Discrete Dynamics in Nature and Society Volume 2012, Article ID 536570,22pages doi:10.1155/2012/536570
Research Article
Local and Global Dynamics in
a Discrete Time Growth Model with Nonconcave Production Function
Serena Brianzoni,
1Cristiana Mammana,
2and Elisabetta Michetti
21Dipartimento di Management, Universit`a Politecnica delle Marche, 60121 Ancona, Italy
2Dipartimento di Economia e Diritto, Universit`a di Macerata, 62100 Macerata, Italy
Correspondence should be addressed to Elisabetta Michetti,michetti@unimc.it Received 18 July 2012; Accepted 7 November 2012
Academic Editor: Juan J. Nieto
Copyrightq2012 Serena Brianzoni et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the dynamics shown by the discrete time neoclassical one-sector growth model with differential savings while assuming a nonconcave production function. We prove that complex features exhibited are related both to the structure of the coexixting attractors and to their basins.
We also show that complexity emerges if the elasticity of substitution between production factors is low enough and shareholders save more than workers, confirming the results obtained while considering concave production functions.
1. Introduction
The standard one-sector Solow-Swan model see 1, 2 represents one of the most used frameworks to describe endogenous economic growth. It describes the dynamics of the growth process and the long-run evolution of the economic system.
Let yt fkt be the production function in intensive form, mapping capital per workerktinto output per workeryt, then the Solow-Swan growth model describing the evo- lution of the state variablektin a discrete time setup is given by
kt1 1 1n
1−δktsfkt
, 1.1
where n ≥ 0 is the constant labor force growth rate,δ ∈ 0,1 is the depreciation rate of capital, ands∈0,1is the constant saving rate.
Most papers on economic growth considering the Solow-Swanor neoclassicalmodel used the Cobb-Douglas specification of the production function, which describes a process with a constant elasticity of substitution between production factors equal to one. It is quite immediate to observe that in this formulation the system monotonically converges to the steady statei.e., the capital per capita equilibriumso neither cycles nor complex dynamics can be exhibited.
More recently, several contributions in the literature have considered the Constant Elasticity of SubstitutionCESproduction function, in order to study growth models with elasticity of substitution that can be either greater or lower than onesee for instance3,4. In fact, as underlined in Klump and de La Grandville5, the elasticity of substitution between production factors plays a crucial role in the theory of economic growth since it represents one of the determinants of the economic growth level. Anyway the long run dynamics is still simple.
Another consideration is that the standard one-sector growth model does not take into account that different groups of agentsworkers and shareholders have constant but dif- ferent saving propensities. Such an issue has been studied by many authors i.e., 6–9 in order to understand if the differential saving might influence the final dynamics of the system. In fact different but constant saving propensities make the aggregate saving pro- pensity nonconstant and dependent on income distribution, so that multiple and unstable equilibria can occur.
B ¨ohm and Kaas 10 investigated the following discrete-time neoclassical growth model with constant but different saving propensities between capital and labor:
kt1 1 1n
1−δktsw
fkt−ktfkt
srktfkt
, 1.2
wheresw ∈0,1andsr ∈0,1are the constant saving rates for workers and shareholders, respectively. They use a generic production function satisfying the weak Inada conditions, that is, limkt→ ∞fkt/kt 0, limkt→0fkt/kt ∞. The authors show that instability and topological chaos can be generated in this kind of model.
Starting from B ¨ohm and Kaas10, recent contributions in this research line take into account other production functions in which the weak Inada conditions are not verified.
Brianzoni et al.11–13 investigated the neoclassical growth model in discrete time with differential savings and endogenous labour force growth rate while assuming CES production function. The authors proved that multiple equilibria are likely to emerge and that complex dynamics can be exhibited if the elasticity of substitution between production factors is sufficiently low. The results obtained prove that production function elasticity of sub- stitution plays a central role in the creation and propagation of complicated dynamics in growth models with differential saving.
As a further step in this field, Brianzoni et al. 14 firstly introduced the Variable Elasticity of SubstitutionVESproduction function in the form given by Revankar15, in the discrete time neoclassical growth model the same setup in continuous time was considered by Karagiannis et al.16. The use of the VES production function allows to take into account that the elasticity of substitution between production factors is influenced by the level of economic development. The authors prove that the model can exhibit unbounded endogenous growth when the elasticity of substitution between labour and capital is greater than one, as it is quite natural while the variable elasticity of substitution is assumed and differently from CES confirming the results obtained by Karagiannis et al. 16 in
the continuous setup. Furthermore, the results obtained aim at confirming that the production function elasticity of substitution is responsible for the creation and propagation of com- plicated dynamics, as in models with explicitly dynamic optimizing behavior by the private agentssee Becker17for a survey about these models.
For many economic growth models based on intertemporal allocation, the hypothesis of a concave production function has played a crucial role. In fact the production function is the most important part of a growth model as it specifies the maximum output for all possible combinations of input factors and therefore determines the way the economic model evolves in time. Usually a production function is assumed to be non-negative, increasing and concave, and also to fulfill the so called Inada Conditions, that is,f0 0, limkt→0fkt ∞and limkt→∞fkt 0.
