 and 1.075, the two-period cycles change their stability property
and we have a point of bifurcation; for lower values of G there may be other points ofbifurcation.
In other words, very complicated dynamics may be possible in this framework andconsequently, the relevance of the Proposition such as the one mentioned above may beascertained only if we are assured that parameters have certain values which guarantee therobust stability of the two-period cycles i.e., G is close to θ − 1. This is robust because there
would be an open set O = (G, θ − 1) and any G ∈ O would ensure stability of the two period
cycles. The lower bound on this interval amounts to where the product A.B becomes unity
3 . To obtain an estimate for G, we may note that the expression in A.B may be written as1) 1(θ − 1)(1 − σ.k2σ = C.D say
G2
where we know that D < 1 so now if C < 1 then we are sure that A.B < 1 and the cycle isstable. Thus a sufficient condition for the stability of the two-cycle is:3

See, Matsuyama (1999), p. 344.; it is clear from this, that the author is aware that there may be ranges

over which the 2-cycles may be unstable; also over this range, the attracting cycles may have periods which
are powers of 2. It appears that the author is only concerned about the empirical plausibility of the existenceof 2-cycles, as his discussion on plausibility considers only the condition for the existence of 2-cycles, viz.,θ − 1 > G > 1, with G being ‘close‘ to θ − 1. The entire plausibility exercise considers G = θ − 1.

11Robust Cyclical Growth
2 If G > Ĝ =√1 )) then the 2-cycle is stable.
((θ − 1)(1 − σ

The claim follows by noticing that whenever the condition is satisfied, the term C < 1. In
fact for σ = 5, for example, we have, Ĝ = 1.07384 so that, whenever G > 1.07384 the twocycle is stable4 . This allows us to obtain an estimate of the open interval O, viz., (Ĝ, θ − 1)
and for any G-value in this interval, the two-cycle is stable. Since we do not know what
the magnitude of the parameter σ will be, let us take the range suggested in Matsuyama(1999), [5, 22]; we examine next, how the range of G-values which imply stability for the
2-cycle, [Ĝ, θ − 1) behaves with a variation in the parameter σ.
3 Let H be defined as the range of G-values which imply stability of the period 2-cycles,i.e., H = θ − 1 − Ĝ. Then H is an increasing concave function of σ in the range [5, 22].
The proof of this claim is contained in the Figure 2 below5
FIGURE 2 HERE
Thus so far as two-cycles are concerned, the range of G-values which ensure stability
increase with the value of σ. Be that as it may, we shall show below, that we do not need toconfine attention to 2-cycles alone to establish the fact that cycles are growth enhancing. In
fact, the range of values for the parameter G for which stable cycles are growth enhancing,as we indicate, cover a much wider range.
4

It should be clear that Ĝ > G. Thus for σ = 5, G ∈ [1.07384, 1.44) implies stability of the two period

cycle. In Matsuyama (1999), as we indicated above, G is taken to be θ−1 = 1.44 while discussing plausibility.
Our analysis thus reveals that the 2-cycles are a lot more “plausible”.
5

Given the nature of the functions involved, we have used Mathematica to generate the diagram which

seems to be enough for the purpose at hand.

12

Robust Cyclical Growth

4

When 2-cycles fail to attract

When the 2-cycle is not attracting, we need to analyze further. In this section, we shallconfine ourselves to the case when 1 < G < θ − 1. Given this restriction, we note that
φ(G) < 1. To proceed formally, note first of all,4 φ : [0, G] → [0, G].
Next, we note that6 :5 [φ(G), G] is an absorbing state (i.e., φ(G) ≤ k ≤ G ⇒ φ(G) ≤ φ(k) ≤ G).

In addition,
6 For any initial point k o ∈ [0, G], k o 6= 0, k ? , ∃t such that φt (k o ) ∈ [φ(G), G] unless
φt (k o ) = 0 for some t.
For, suppose to the contrary, there is some k o ∈ [0, G] such that for no t is φt (k o ) ∈1−1/σ[φ(G), G]; then φt (k o ) ∈ (0, φ(G))∀t i.e., kt+1 = φ(t+1) (k o ) = φ(kt ) = Gkt> kt . Thus{kt } is a monotonically increasing sequence bounded above and hence must converge tosome k, say, k ∈ [0, φ(G)]. Note that since kt+1 and kt both converge to k ∈ [0, φ(G)], wehave φ(k) = k ⇒ k = 0 which is a contradiction, since a monotonically increasing sequenceof positive numbers cannot converge to zero.
Thus the structure of an arbitrary trajectory is that apart from hitting the unstableequilibrium 0 accidentally, it will enter the interval [φ(G), G] in finite time and remain
inside, thereafter. Limit points for the trajectory will thus exist and will be located in theinterval [φ(G), G]. Our next task is to locate these limit points, if possible.
We note, however, that cyclical orbits, if these exist, have a special structure:
6

See, for instance, Matsuyama (1999), p. 342.

