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The main goal of the present work is to analyze the evolution of non-compliant firms in public procurement. The dynamic setting is formalized by an evolutionary adaptation process which describes whether honest or dishonest behavior prevails in society at any given time. Firms will either behave in an honest or dishonest way depending on the payoffs and the type of firms they meet, through a word-of-mouth process. The honesty propensity assumption is introduced. By making use of both analytical tools and numerical techniques, we show that monomorphic configurations may represent stable or unstable equilibria, while coexistence of both groups in the steady state is possible and its asymptotic stability is sensitive to parameter modifications. Moreover, a rich variety of integrated dynamic behavior can be observed due to different kinds of bifurcations, smooth or border collision. A social planner can evaluate social and/or fiscal policies to manipulate preferences for honest behavior.

In recent years, the study of the evolution of non-compliant behaviour in public procurement has been widely developed due to the growing economic relevance of this phenomenon. When such a question is formalized in terms of a dynamical model, new insights can be pursued, related to the possible evolution from a situation with low dishonesty level to high dishonesty level or vice versa. The present model considers an evolutionary adaptation process explaining whether honest or dishonest behaviour prevails in society at any given time by assuming endogenous monitoring by the State. We will distinguish between a scenario in which firms converge to monomorphic configurations (all honest or all dishonest) and a scenario in which firms converge to polymorphic compositions (that is with coexistence of both groups), depending on the relevant parameters. By making use of both analytical tools and numerical simulations, the present work aims at explaining the effectiveness of economic policies to reduce or eliminate non-compliant behaviour. Social stigma is found to play a key role: if the “inner attitude toward honesty” of a country is not strong enough, then dishonesty cannot be ruled out. However, increasing both the fine level attached to dishonest behaviour and the monitoring effort by the State can reduce asymptotic dishonesty levels and escape form the dishonesty trap.

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 economic theory the majority of macroeconomic models describing economic growth employ differential equations or sets of differential equations (see, among all, Solow, 1956 and Haavelmo, 1954). Nevertheless, economic data are usually available in discrete time. Therefore, when attempting to apply these models it is often necessary to use their discrete form, i.e. difference equations. To this aim, more and more often authors propose and analyse discrete versions of the models originally built with the assumption of time continuity. Despite many standard numeric techniques and ready-made software, obtained discrete models do not always maintain model characteristics in continuous time and the long run behaviours of the discretized model could differ from the original one. In this work, we present a modification of non-standard discretization method related to the methodology proposed by Mickens (2000), (2003) and its revisions (see Kwessi et al., 2018) for converting economic models from continuous time to discrete time. Such a discretization method preserves the original dynamic properties of the continuous model, in the sense of equilibria, their stability and bifurcation characteristics. Furthermore, the discretization produces solution trajectories in qualitative and quantitative agreement with those of the continuous model. An example of economic model described by a system of nonlinear differential equations is studied: we applied the discretization method to the Goodwin model (Goodwin, 1967) and provided a comparative analysis for qualitative and quantitative long run behaviour of the continuous and discrete version of the system.

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.

The choice of time as a discrete or continuous variable may radically affect equilibrium stability in an endogenous growth model with durable consumption. In the continuous-time Ramsey model [F. P. Ramsey, Econ. J. 38(152), 543-559 (1928)], the steady state is locally saddle-path stable with monotonic convergence. However, in the discrete-time version, the steady state may be unstable or saddle-path stable with monotonic or oscillatory convergence or periodic solutions [see R.-A. Dana et al., Handbook on Optimal Growth 1 (Springer, 2006) and G. Sorger, Working Paper No. 1505 (2015)]. When this occurs, the discrete-time counterpart of the continuous-time model is not consistent with the initial framework. In order to obtain a discrete-time Ramsey model preserving the main properties of the continuous-time counterpart, we use a general backward and forward discretisation as initially proposed by Bosi and Ragot [Theor. Econ. Lett. 2(1), 10-15 (2012)]. The main result of the study here presented is that, with this hybrid discretisation method, fixed points and local dynamics do not change. For what it concerns global dynamics, i.e., long-run behavior for initial conditions taken on the state space, we mainly perform numerical analysis with the main scope of comparing both qualitative and quantitative evolution of the two systems, also varying some parameters of interest.