Let us focus on the meaning of condition limkt→0fkt ∞from an economic point of view. We take into account a region with almost no physical capital, that is there are no machines to produce goods, no infrastructure, and so forth. Then the previous condition states that it is possible to gain infinitely high returns by investing only a small amount of money. This obviously cannot be realistic since before getting returns it is necessary to create prerequisites by investing a certain amount of money. After establishing a basic structure for production, one might still get only small returns until reaching a threshold where the returns increase greatly to the point where the law of diminishing returns takes effect. In literature this fact is known as poverty traps. In other words, we should expect that there is a critical level of physical capital having the property that, if the initial value of physical capital is smaller than such a level, then the dynamic of physical capital will descend to the zero level, thus eliminating any possibility of economic growth. Thus we introduce the before mentioned threshold k in order to consider what happens to the Solow model if we do not assume a concave production function. In fact concavity assumptions provide a good approximation of a high level of economic development but is not always applicable to less- developed countries. A small amount of money may have an effect in the short run but this effect will tend to zero in the long run if there are no more investments. Thus it makes sense to assume that only an amount of money larger than some threshold will lead to returns.
The first model with nonconcave production function was introduced by Clark18 and Skiba19. Following such works several contributions have then focused on the exis- tence and implications of critical levelssee, among others,20–22. Capasso et al.23focu- sed on a parametric class of nonconcave production functions which can be considered as an extension of the standard Cobb-Douglas production function; the authors study the Solow growth model in continuous time and show the existence of rich dynamics by mainly using numerical techniques.
In the present work we study the discrete time one sector Solow-Swan growth model with differential savings as in B ¨ohm and Kaas10, while assuming that the technology is described by a nonconcave production function in the form given by Capasso et al.23. Our main goal is to describe the qualitative and quantitative long run dynamics of the growth model to show that complex features can be observed and to compare the results obtained with the ones reached while considering the CES or the VES technology.
The results of our analysis show that our model can exhibit complexity related both to the structure of the attractors of the system passing from locally stable fixed points to bounded fluctuations or, even, to chaotic patterns, to the coexistence of attractors giving rise to multistability phenomenon and, finally, to the structure of the basins of attractionfrom a simple connected to a non-connected one.
The role of the production function elasticity of substitution has been related to the creation and propagation of complicated dynamics. In fact, similarly to what happens with the CES and VES production functionsee Brianzoni et al.11,13,14, if shareholders save more than workers and the elasticity of substitution between production factors is low, then fluctuations may arise.
The paper is organized as follows. InSection 2we introduce the model. InSection 3we perform the dynamic analysis. InSection 4we deal with the case in which complex dynamics are exhibited.Section 5concludes our paper.
2. The Model
Letyt fktbe the production function in intensive form, mapping capital per workerkt
into output per workeryt, then, the neoclassical one-sector growth model with different but constant saving rates between worker and shareholders as proposed by B ¨ohm and Kaas10 is given by:
kt1 1 1n
1−δktswwkt srktfkt
, 2.1
whereδ ∈0,1is the depreciation rate of capital,sw∈0,1andsr ∈0,1are the constant saving rates for workers and shareholders, respectively, andn≥0 is the constant labor force growth rate.
The wage ratewktmust equal the marginal product of labor, that is,wkt:fkt− ktfkt, furthermore shareholders receive the marginal product of capitalfkthence the total capital income per worker isktfkt, then2.1may be re-written as follows:
kt1 1 1n
1−δktswfkt sr −swktfkt
. 2.2
In order to obtain the dynamic system describing the evolution of the capital per capita as given by2.2we have to specify the production function.
Economic growth models used to consider the hypothesis of a production function satisfing the following standard economic properties:fk>0,fk>0, andfk<0, for all k > 0; observe that such properties hold for the Cobb-Douglas, CES, and VES production functions. In addition both the VES and the Cobb-Douglas production functions verify one of the Inada Conditions, that is limkt→0fkt ∞.
According to the previous condition an economy with no physical capital can gain infinitely high returns by investing only a small amount of money, hence it cannot be con- sidered a realistic assumption. In fact, it is quite obvious to assume that a certain amount of investment is needed before reaching a threshold capital levelksuch that great returns are obtained only forkt≥k. To be more precise, a more realistic economic assumption might take into account that ak >0 does exist such that the production functionfkis characterized by increasingdecreasingmarginal returns forkt< kkt≥k. This fact is known as the poverty trap as, if the initial value of capital is sufficiently small, then the dynamic law of capital accumulation will push the capital level to zero and no economic growth will take place.
Given such considerations, the concavity assumption provides a good approximation for
a production function only at high level of economic development while, if a less developed country is considered, then a nonconcave production function has to be taken into account.
Following Capasso et al. 23, we consider a sigmoidal production functioni.e., it shows an S-shaped behaviorgiven by
fkt αkpt
1βkpt, 2.3
whereα >0,β >0, andp≥2. Observe that
fkt αpkp−1t
1βkpt2,
fkt
pαktp−2 p
1−βkpt
− 1βkpt
1βkpt3 ,
2.4
and recall that the Inada Conditions aref0 0, limkt→0fkt ∞and limkt→∞fkt 0.
