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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.

This review analyses the influence of technologies and saving propensities of workers and shareholders on economic growth, considering the \cite{bk2000} model. We show how investing behaviours and production peculiarities condition the evolution of capital over time. We highlight that fluctuations and multiple equilibria arise only when the elasticity of substitution between capital and labour is lower than one. Moreover, only production functions with variable elasticity of substitution between inputs are able to describe the poverty trap phenomenon. Complex dynamics emerge when the difference between the saving propensity of the two income groups is sufficiently high.

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.