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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.

The standard one-sector Solow-Swan model (see [

Let

Most papers on economic growth considering the Solow-Swan (or neoclassical) model 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 state (i.e., the capital per capita equilibrium) so neither cycles nor complex dynamics can be exhibited.

More recently, several contributions in the literature have considered the Constant Elasticity of Substitution (CES) production function, in order to study growth models with elasticity of substitution that can be either greater or lower than one (see for instance [

Another consideration is that the standard one-sector growth model does not take into account that different groups of agents (workers and shareholders) have constant but different saving propensities. Such an issue has been studied by many authors (i.e., [

Böhm and Kaas [

Starting from Böhm and Kaas [

Brianzoni et al. [

As a further step in this field, Brianzoni et al. [

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,

Let us focus on the meaning of condition

The first model with nonconcave production function was introduced by Clark [

In the present work we study the discrete time one sector Solow-Swan growth model with differential savings as in Böhm and Kaas [

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 attraction (from 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 function (see Brianzoni et al. [

The paper is organized as follows. In Section

Let

The wage rate

In order to obtain the dynamic system describing the evolution of the capital per capita as given by (

Economic growth models used to consider the hypothesis of a production function satisfing the following standard economic properties:

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 considered a realistic assumption. In fact, it is quite obvious to assume that a certain amount of investment is needed before reaching a threshold capital level

Following Capasso et al. [

Observe also that the elasticity of substitution between production factors of function (

if

if

if

By substituting (

We first consider the question of the existence and number of fixed points or steady states of map (

The estabilishment of the number of steady states is not trivial to solve, considering the high variety of parameters. As a general result, the map

In order to determine the positive fixed points of

Let

if

if

if

Let

Making use of relation

The critical points of

According to the previous proposition, if

Let

Firstly notice that function

Observe that as in Capasso et al. [

Furthermore, from the previous proposition it follows that the fixed point

Map

Map

Since

We now focus on the case stated in Proposition

Let

Recall that

From the previous proposition it follows that

Let

Similarly to the considerations used to prove Proposition

As a consequence, the steady state

In order to obtain results concerning the local stability of

Recall that

hence the critical points of

In order to obtain a sufficient condition for

Assume

We first compute the derivative of function

The previous considerations prove the following proposition.

Let

Let

Let

if

if

We first focus on case (i) of Proposition

When

Finally, in case (iii) of Proposition

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 zero (poverty trap) or a positive capital per capita growth rate. Observe that, according to condition (i) of Proposition

In order to find more complex long run growth patterns we have to focus on case (ii) of Proposition

First notice that if

Observe that, for any given value of the other parameters, if

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 neighborhood of an attractor may have preimages far from it. This may give rise to complexity both in the qualitative structure of the attractor (that can be periodic or chaotic) and in the topological structures of the basins (that can even be formed by the union of several disjoint portions).

Condition

Let

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 if

We now want to study the qualitative asymptotic properties of the sequence generated by

We will prove that a generic trajectory may converge to a given steady state or to a more complex attractor, that may be periodic (an

Furthermore, we will prove that multistability, that is, the existence of many coexisting attractors (that may be periodic or even chaotic sets) emerges.

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 attraction (that 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.

Recall that function

In order to assess the possibility of complex dynamics arising, we have to consider the case in which Proposition

Let

Let

In the case previously described

Being

Let

In fact

According to the previous proposition every i.c.

As

If

If

If

Map

In Figure

Coexisting attractors of

Recall that we are considering the cases in which the unstable fixed point

In order to discuss the bifurcations leading to chaos we present some numerical simulations. Hence we fix the following parameter values:

As we have discussed, the system becomes more and more complex as

One dimensional bifurcation diagram of map

Obverve first that the trajectory converges to

The following statement summarizes our previous considerations.

Assume the same hypotheses of Proposition

As function

In the previous subsection we described the structure of set

A completely different situation appears if

Let

(a) Unconnected basin of attraction of the origin for the following parameter values:

In Figure

K-L staircase diagram for the following parameter values:

Finally, at

The joint analysis of the map w.r.t.

Two dimensional bifurcation diagrams of map

Firstly observe that once

Bifurcation diagrams w.r.t.

Secondly, parameter

In Figure

Two dimensional bifurcation diagrams of map

In Figure

Bifurcation diagrams w.r.t.

In this work we considered a Solow-Swan growth model in discrete time with differential savings between workers and shareholders as in Böhm and Kaas [

The study conducted represents a further step in the knowledge of the role played by the elasticity of substitution and the difference between saving rates in generating cycles or complex dynamics in simple neoclassical growth models. In fact, the results herewith obtained can be compared with the ones reached in Brianzoni et al. [

Our study aims at confirming that the elasticity of substitution between production factors plays a crucial role in economic growth theory (see Solow [

As with the CES (and differently from the VES) unbounded endogenous growth cannot be observed as the coexisting attractors of the system consist of fixed points or compact sets.

Anyway, differently from the other cases studied, with nonconcave production function the origin is always a locally stable fixed point so that the system may converge to the poverty trap.

Finally, the existence of such an attractor for our bimodal map implies multistability and basins complexity.

The authors acknowledge financial support by Minister of Education, Italy (MIUR) through the 2009 Research Project of National Relevance (PRIN) on Structural Change led by Neri Salvadori.