Then function2.3is positive for allkt>0, strictly increasing and it is a convex-concave pro- duction function. In factfkt>0, for allkt>0 while ak >0 exists such thatfkt><0, if 0 < kt < kkt > k, beingk p−1/βp11/pthe inflection point off. Furthermore the production function2.3does not satisfy one of the Inada Conditions, in fact
klimt→0fkt lim
kt→0
αpkp−1t
1βkpt2 0. 2.5
Observe also that the elasticity of substitution between production factors of function 2.3depends on the level of the capital per-capitaktas it is given by
σk 1 βpkp
p
1−βkp
−
1βkp 2.6
so that also this function belongs to the class of VES production functions, that is, the elasticity of substitution depends onk. Observe the role played by the constantp: ifpis great enough thenσkdecreases w.r.t.pwhere
σpk βkp
p2lnk−plnk−1−βkp p−1−βkp
p12 2.7
and
iifk >1 then limp→∞σk 0;
iiifk <1 then limp→∞σk 1;
iiiifk1 then limp→∞σk 1/1−β.
By substituting2.3and2.4into2.2we obtain the following one dimensional map describing the capital accumulation:
kt1Fkt 1 1n
1−δkt αktp 1βkpt
swpsr −sw
1βktp
2.8
withFcontinuous and smooth function for allkt>0.
3. Stability of Steady States
We first consider the question of the existence and number of fixed points or steady states of map2.8and then we discuss about their stability depending on all the parameter values.
The estabilishment of the number of steady states is not trivial to solve, considering the high variety of parameters. As a general result, the mapFalways admits one fixed point characterized by zero capital per capita, that is,k 0 is a fixed point for any choice of para- meter values. Anyway steady states which are economically interesting are those characte- rized by positive capital per worker.
In order to determine the positive fixed points ofF, let us define the following function:
Gkt: ktp−1 1βkpt
swpsr−sw
1βktp
, kt>0, 3.1
then solutions ofGkt nδ/αare positive fixed points ofF. The following proposition states the number of fixed points ofFaccording to the parameter values.
Proposition 3.1. LetFbe given by2.8andsr > sw. Then ak0>0 does exist such that:
iifnδ/α > Gk0,Fhas a unique fixed point given bykt0;
iiifnδ/αGk0,Fhas two fixed points given bykt0 andktk∗>0;
iiiifnδ/α < Gk0,Fhas three fixed points given bykt0 and 0< k1< k2.
Proof. kt 0 is a solution of kt Fkt for all parameter values hence it is a fixed point.
Function 3.1 is such that Gkt > 0, for all kt > 0, furthermore limkt→0Gkt 0 and limkt→∞Gkt 0. Hence Gkt has at least one critical point being a maximum point namelyk0. We prove that it is unique.
LetGk:Gktthen, given the production functionfas defined in2.3, we obtain
Gk f
αk
swpsr−sw
1βkp
. 3.2
DefineΔssr−swandMk βkpthen we obtain
G 1 α
fk−f k2
swp Δs 1M
−f k
p2ΔsM k1M2
. 3.3
Making use of relationfpf/k1Mwe reach the following expression:
G f
α1M2k2
aM2bMc , 3.4
wherea−sw,bswp−2−Δsp2pandcswp−1 Δsp2−p.
The critical points ofGare the positive solutions ofaM2bMc0; beinga <0 and c >0 then it admits a unique positive solution given byM −b−√
b2−4ac/2a >0. Hence functionGhas a unique maximum pointk0 M/β1/p and consequently it intersects the positive and constant functiong nδ/αin two, one, or no points ifg < Gk0,gGk0 org > Gk0, respectively.
According to the previous proposition, ifsr > sw i.e., shareholders save more than workersthe map always admits the equilibriumk0, moreover up to two additionalposi- tivefixed points can exist according to the parameter values hence multiple equilibria are exhibited. We now wish to study the local stability of these equilibria. Let us consider first the origin.
Proposition 3.2. LetFbe as given in2.8andsr > sw. Then the equilibriumk0 is always locally stable.
Proof. Firstly notice that functionFmay be written in terms of functionGbeing
Fk 1
1n1−δkαkGk, 3.5
henceFk 1/1n1−δαGkkGk. Being limk→0Gk 0 and limk→0kGk 0, thenF0 1−δ/1n∈0,1and consequently the origin is a locally stable fixed point for mapF.
Observe that as in Capasso et al. 23the use of the S-shaped production function implies the existence of a poverty trap. Recall that in models previously proposed in which production function is concaveas the CES production function in Brianzoni et al.11,13or the VES production function in Brianzoni et al.14the origin is always a locally unstable fixed point hence the economy will converge in the long run to positive level growth rates eventually with periodic or evena-periodic dynamic features. Differently, in our new setup, the origin is always locally stable hence economies starting from a sufficiently low level of capital per-capita may be captured by a poverty trap. More precisely, there exists a critical level of physical capital having the property that, if the initial value of physical capital is smaller than such a level, then the dynamic of physical capital will converge to zero, thus eliminating any possibility of economic growth.