13

Robust Cyclical Growth

7 Consider any cycle of period n, k1 , k2 , ..., kn ∈ [0, G], say, where φ(kn ) = k1 with k1 =min1≤j≤n kj must have k1 < 1 and kn > 1, with kn = maxj kj .
First of all note that kn > k ? since otherwise k1 = φ(kn ) ≥ kn : a contradiction. Thus
1 < k ? < kn ≤ G. Next suppose that to the contrary, kr = maxj kj , r 6= n; then sinceφ(kr ) = kr+1 < kr given the maximal nature of kr , it follows that kr > kn > k ? > 1 andthus, kr+1 < k1 : a contradiction to the definition of k1 .
We have thus, φ(1) = G > k ? > k1 > φ(G). Suppose that k1 ≥ 1 that is, we have:1 ≤ k1 < k ? , kn . Then, φ(k1 ) = k2 > φ(k ? ) = k ? ⇒ k3 = φ(k2 ) < k ? but k3 > k1 sincerecall that k1 was the minimum. Thus k ? > k3 > k1 ≥ 1 and G > k2 > k ? ; proceeding inthis manner, it may be concluded that n must be even i.e., n = 2s say and we must havek1 < k3 < k5 ....k ? < k2s=n < k2(s−1) < ...k2 ≤ G
But now k1 6= φ(kn ): a contradiction. Hence k1 < 1 as claimed. •Thus any n-cycle must spend at least one period in the Solow-Regime. This fact allows
us to note the following property of growth rates along any n-period cycle. Consider any
such cycle k1 , k2 , ..., kn ; we shall write kn+1 = k1 and we note the following: If kj < 1 then1−1/σkj+1 = Gkj; thus we have
kj+1
−1/σ
= Gkj
kj

which means that
gK =

Kj+1
−1/σ
= Gkj
>G
Kj

since no innovation occurs and Nj+1 = Nj . On the other hand, if kj > 1, we have
kj+1
G
=
kj
1 + θ(kj − 1)

14

Robust Cyclical Growthand further since innovation takes place , we have from equation (5)

gN =

Nj+1
= 1 + θ(kj − 1)Njconsequently, we have:gK
G
Kj+1 Nj.
==
Kj Nj+1gN
1 + θ(kj − 1)which means that gK = G, given the expression for gN obtained above. With these preliminaries, let us return to the n-cycle k1 , k2 , ..., kn with kn+1 = φ(kn ) = k1 which means
that
Kn+1
K1
Kn+1
Nn+1
=
⇒
=
Nn+1
N1
K1
N1
in other words, we have:
Kn+1 K2
Nn+1 N2
...
=
...
Kn
K1
Nn
N1
−1/σ

on the left hand side, each term is either G if the corresponding kj ≥ 1 or is G.kj

if

the corresponding kj < 1. Thus the product on the left hand side is greater than Gn , since
we have shown that at least one of the kj ’s namely k1 is less than 1. Consequently, theaverage gross growth rate gK = gY = gN along the cycle must be greater than G.
We note this in the form of the following claim:

8 Along any n-period cycle, the average gross growth rate gK = gY = gN > G.

Let us return to the Example 2 , where the 2-cycle had been shown to be unstable. It
may be shown that in that context viz., G = 1.07, σ = 5: there is a stable 4-period cycle
k1 < 1, k2 > 1, k3 < 1, k4 > 1, φ(ki ) = ki+1 , i = 1, 2, 3, 4 with k5 = k1 . k1 = 0.9888, k2 =1.060402, k3 = 0.988814, k4 = 1.0604139; further it may be checked that |φ0 (k1 )....φ0 (k4 )|= 0.000583969 < 1 which signifies that the 4 period cycle is stable. Figure 3 contains ademonstration of the existence and convergence to this cycle.