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.

This paper introduces a sigmoidal production function that considers production possible even when the only input is labour. The long-run behaviour of an economy described by the neoclassical Solow-type growth model with differential savings is investigated considering the technology presented. It is found that labour productivity influences the existence of boom and bust periods as well as the level of capital per capita in equilibrium.

This work investigates the qualitative and quantitative dynamics of a Solow–Swan growth model with differential savings as proposed by Bohm and Kaas (J Econ Dyn Control 24:965–980, 2000) assuming the shifted Cobb–Douglas (SCD) production function (see Capasso et al. in Nonlinear Anal. 11:3858–3876, 2010) which makes it possible to consider the long-run dynamics of non-developed and developing countries as well as that of developed economies. The resulting model is described by a nonlinear discontinuous map generating both a poverty trap and complex dynamics. Furthermore, multistability phenomena may emerge: besides the “vicious circle of poverty”, long-run behaviours may include boom and bust periods. Complex basins can emerge, hence, economic policies trying to raise the capital per capita may fail and economies may be captured by the poverty trap.

This research aims at studying a general equilibrium closed economy with overlapping generations and inherited tastes (aspirations), as in de la Croix (Econ Lett 53(1):89–96, 1996). It shows that the interaction between the intensity of aspirations and the elasticity of substitution of effective consumption affects the qualitative and quantitative long-term dynamics from both local and global perspectives. The related literature is extended by showing that (1) the Neimark–Sacker bifurcation found by de la Croix (1996) does not necessarily give rise to fluctuations and (2) endogenous (long-lasting) fluctuations occur through the emergence of period-doubling bifurcations.

This paper presents a new stochastic asset pricing model in a context of bounded rationality, where beliefs about future prices are formed via an expectations updating rule characterized by a stochastic multiplicative random variable, working as an agent-based time dependent weight of the conditional expectation of the fundamental. The agent's belief about future prices depends on his confidence in the forecasts made by other agents, measured by the distribution type of agents and by a confidence parameter. The resulting stochastic dynamical system is firstly analyzed in a deterministic setting, deriving conditions for uniqueness and stability of steady states and proving that, for high values of the confidence parameter, no complicated dynamics can be exhibited, hence the new component has a stabilizing effect on the qualitative dynamics. Differently, for small values of the confidence parameter, we prove the existence of a stability region in the parameters plane where the only possible dynamics is convergence to a steady state, while complexity is exhibited outside such region. Starting from the results obtained in the deterministic case, the model is then explored by reintroducing randomness. More specifically, we analyze the stability region in three directions: first of all, a robust estimate of the stability region's measure is provided; second, a long run equilibrium relation between the parameters of the system is obtained; third, the persistence properties of the series describing the bifurcation curves is performed. We finally underline some economic implications.

In contrast with the existing literature with overlapping generations, aspirations and exogenous fertility, this article shows that endogenous fertility prevents endogenous fluctuations in a general equilibrium economy.

In this article, we investigate the local and global dynamics of a nonlinear duopoly model with price-setting firms and managerial delegation contracts (relative profits). Our study aims at clarifying the effects of the interaction between the degree of product differentiation and the weight of manager's bonus on long-term outcomes in two different states: managers behave more aggressively with the rival (competition) under product complementarity and less aggressively with the rival (cooperation) under product substitutability. We combine analytical tools and numerical techniques to reach interesting results such as synchronisation and on-off intermittency of the state variables (in the case of homogeneous attitude of managers) and the existence of chaotic attractors, complex basins of attraction, and multistability (in the case of heterogeneous attitudes of managers). We also give policy insights.