Furthermore, from the previous proposition it follows that the fixed point k 0 is globally stable when it is the unique steady state owned by the map i.e., case i of Proposition 3.1, see Figures 1 and 2a. According toProposition 3.1, this happens when nδ/α > Gk0.
SinceGk0does not depend onαand beingα >0 we can conclude that∃αsuch that k 0 is the unique steady state owned by the system for all 0 < α < αthis means that the production function upper bound is small enough. Similarly, beingnupper unbounded,
0 3 0
3
kt+1
kt
a
0 2
0 2
kt+1
kt
b
0 2
0 2
kt+1
kt
c
Figure 1: MapFand its fixed points in the case of strict monotonicity for the following parameter values:
β0.5,sw0.2,sr1,p3,δ0.2,α0.3. InbnαGk0−δ0.1606n; inan > n, incn < n.
0 3
0 3
kt+1
kt
a
0 3
0 3
kt+1
kt
b
0 3
0 3
kt+1
kt
c
Figure 2: MapFand its fixed points in the case of non-invertibility for the following parameter values:
β0.5,sw0.2,sr1,p8,δ0.2,α0.3. Inbnn0.7444; inan > n, incn < n.
then andoes exist such thatk0 is the unique steady state of the model for alln > n,i.e., if population growth rate is sufficiently highgiven the other parameter values. In such cases the poverty trap cannot be avoided and the system will converge to a zero growth rate.
We now focus on the case stated inProposition 3.1iii, that is, mapFhas three fixed points given bykt 0,k1 >0 andk2 > 0, beingk1 < k2. About the local stability of the two positive fixed points, we start proving the following proposition.
Proposition 3.3. LetFbe as given in2.8,sr > sw, andnδ/α < Gk0. Then the equilibrium k1is always locally unstable.
Proof. Recall that
Fk 1
1n1−δkαkGk, 3.6
henceFk1 1−δ/1n α/1nGk1 k1Gk1. FromProposition 3.1it follows thatGk1 nδ/α, hence, by substituting in the previous formula, we obtain that
Fk1 1 α/1nk1Gk1 >1 sinceGk1> 0in factk1 is located on the increasing branch of functionG.
From the previous proposition it follows thatk1is a locally unstable fixed point; fur- thermore mapFis increasing in such a point. Let us now focus on the local stability ofk2by proving the following proposition.
Proposition 3.4. LetFbe as given in2.8,sr > sw, andnδ/α < Gk0. ThenFk2<1.
Proof. Similarly to the considerations used to proveProposition 3.3we obtain thatFk2
1 α/1nk2Gk2 < 1 beingGk2 < 0 ask2 is located on the decreasing branch of functionG.
As a consequence, the steady statek2may be locally stable ifFk2∈−1,1or locally unstable ifFk2 < −1; in this last casek2 loses its stability via flip bifurcation and a two period-cycle may be created proving that our model may produce fluctuations in economic growthwe will explain the route to chaos in the following section.
In order to obtain results concerning the local stability ofk2we use the following argu- ments to prove that functionFmay be strictly increasing or bimodal, according to the para- meter values.
Recall thatFmay be written in terms ofG, and consider the expression forFas given in the proof ofProposition 3.2, that is,
Fk 1 1n
1−δα
Gk kGk
, 3.7
then F 0 if and only ifGk kGk δ−1/α < 0. Observe that Gk kGkcan be written in terms ofMk βkp ≥ 0 beingMa continuous, differentiable and strictly increasing function ofkas follows:
Gk kGk pMp−1/p
βp−1/p1M3
sw−pΔs
MswpΔs
HMk, 3.8
hence the critical points ofFare solutions of
HMk δ−1/α. 3.9
Observe that ifsw−pΔs ≥ 0 thenHMkis positive for allk > 0 and consequently3.9 has no solution. In such a caseFis strictly increasing and no complicated dynamics occurs.
In order to obtain a sufficient condition forFbeing non-invertible, we study function HMto determine the number of solutions of3.9.
Assumesw−pΔs < 0. FunctionHMis such that H0 0, limM→ ∞HM 0 andHM > <0 ifM < >swpΔs/−swpΔsbeingswpΔs/−swpΔs >1.
As a consequence HM can intersect the constant and negative function h δ −1/α only in points belonging to the intervalI swpΔs/−sw pΔs,∞. ObviouslyH admits at least one maximum pointMMless thanswpΔs/−swpΔsand at least one minimum pointMmbelonging to the intervalI. We now want to prove thatHadmits only
one minimum pointMm, so that it may intersect functionhat most in two pointsM1and M2in such a case funtionF is bimodal, that is it admits both a local maximum and a local minimum point.