15

Robust Cyclical Growth
FIGURE 3 HERE

For the case σ = 5, it has not been possible to locate any more points of bifurcation;that is on the basis of simulation exercises, and the bifurcation diagram (Figure 4) it seems
reasonable to conclude that there may be at the most a stable 4-period cycle.

FIGURE 4 HERE

The bifurcation diagram7 plots for values of G between θ − 1 = 1.4 and 1, the last 100
points in an iteration of 1000 points from an arbitrary initial point. For G between 1 and
1.4, beginning with the high G values, we note the first approach to a two period cycle; forlower values of G, the two period cycle disappears and a stable four period cycle appears.
This happens for values of G close to 1 and it has not been possible to uncover other pointsof bifurcation8 . Outside the range, we note convergence to a steady state, as the results
indicate.
We consider next bifurcation diagrams for narrower ranges of the parameters but stillfor σ = 5: first consider the range of G values between 1.4 and 1.48.
FIGURE 5 HERE
Notice the shift from convergence to the fixed point to convergence to the two-period cycle
as the value of G decreases. And consider next the range of G values between 1 and 1.08,
still for σ = 5.
FIGURE 6 HERE
7

All diagrams were prepared on Mathematica Version 4.1 .

8The exercises we have conducted seem to indicate that bifurcations leading to cycles with period 8 andperiod 16 and more are non-existent, at least when σ is small; see Matsuyama (1999), p.344. For suchpossibilities, σ has to exceed 22. See below.

16

Robust Cyclical Growth

Notice here the shift from convergence to a two period cycle to convergence to a four period
cycle as G attains the value unity. Thus Figures 5 and 6 focus attention on the two extreme
ranges of the parameter G contained in Figure 4.
We present next, an example of topological chaos when G is small and σ is large. For
this purpose it would be helpful to note the following:

9 φ2 (1) = φ(φ(1)) < 1 ⇔ 1 < G < θ − 1.

Thus a necessary condition for topological chaos9 is always satisfied, for every value of
G whenever, the steady state is unstable. But to clinch matters and to show the presenceof complicated dynamics, we need to show that φ3 (1) < k ? 10

Example 3 Consider σ = 22 and G = 1.001; one may then check that φ3 (1) < k ? . We
thus have topological chaos for this map. For values of σ ≥ 22, low values of G imply theexistence of topological chaos11 .

The bifurcation diagram for σ = 22 is provided below.

FIGURE 7 HERE

The above set of exercises allows us to draw the following conclusions:• When the steady state is unstable, two-cycles exist; further, any G value in the set[Ĝ, θ − 1) implies that the two-cycle is stable. This range increases with σ over therange [5, 22] of plausible σ-values.
9

See, for instance, Mitra (2001), p. 140, (2.5).

10

Mitra (2001), Proposition 2.3, p. 142.

11

See Mitra (2001), p.142 where he looks at the case when σ = 50, G = 1.01 to apply his result.

Robust Cyclical Growth17• When σ is small (around 5), the steady state and the two period-cycle is unstable,
there would be a stable cyclical orbit with period 4; cycles with higher periods do notappear possible.
• When σ is large (≥ 22), possibilities for chaos exist particularly if the value of G issmall enough.
• Whenever the steady state is unstable and a stable cycle exists, the rate of growthalong the cycle will be greater than that along the unstable steady state.

5Concluding Remarks

We examined whether a growth cycle could be obtained as limiting behavior; it turned
out this led us to a consideration of what could be the parameter values which would
imply such behavior. We also noted the existence of bifurcation points around which thestability properties of these cycles change sharply and hence it is important to examine therobustness of the claims. For meaningful cyclical behavior of period 2, the parameters had
to be in the right range so the cycles could be attractive. We computed a range of values
which imply the stability of period 2 cycles. We also investigated how this range changes
when the parameter σ varies. It turns out that large σ values enhances the chances of thestability of the 2-cycles.
With low G-values, outside the range, 2-cycles may lose stability but stable cycles withlarger periods may exist. It was of some interest to note that the possibilities of higher rates
of growth along period 2-cycles are retained by the cycles with larger periods. Thus theconclusion that growth paths, which are cyclical in nature, are capable of providing higher
rates of growth turns out to be a quite robust conclusion.