We analyze the dynamics of a generalized discrete time population model of a two-stage species with recruitment and capture. This generalization, which is inspired by other approaches and real data that one can find in literature, consists in considering no restriction for the value of the two key parameters appearing in the model, that is, the natural death rate and the mortality rate due to fishing activity. In the more general case the feasibility of the system has been preserved by posing opportune formulas for the piecewise map defining the model. The resulting two-dimensional nonlinear map is not smooth, though continuous, as its definition changes as any border is crossed in the phase plane. Hence, techniques from the mathematical theory of piecewise smooth dynamical systems must be applied to show that, due to the existence of borders, abrupt changes in the dynamic behavior of population sizes and multistability emerge. The main novelty of the present contribution with respect to the previous ones is that, while using real data, richer dynamics are produced, such as fluctuations and multistability. Such new evidences are of great interest in biology since new strategies to preserve the survival of the species can be suggested.

This work models and analyzes the dynamics of a two-stage species with recruitment and capture factors. It arises from the discretization of a previous model developed by Ladino and Valverde (2013), which represents a progress in the knowledge of the dynamics of exploited populations. Although the methods used here are related to the study of discrete-time systems and are different from those related to continuous version, the results are similar in both the discrete and the continuous case what confirm the skill in the selection of the factors to design the model. Unlike for the continuous-time case, for the discrete-time one some (non-negative) parametric constraints are derived from the biological significance of the model and become fundamental for the proofs of such results. Finally, numerical simulations show different scenarios of dynamics related to the analytical results which confirm the validity of the model.

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.

The article is devoted to the study of global attractors of quasi-linear non-autonomous difference equations. The results obtained are applied to the study of a two-dimensional triangular growth model of Solow type with Variable Elasticity of Substitution production function and endogenous population growth rate described by the Beverton-Holt equation.

We analyze the dynamics produced by the neoclassical one-sector growth model with differential savings as in [3] while assuming a sigmoidal production function as in [8] and the labor force dynamics described by the Beverton Holt equation, as proposed in [2]. The resulting discrete time dynamic system is bidimensional, autonomous and triangular: we determine its invariant sets and study their local stability. We also perform a mainly numerical analysis to show that complex features are exhibited, related both to the structure of the coexixting attractors and to their basins. The study presented shows that complexity emerges if the elasticity of substitution between production factors is low enough and shareholders save more than workers, thus confirming the results obtained with concave production functions (see [6], [7] and [9]).

In this paper we analyze the dynamics shown by the neoclassical one-sector growth model with differential savings as in Bohm and Kaas [J. Econom. Dynam. Control, 24 (2000), pp. 965--980] while assuming a sigmoidal production function as in [V. Capasso, R. Engbers, and D. La Torre, Nonlinear Anal., 11 (2010), pp. 3858--3876] and the labor force dynamics described by the Beverton--Holt equation (see [R. J. H. Beverton and S. J. Holt, Fishery Invest., 19 (1957), pp. 1--533]). We prove that complex features are exhibited, related both to the structure of the coexisting attractors (which can be periodic or chaotic) and to their basins (which can be simple or nonconnected). In particular we 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 with concave production functions. Anyway, in contrast to previous studies, the use of the S-shaped production function implies the existence of a po...

We study the relationship between corruption in public procurement and economic growth within the Solow framework in discrete time, while assuming that the public good is an input in the productive process and that the State fixes a monitoring level on corruption depending on the tax revenues. The resulting model is a two-dimensional, continuous and piecewise smooth dynamic system which describes the evolution of the capital per capita and that of the corruption level. We prove that the model admits multiple equilibria: their stability and the structure of their basins is studied. We also present numerical simulations for economic meaningful parameter values, useful to explain the role of parameters in the long–run path of the model. Our study aims at demonstrating that stable equilibria with positive corruption may exist (according to empirical evidence), even though the State may reduce corruption by increasing the wage of the bureaucrat or by increasing the amount of tax revenues used to monitor corruption.

This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their substitutability. This means that when products are complements, players may coordinate their behaviour even if initial conditions are different. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very different dynamic patterns.