We first compute the derivative of functionH. We rewrite functionHas follows:
HM AM2BM
M1/p1M3, 3.10
whereA p/βp−1/psw−pΔs<0 andB p/βp−1/pswpΔs>0, hence
HM 1
pM1/p1M4
−1−p
AM2 2p−1
A− 2p1
B
M
p−1
B 3.11
so that it admits two positive zeroes that areMMandMm, respectively, where
Mm − 2p−1
A− 2p1
B
2p−1 A−
2p1
B24 p2−1
AB 2
−1−p
A 3.12
and consequently ifHMm<δ−1/αthen3.9has two solutionsM1andM2. In such a case functionFis bimodal hencekM M1/β1/pandkm M2/β1/pare, respectively, its maximum and minimum points.
The previous considerations prove the following proposition.
Proposition 3.5. LetFbe as given in2.8,sr > sw.
iLetsw−pΔs≥0, then mapFis strictly increasing.
iiLetsw−pΔs < 0 and defineA p/βp−1/psw−pΔs < 0,B p/βp−1/psw pΔs > 0 andHM AM2BM/M1/p1M3. ThenMm > 1 does exist such that:
aifHMm≥>δ−1/αthenFis (strictly) increasing;
bifHMm<δ−1/αthenFadmits a maximum pointkMand a minimum point kmsuch that 1< kM< km.
We first focus on caseiofProposition 3.5stating a sufficient condition such that the map is increasingseeFigure 1. Obviously in such a case, if the origin is the unique fixed point, then it is globally stablecaseiofProposition 3.1.
WhenGk0crosses the valuenδ/αthen a fold bifurcation occurs creating a new fixed point namelykcaseiiofProposition 3.1. Such a steady state is a non-iperbolic fixed point being its eigenvalue equal to one. Anyway, asFis strictly increasing, thenkattracts trajectories having initial condition i.c. k0 ≥ k while the interval0, k is the basin of attraction of the origin. Observe that in such a case the poverty trap can be avoided if the economy starts from a sufficiently high level of capital per capita.
Finally, in caseiiiofProposition 3.1three fixed points are owned byF; furthermore Fk2 ≥ 0. Hence, taking into account the result proved in Proposition 3.4, it must be Fk2 ∈ 0,1 seeFigure 1cand consequently the steady state k2 is locally stable. The unstable fixed pointk1separates the basin of attraction of the origin and that ofk2. In this case both the structure of the attractorsfixed pointsand that of their basinsconnected setsare simple.
In all the above mentioned cases only simple dynamics is presented: that is, the economic system monotonically converges to a steady state characterized by a zeropoverty trapor a positive capital per capita growth rate. Observe that, according to conditioniof Proposition 3.5 aΔs does exist such thatF is increasing for allΔs < Δs. This means that no cycles or complex features are observed if the difference between the two propensities to save is low enough, confirming what proved in Brianzoni et al.11,13,14in which concave production functions were taken into accountthe CES and the VES function, resp..
In order to find more complex long run growth patterns we have to focus on caseii ofProposition 3.5stating conditions such thatFis non-invertibleseeFigure 2.
First notice that ifFhas only the origin as a fixed point then, as in the previous case, it is globally stablecaseiofProposition 3.1. Hence we want to consider the case in which more than one fixed point is owned byF, that is, conditioniioriiiofProposition 3.1holds.
Observe that, for any given value of the other parameters, ifαis great enough then nδ/α≤Gk0, henceFhas two or three fixed points: multiple equilibria exist and their basins of attraction have to be discussed. Furthermore, sinceFcan be bimodal, then complex dynamics can be exhibited, so that the role of the parameters of the model and the route to complexity must be described.
In fact the iterated application of a noninvertible map repeatedly folds the state space allowing to define a bounded region where asymptotic dynamics are trapped. Furthermore, the iterated application of the inverses repeatedly unfolds the state space, so that a neigh- borhood of an attractor may have preimages far from it. This may give rise to complexity both in the qualitative structure of the attractorthat can be periodic or chaoticand in the topological structures of the basinsthat can even be formed by the union of several disjoint portions.
Condition sw−pΔs < 0 in Proposition 3.5iiis necessary for F being bimodal. In such a case limp→∞HMm −∞so that ap1 >0 does exist such thatFis bimodal for all p > p1. Being the non-invertibility ofFa necessary condition for our model having complex dynamics, the previous condition is necessary for cycles of chaotic dynamics to be observed.
In addition another necessary condition for our setup showing nontrivial dynamics is that three fixed points are owned byFthat isnδ/α < Gk0. Similarly to the previous con- sideration, we observe that limp→∞Gk0 ∞and consequently ap2 > 0 does exist such that F admits three fixed points for all p > p2. Let p max{p1, p2,2}, then the following proposition trivially holds.
Proposition 3.6. LetFbe as given in2.8,sr > sw. Then ap >0 does exists such thatFis bimodal and admits three fixed points for allp > p.
The previous condition is necessary for cycles or chaos to be observed in our model.
It is straightforward to observe that complex features can emerge ifpis great enough as long assr > sw. This fact proves that both the elasticity of substitution and the different saving propensities contribute to generate complexity in economic growth, as in models with con- cave production functions.
4. Complex Dynamics
We now want to study the qualitative asymptotic properties of the sequence generated byF in the case of non-invertibility, by combining analytical tools and numerical simulations.