Robust Cyclical Growth18

Our analysis also indicates that the chances of encountering chaotic trajectories arise
only when G is “small” and σ is “large”. Recall that if σ is large, there would be a larger
range of possible G-values which imply stability of the 2-cycles. So outside this range,
we shall have higher period cycles and only for G values close to 1 there maybe chaotic
trajectories. The simulation exercises seem to indicate that unless σ was greater or equal
to 22 and G was “small”(close to 1), we would not encounter this phenomenon; how small
G must be depends on the value of σ, of course. But so long as cycles were stable, theconclusion that cyclical paths lead to enhanced growth, remains valid.
It may be recalled that the parameter σ has two roles in the analysis: first of all in thedetermining the share of labor (1/σ) and secondly, in fixing the demand elasticity of the theintermediate goods (σ) and thereby the monopoly margin. The other parameter of interest
is G, the gross rate of growth along the steady state in the Romer regime. Thus increasing
σ amounts to lowering the share of labor and at the same time, increasing the monopoly
margin. This has been shown to lead to two effects: first, this makes the chances of a stable
2-cycle larger that is for a wider set of G values, the two cycles will be stable; secondly,
if the 2-cycles were unstable that is, if G were to be low enough, we may also encounterchaotic trajectories. But so long as there are stable cycles, growing along them would leadto larger rates of growth12 .
12Even in the presence of topological chaos, there may be a stable cycle with period which is not a powerof 2; this is the case when chaos may not be observable as in the case of the logistic map f (x) = Kx(1 − x);for K ≈ 3.83, one may show that f 3 (1/2) = 1/2 and so satisfies the conditions of topological chaos; but
almost all trajectories converge to this 3-period cycle, see, for example Day and Pianigiani (1991) in thisconnection; our results show that even along such a cycle, the rate of growth would be larger than the one
along the unstable steady state. The fundamental point, thus, is that there should be a stable cycle. The
periodicity is of little concern in this connection.

Robust Cyclical Growth19

Finally in any dynamic economic model, there are several aspects in the specification of
the laws of change: one is the functional form; for example, whether processes are linear or
non-linear; and if non-linear, what terms are present. Quite apart from these issues, there
are, in addition parameters of various kinds. In economic theory, the problem has been thatthese parameters can never be specified with any degree of exactitude. It is because of thisreason that the exercises carried out above assume significance.

Robust Cyclical Growth

20

References
Day, R. H. and G. Pianigiani (1991), “Statistical Dynamics and Economics”, Journal of
Economic Behavior and Organization, 16, 37-83.
Matsuyama, K. (2001),“Growing through Cycles in an Infinitely Lived Agent Economy”,
Journal of Economic Theory, 100, 200-234.
———-, (1999), “ Growing through Cycles”, Econometrica, 67, 335-348.
Mitra, T.(2001), “A Sufficient Condition for Topological Chaos with an Application to a
Model of Endogenous Growth”, Journal of Economic Theory, 96, 133-152.

Robust Cyclical Growth21

Figure 1A: The Map with G > 1
1.2
1
0.8
0.6
0.4
0.2
0.2

0.4

0.6

0.8

1

1.2

Figure 1B: The Map with G < 1
1.4
1.2
1
0.8
0.6
0.4
0.2
0.2

0.4

0.6

0.8

1

1.2

1.4

Robust Cyclical Growth22

Figure 2: Range of G-values implying stability of 2-cycles
H: Stability

Range

0.395
0.39
0.385
0.38
0.375
7.5

10

12.5

15

17.5

20

σ

Figure 3: Convergence to a 4-cycle (σ = 5, G = 1.07)
1.1

1.05

1

0.95

0.95

1
k

1.05

1.1

Robust Cyclical Growth23

Figure 4: The Bifurcation Map
s=5
1.5

1.4

k

1.3

1.2

1.1

1

1.1

1.2

G

1.3

1.4

1.5

Figure 5: The Bifurcation Map
s = 5, 1.4 £ G £ 1.48
1.5

1.4

k

1.3

1.2

1.1

1

1.43

1.44

1.45
G

1.46

1.47

1.48

24

Robust Cyclical Growth

Figure 6: The Bifurcation Map
s = 5, 1.0 £ G £ 1.08
1.075
1.05
1.025

k

1
0.975
0.95
0.925

1.02

1.04
G

1.06

1.08

Figure 7: The Bifurcation Map s = 22

1.6
1.5

k

1.4
1.3
1.2
1.1
1

1.1

1.2

1.3

G

1.4

1.5

1.6