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This work investigates the economic growth problem of establishing a relation between the elasticity of substitution between production factors, capital and output per-capita levels when dealing with a non constant elasticity of substitution production function. Starting from a discrete-time setup, some definitions of elasticity of substitution associated to an attractor are proposed and a general method to measure it is suggested. Thanks to this methodology a government may select a proper economic policy in order to reduce production costs without decreasing the capitalisation trend of the economy. The method proposed is applied to the Solow’s type growth model with differential savings using a Variable Elasticity of Substitution (VES) production function with constant returns to scale. It is found that when shareholders save more than workers or the elasticity of substitution is higher than one, a country characterised by production functions with higher elasticity of substitution experiences higher capital and output per-capita equilibrium levels. On the other hand, when the elasticity of substitution is lower than one and workers save more than shareholders, an ambiguous relation between elasticity of substitution and asymptotic dynamics is shown.

The Kaldor model describes the dynamics of economic growth over time. Many works showed that a large variety of qualitative dynamics can be produced depending on the economic assumptions. This work provides conditions to mark boundaries during boom and bust cycles, investigating how saving behaviors of workers and shareholders and elasticity of substitution between capital and labor influence the growth dynamics of non‐developed, developing and developed countries considering general and specific technologies of production. The results obtained can be used by policy‐makers to increase the lower level an economy can reach during boom and bust periods and reduce fluctuations.

This article aims at studying a general equilibrium model with overlapping generations that incorporates inherited tastes (aspirations) and endogenous longevity. The existence of standard-of-living aspirations transmitted between two subsequent generations in a context where the individual state of health depends on public investments in health has some remarkable consequences at the macroeconomic level. First, aspirations allow escaping from the well-known poverty trap scenario described by Chakraborty (2004). Second, the steady-state equilibrium may be destabilized through a super-critical Neimark–Sacker bifurcation when the health tax rate is set at too high or too low a level. This causes endogenous fluctuations in income and longevity.

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of discrete autonomous gradient-like dynamical systems under the assumption that the original discrete autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular τ-periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of τ-periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown that the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.

We consider a class of discrete time two-dimensional dynamic systems that are symmetric, i.e., (x’,y’) = T(x,y) such that ToS = SoT, where S: (x,y)→(y,x) is the reflection through the diagonal. Symmetry implies some properties in terms of qualitative and quantitative dynamics, for instance there exist synchronized trajectories, while fixed points and other invariant sets are symmetric w.r.t. the main diagonal or they are invariant as well. Synchronization may also emerge. After showing some of the main features related to this kind of systems (such as attractors and their basins), some numerical examples obtained using software MatLab will be presented and the related algorithms will be discussed.

The paper is dedicated to the study of the problem of continuous dependence of compact global attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems (IIFS). We prove that if a family of non-autonomous dynamical systems ‹ $(X,\mathbb T_1,\pi_{\lambda}),(Y,\mathbb T_{2},\sigma),h $ › depending on parameter $\lambda\in\Lambda$ is uniformly contracting (in the generalized sense), then each system of this family admits a compact global attractor $J^{\lambda}$ and the mapping $\lambda \to J^{\lambda}$ is continuous with respect to the Hausdorff metric. As an application we give a generalization of well known Theorem of Bransley concerning the continuous dependence of fractals on parameters.

In this study we will research the dynamics shown by a cobweb-type model with hyperbolic demand, sigmoidal supply and with backward-looking mechanism of expectation creation, whereby the new state of the system is obtained from all the previous states observed by weighted arithmetical mean with exponentially decreasing weights in the ρ region. The study herewith presented aims at confirming the existence of a stabilising effect due to the presence of infinite memory since, with all the other conditions begin the same, a memory rate $${\bar \rho }$$ > exists at which market equilibrium is a sink. An unstable system, therefore, becomes stable in the presence of sufficiently resistant expectations with infinite historical memory, although this transition to stability is accompanied by the onset of chaos. The resulting effect, therefore is one of “qualitative destabilisation,” that is with reference to the qualitative dynamic performance produced, associated to a “quantitative stabilisation,” that is to say with reference to the decreasing width of the invariant sets within which relevant dynamics occur.