We will prove that a generic trajectory may converge to a given steady state or to a more complex attractor, that may be periodicanm-period cycleor chaotic. In this last case we will determine the bounded set of the line where the system’s dynamics are trapped and we will describe the complexity of the attractors belonging to these sets.
Furthermore, we will prove that multistability, that is, the existence of many coexisting attractorsthat may be periodic or even chaotic setsemerges.
Finally, as our map is characterized by coexisting attractors, we will study global bifurcations occurring as some parameters are varied that are responsible for changing in the properties of the attracting sets and of their basins of attractionthat may consist of infinitely many unconnected sets.
These problems lead to different routes to complexity, one related to the complexity of the attracting sets which characterize the long run time evolution of the dynamic process, the other one related to the complexity of the boundaries which separate the basins when several coexisting attractors are present. These two different kinds of complexity are not related in general, in the sense that very complex attractors may have simple basin boundaries, whereas boundaries which separate the basins of simple attractors, such as coexisting stable equilibria, may have very complex structures.
4.1. Multistability and Complex Attractors
Recall that functionFalways admits a locally stable fixed point given byk0, hence the set A0{0}is an attractor forFfor all parameter values.
In order to assess the possibility of complex dynamics arising, we have to consider the case in whichProposition 3.6holds, that isFis bimodalit has two critical pointskMandkm and admits three fixed points. Let us start considering the stability of the fixed pointk2and the bifurcations it undergoes as some parameters vary. As we pointed out, the eigenvalue associated tok2can be positive or negative depending on the position ofk2w.r.t. the critical points, more preciselyFk2<0 if and only ifk2 ∈kM, km. Obviously ifk2≤kMork2 ≥km no complex dynamics can arise.
Let p > p, then after the fold bifurcation occurring at n δ/α Gk0 see Proposition 3.1, two fixed points k1 and k2 are created for instance when n decreases crossing a given value n or α increases crossing a given value α. Immediately after this bifurcation the two fixed points born from a little perturbation of the parameters both belong to the increasing branch ofFas it is continuous and differentiable also w.r.t. the parameters of the model, hence 0< k1< k2 < kM < km. As a consequencek1is unstable while the other fixed points arelocallystable.
LetFkm> k1, then the unstable fixed pointk1separates the basin of attraction ofA0
given byBA0 0, k1from the basin of attraction of the positive steady statek2given by Bk2 k1,∞. In this situation an economic policy trying to increase investment can be able to push the economy out of the poverty trap toward a positive long run economic growth rate.
In the case previously describedk2is a locally stable fixed point. Anyway, as under- lined,k2 may be locally unstable and in such a case a more complex attractor may appear around k2; we call such an attractor A. Obviously, given the shape of mapF, a necessary
1 1.5 1
1.5
kt+1
kt
a
0.8 1.8
0.8 1.8
kt+1
kt
b
0.5 2.4
0.5 2.4
kt+1
kt
c
Figure 3: MapFand setSforβ 0.9,α1,n0.5,sw 0.1 andsr 0.7.ap 6 andFSis strictly decreasing;bp8 andFSis unimodalthe maximum point belongs toS;cp12 andFSis bimodal bothkMandkmbelong toS.
condition forAhaving a complicated structure iskM < k2 < km so that the fixed pointk2 belongs to the decreasing branch ofF. Given the analytical form ofF we cannot give con- ditions for such a case, anyway we can describe the transition to chaos by means of numerical simulations. In this subsection we want to describe the structure ofAby focusing on the case in whichFkm> k1. A bimodal map having a similar behaviour to the one we are studing is in Liz24.
BeingFbimodal it admits a forward invariant interval bounded by the local maximum and the local minimum. The following proposition states the existence of a trapping set for the mapFfor economically meaningful parameter range related to nontrivial dynamics.
Proposition 4.1. LetFbe as given in2.8,sr > sw,p > pandFkm> k1. IfkM< k2< kmthen the setS Fkm, FkMis trapping.
In fact F acts from the interval S into itself, in other words S is a closed region positively invariant. Recall that setE ⊆ Ris positivelynegativelyinvariant ifFtE ⊆ E FtE ⊇ E, for allt ∈ Z.Eis called invariant when it is both positively and negatively invariant. Moreover any trajectory generated from an i.c. ink1,∞enters inSafter a finite number of iterations, that is,Sis absorbing. AsSis trapping then the attractorAdifferent fromA0ofF must belong toS. Obviously it may consists of a fixed point k2or a more complex set or two coexisting attractors may be owned. Anyway, observe that the only way in whichk2may lose stability is via period-doubling bifurcation.
According to the previous proposition every i.c.k0> k1generates bounded trajectories converging to an attractor included into the trapping intervalS.
AsSis trapping, we can restrict mapF to such a set; we callFSthe mapF defined inSand we describe the properties ofFS. Observe that the following cases may occursee Figure 3.
iIfFkM< kmandFkm > kM thenkM ∈/Sandkm /∈ Swe are considering the case in which the critical points do not belong to S. In such a caseFSis strictly decreasing and Aconsists of a fixed point or of a 2-period cycle; more complex dynamics are ruled out.