This paper aims at studying a nonlinear dynamic duopoly model with price competition and horizontal product differentiation augmented with managerial firms, where managers behave according to market share delegation contracts. Ownership and management are then separate and managers are paid through adequate incentives in order to achieve a competitive advantage in the market. In this context, we show that complexity arises, related both to the structure of the attractors of the system and the structure of their basins, as multistability occurs. The study is conducted by combining analytical and numerical techniques, and aims at showing that slight different initial conditions may cause very different long-term outcomes.

We study the dynamics of a duopoly game a la Bertrand with horizontal product differentiation as proposed by Zhang et al. (2009) [35] by introducing opportune microeconomic foundations. The final model is described by a two-dimensional non-invertible discrete time dynamic system T . We show that synchronized dynamics occurs along the invariant diagonal being T symmetric; furthermore, we show that when considering the transverse stability, intermittency phenomena are exhibited. In addition, we discuss the transition from simple dynamics to complex dynamics and describe the structure of the attractor by using the critical lines technique. We also explain the global bifurcations causing a fractalization in the basin of attraction. Our results aim at demonstrating that an increase in either the degree of substitutability or complementarity between products of different varieties is a source of complexity in a duopoly with price competition.

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.

We study the dynamics of a one-dimensional piecewise smooth map defined by constant and logistic functions. This map has qualitatively the same dynamics as the one defined by constant and unimodal functions, coming from an economic application. Namely, it contributes to the investigation of a model of the evolution of corruption in public procurement proposed by Brianzoni et al. [4]. Bifurcation structure of the economically meaningful part of the parameter space is described, in particular, the fold and flip border-collision bifurcation curves of the superstable cycles are obtained. We show also how these bifurcations are related to the well-known saddle-node and period-doubling bifurcations.

In recent years, several models have focused on the study of the asset price dynamics and wealth distribution when the economy is populated by boundedly rational heterogeneous agents with Constant Relative Risk Aversion (CRRA) preferences. Chiarella and He (2001) study an asset pricing model with heterogeneous agents and fixed population fractions. In order to obtain a more appealing framework, Chiarella and He (2002) allow agents to switch between different trading strategies. More recently, Chiarella et al. (2006) consider a market maker model of asset pricing and wealth dynamics with fixed proportions of agents. A large part of contributions to the development and analysis of financial models with heterogeneous agents and CRRA utility do not consider that agents can switch between different predictors. Moreover, the models which allow agents to switch between different trading strategies make the following simplified assumption: when agents switch from an old strategy to a new strategy, they agree to accept the average wealth level of agents using the new strategy. Motivated by such considerations, we develop a new model based upon a more realistic assumption. In fact, we assume that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). The most important fact is that the wealth of the new group takes into account the wealth realized in the group of origin, whenever agents switch between different trading strategies. This leads the final system to a particular form, in which the average wealth of agents is defined by a continuous piecewise function and the phase space is divided into two regions. Nevertheless, our final dynamical system is three dimensional and we can find all the equilibria. We will prove that it admits two kinds of steady states, fundamental steady states (with the price being at the fundamental value) and non fundamental steady states. Several numerical simulations supplement the analysis.

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.

Authors describe a hyoid bone body, without horns, attributed to Homo erectus from Castel di Guido (Rome, Italy), dated to about 400,000 years BP. The hyoid bone body shows the bar-shaped morphology characteristic of Homo, in contrast to the bulla-shaped body morphology of African apes and Australopithecus. Its measurements differ from those of the only known complete specimens from other extinct human species and early hominid (Kebara Neandertal and Australopithecus afarensis), and from the mean values observed in modern humans. The almost total absence of muscular impressions on the body’s ventral surface suggests a reduced capability for elevating this hyoid bone and modulating the length of the vocal tract in Homo erectus. The shield-shaped body, the probable small size of the greater horns and the radiographic image appear to be archaic characteristics; they reveal some similarities to non-humans and pre-human genera, suggesting that the morphological basis for human speech didn’t arise in Homo erectus.