A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T :Ox; yU!O r 1O yU ; r 2O xUU, is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map FOxUaO r 1 r 2UOxU. The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031‐2048] as a model of an economic system, in which the reaction functions r1 and r2 are logistic maps. ” 2000 Elsevier Science Ltd. All rights reserved.

We study the dynamics shown by the discrete time neoclassical one-sector growth model with differential savings as in Bohm and Kaas [4] while assuming VES production function in the form given by Revankar [24] . It is shown that the model can exhibit unbounded endogenous growth despite the absence of exogenous technical change and the presence of non-reproducible factors if the elasticity of substitution is greater than one. We then consider parameters range related to non-trivial dynamics (i.e. the elasticity of substitution in less than one and shareholders save more than workers) and we focus on local and global bifurcations causing the transition to more and more complex asymptotic dynamics. In particular, as our map is non-differentiable in a subset of the states space, we show that border collision bifurcations occur. Several numerical simulations support the analysis.

Si considerino fluttuazioni e crescita in un modello neo-keynesiano in tre diverse ipotesi macroeconomiche, che danno origine a diversi modelli matematici in tempo discreto, o mappe. In particolare si analizza un “modello di crescita” bidimensionale, rappresentato da un endomorfismo lineare a tratti. Separando il comportamento dinamico delle pendenze di rette che escono dall'origine, dalla dinamica dei punti su tali rette, si mostra che il comportamento dinamico del modello e governato dalla dinamica di un endomorfismo unidimensionale. Cio semplifica lo studio dei valori di biforcazione che creano rette invarianti, su cui le traiettorie sono divergenti, e consente di determinare il bacino di attrazione di regioni assorbenti che contengono i cammini di crescita, regolari o caotici.

In this paper we study an asset pricing model with heterogeneous beliefs, wealth dynamics and time varying proportions. We derive the resulting deterministic nonlinear dynamic system and analyze the model with adaptive beliefs and switching mechanism in order to investigate bifurcations and global dynamics and thus consider the effects on wealth distribution among different types of agents. Our study aims at confirming that heterogeneity in beliefs may lead to complexity.

In this paper we study the dynamics of a discrete triangular system T in capital per capita and population growth representing the neoclassical growth model with CES production function and differential savings, under the assumption that the labor force growth rate is endogenous and described by a generic iterative scheme having a unique positive globally stable equilibrium. The study herewith presented aims at confirming the existence of a compact global attractor for system T along the invariant line. Consequently asymptotic dynamics of growth models with constant population growth rate can be related to those with non-constant population growth if the steady state rate is globally stable. Furthermore we prove that the system exhibits cycles or even chaotic dynamics patterns if shareholders save more than workers, when the elasticity of substitution between production factors drops below one (so that capital income declines). The analytical results are supplemented by numerical simulations.

In this study we research the dynamics shown by a cobweb-type model with linear demand, non-invertible supply function and with forward-looking expectations associated to backward looking ones. The study of the dynamics exhibited by our model will show how the presence of forward-looking expectations represents a stabilising factor: As the weight attributed to the above mentioned foreseeing form increases, the system becomes less and less complex until it only generates orbits converging either to the fixed point or to a cycle-2. It is particularly interesting to analyse at the same time the role played by the two forms of expectations considered, both of which contribute to market stabilization.

In this study we will research the dynamics shown by a cobweb-type model with hyperbolic demand, sigmoidal supply and with backward-looking mechanism of expectation creation, whereby the new state of the system is obtained from all the previous states observed by weighted arithmetical mean with exponentially decreasing weights in the rho region. The study herewith presented aims at confirming the existence of a stabilising effect due to the presence of infinite memory since, with all the other conditions begin the same, a memory rate ('rho) exists at which market equilibrium is a sink. An unstable system, therefore, becomes stable in the presence of sufficiently resistant expectations with infinite historical memory, although this transition to stability is accompanied by the onset of chaos. The resulting effect, therefore is one of "qualitative destabilisation," that is with reference to the qualitative dynamic performance produced, associated to a "quantitative stabilisation," that is to say with reference to the decreasing width of the invariant sets within which relevant dynamics occur.