0.5 2.4 0.5
2.4
kt+1
kt
a
0.5 2.4
0.5 2.4
kt+1
kt
b
Figure 4: Coexisting attractors ofFfor the parameter values considered inFigure 3andp12.a4-pieces chaotic attractor for the initial conditionkM;bcycle-2 for the initial conditionkm.
iiIfFkM><kmandFkm><kMthenkm∈/∈SwhilekM∈/∈S. This means that only the minimummaximumpoint belongs to the trapping setShenceFSis unimodal. In such a caseAconsists of a fixed point, anm-period cycle or a more complex attractor. The bifurcation structure eventually leading to complexity is the period-doubling bifurcation cascadeseeFigure 3b.
iiiIfFkM> kmandFkm< kMthen the two critical points belong toFS; this means that F is bimodal inS seeFigure 3cso that two coexisting attractors may be presented, namely,AMandAm, whereAMAmis the set attracting the trajectory starting from the maximumminimumpoint. In such a case the attractorAinside Sis given byA AM∪Am. About the coexistence of attractors in bimodal maps see Mira et al.25.
InFigure 4we show the two coexisting attractorsAmandAMby choosing parameter values such thatFSis bimodal. Observe thatAMis a 4-pieces chaotic attractor whileAmis a 2-period cycle.
Recall that we are considering the cases in which the unstable fixed pointk1does not belong to the trapping setS, and consequently the basins of attraction ofA0and ofAhave a simple structuresimply connected basins, that is,BA0 0, k1andBA k1,∞.
In order to discuss the bifurcations leading to chaos we present some numerical simu- lations. Hence we fix the following parameter values:δ0.2,α1,β0.9,sw0.1,n0.5 and we let parameterspandΔs sr −swvary. In fact we are mostly interested in the role played by parameterspandΔs: the former informs about the elasticity of substitutionthat decreases aspincreases, while the latter takes into account the difference between the two saving ratesonceswis fixed.
As we have discussed, the system becomes more and more complex aspincreases.
This consideration is also supported by looking atFigure 3showing thatFSbecomes bimodal asp increases. In order to better understand the role of parameterpthat is strictly related to the elasticity of substitution in our model, we describe the sequence of bifurcations ofF
6 8 10 12 14 16 18 0
0.5 1 1.5 2 2.5 3
p kt
a
6 8 10 12 14 16 18
0 0.5 1 1.5 2 2.5 3
p kt
b
Figure 5: One dimensional bifurcation diagram of mapFw.r.t.pforsr0.7.ak0kMand the attractor AMis presented;bk0kmand the attractorAmis presented.
w.r.t.pand we show that multistability occursabout multistability see, among others, Bischi et al.26and Sushko et al.27. The bifurcation diagrams inFigure 5show how dynamics are increasingly complex if the elasticity of substitution between production factors declines i.e.,p increases. InFigure 5awe depicted the attractor fork0 kM while inFigure 5b the initial condition is given by the minimum point.
Obverve first that the trajectory converges to k2 if p < 6.77; if p still increases a period doubling bifurcation occurs and a 2-period cycle appears. After such a bifurcation inFigure 5awe observe a sequence of period doubling bifurcations as the maximum point entered inSatp 10, while inFigure 5bwe observe a break in the bifurcation diagram atp 12 as also the minimum point entered inS, hence multistability can be observed for suitable values of the parameters. Observe that forp < 16.46 condition Fkm > k1 holds so that every initial condition withk0 > k1 generates bounded trajectories converging to an attractor insideS. On the contrary, ifFkm < k1 the unstable pointk1 belongs toS and a global bifurcation occurswe will explain such a case later. This kind of bifurcation requires an analysis of the global dynamical properties of the system, that is, an analysis which is not based on the linear approximation of the map.
The following statement summarizes our previous considerations.
Proposition 4.2. Assume the same hypotheses ofProposition 4.1. ThenA⊆Sattracts all trajectories starting fromBA k1,∞(it may consist of a fixed point, ann-period cycle or a strange attractor or it can be the union of two coexisting attractorsAMandAm) whileA0{0}attracts all trajectories starting fromBA0 0, k1.
4.2. Contact Bifurcations and Complex Basins
As functionF is bimodal then it admits two critical pointskMandkmsuch thatFkMand Fkmseparates the setRinto two subset:Z1 0, Fkm∪FkM,∞whose points have one rank-1 preimage and Z3 Fkm, FkMwhose points have three rank-1 preimages
in the local maximum and minimum the two merging preimageskmandkMare located.
As a consequenceFis aZ1−Z3−Z1non invertible map whereZ3is the portion ofRbounded by the relative minimum value and the relative maximum value. Furthermore, as three fixed points are owned byF, then the origin is a locally stable fixed point,k1 is a locally unstable fixed point whilek2may be locally stable or unstable and oneAor twoAMandAmmore complex attractors may exist aroundk2. As our mapFadmits more then one attracting set, each with its own basin of attraction, we have to describe their changes as the parameter of the model vary. About the structure and the bifurcations related to the basins of attraction see Mira et al.25.
In the previous subsection we described the structure of setAwhenFkm > k1and we showed that multistability due to the presence of two coexisting attractors occurs. Notice also that ifFkm k1then a critical point ofFis pre-periodic. Thus no attracting cycles exist since the basin of attraction of these cycles cannot contain the critical pointk1, providing the evidence of the existence of parameter values such that the map is chaotic.
A completely different situation appears ifFkmcrossesk1 as a contact bifurcation occurs. Such a bifurcation happens if a parameter variation causes a crossing between a basin boundary and a critical set so that a portion of a basin enters in a region where an higher number of inverses is defined, then new components of the basin suddenly appear after the contact see Bischi et al. 28. Obviously trajectories starting from BA0converge to the originimmediate basin, hence if the economy starts from a low level of economic growth it will fall in the poverty trap. On the other hand, if some parameters change this causes the minimum valueFkmto crossk1, then the portionFkm, k1enters inZ3 so that new preimages appear. In factk1has two new preimages given byka1−1< kmandk1b−1> kmand consequently initial conditions belonging toB−1 ka1−1,kb1−1also generate trajectories converging to zero asB−1is mapped into setBA0after one iteration.
LetF2kM < k1, so that the second iterate of the maximum point does not belong to the immediate basin of the origin BA0, then the previous procedure can be repeated while considering the preimages of rank-2 of the unstable fixed pointk1. Again, letk1a−2 andkb1−2be the preimages ofk1a−1andkb1−1, then initial conditions belonging to the set B−2 k1a−2,k1b−2generate trajectories converging toBA0after two iterations. The story repeates and a set of non-connected portion is created, so that the contact between the critical set and the basin boundary marks the transition between simple connected to non-connected basins. Finally the basin of attraction of the origin is given by
B0 BA0∪i≥1 ka1
−i, kb1
−i
. 4.1
In such a case an economic policy trying to push up the investment does not guarantees the exit from the poverty trapseeFigure 6a.
InFigure 7we present two trajectories converging to different attractors for close initial conditions: inFigure 7ak0 1.26 belongs to the basin ofAandAconsists in a very high period cycle or in a chaotic set; inFigure 7bthe initial condition isk0 1.27 belonging to the basin of the origin. In fact in such a case beingFkm< k1the basin structure is complex as it consists of infinitely many non-connected sets.
Finally, at F2kM k1 a final bifurcation occurs such that the maximum point is attracted by the unstable fixed pointk1and the map is chaoticsee Devaney29. After such a final bifurcationF2kM< k1andAis a Cantor setthe origin attracts almost all trajectories, seeFigure 6b.
0 3 0
3
kt
kt+1
B(A0) B−1 B−2
a
0 2
0 2
kt
kt+1
b
Figure 6:aUnconnected basin of attraction of the origin for the following parameter values:β 0.9, sw0.1,sr0.7,p15,δ0.2,n0.5, andα0.32.bAfter the final bifurcation the origin attracts the trajectory starting from the maximum point forα0.34.
0.6 1.26 1.5
0.6 1.5
kt+1
kt
a
0 1.27 1.5
0 1.5
kt+1
kt
b
Figure 7: K-L staircase diagram for the following parameter values:δ 0.2,α 0.4,β 0.9,n 0.5, sw0.1,sr0.9, andp11.ak01.26 generates a trajectory converging toAconsisting in a complex set;bk01.27 generates a trajectory converging toA0.
4.3. Numerical Simulations
The joint analysis of the map w.r.t. pand Δs explains how the elasticity of substitution in the nonconcave production function affects the final long run dynamics of the growth model for different values of the difference between the two saving rates. In Figure 8we present two cycle cartograms showing a two parametric bifurcation diagram qualitatively: each color represents a long-run dynamic behaviour for a given point in the parameter planeΔs, p and for the initial conditionsk0 kMandk0 km, respectively. A large diversity of cycles of different order is exhibited. The red region represents the parameter values for which
0.2 0.9 2
35
∆s p
fp-0
fp-0 fp-k2 Otherc-2 c-3 c-4 c-6 c-8 c-16 c-32
a
0.2 0.9
2 35
∆s p
fp-0
fp-0 fp-k2 Otherc-2 c-3 c-4 c-6 c-8 c-16 c-32
b
Figure 8: Two dimensional bifurcation diagrams of mapFin the plainΔs, pfor the following parameter values:δ0.2,α1,β0.9,n0.5,sw0.1;athe initial condition iskM;bthe initial condition iskm.
3
0
0.2 0.9
∆s kt
a
3
0
0.2 0.9
∆s kt
b
Figure 9: Bifurcation diagrams w.r.t.Δsfor the following parameter values:δ0.2,α1,β0.9,n0.5, sw0.1, andp12;athe initial condition iskM;bthe initial condition iskm.
the function is monotonic and the origin is globally stable; this situation occurs for small values ofp. Also in the dark blue region the system fails to converge to the originpoverty trap, anyway we have entered into the region of bimodality.
Firstly observe that oncepis fixed at an intermediate level, the final dynamics of the system becomes more complex as the difference between the two saving rates increases. In Figure 9we present two bifurcation diagrams w.r.t.Δsshowing how dynamics is increasingly