Adam Krawiec

The aim of this article is to generalise the Solow growth model and to show that, in such a generalised model, there can be more than one non-trivial stationary point. In the neoclassical theory of economic growth, based on the Solow-Swan model, technology is exogenous. Growth models are characterised by a different approach, in which technology (or knowledge) is developed endogenously. Exogenous technology is introduced to the economy ‘from the outside’, while endogenous technology applies when its level depends on another macroeconomic variable in the model. This paper proposes an alternative generalisation of the Solow model with endogenous technology, which is related to the capital-labour ratio in such a way that the level of technology increases as the value of this variable grows. We also analysed the dynamics of the model, determined the stationary points and analysed their stability. Furthermore, we demonstrated the conditions allowing the existence of a stable stationary state. Numerical simulations were carried out to determine the trajectories of the macroeconomic variables examined in the study. We showed that there are two non-trivial stationary points above the savings rate threshold value. The unstable stationary point defines the poverty limit. If the initial level of capital per worker in an economy is below the value of this point, the macroeconomic variables converge to 0 in the long run. If the initial level of capital per worker is greater than this value, the macroeconomic variables converge to a stable stationary point. Moreover, since in general, the value of the capital-labour ratio determines both the level of technology and labour productivity, these variables behave in the same way as the former.

We investigate the dynamics of dust matter with bulk viscosity effects. We explored the analogy dynamical problem to Chaplygin gas. Due to this analogy we give exact solutions for the FRW cosmology with viscosity coefficient parameterized by the Belinskii–Khalatnikov power law dependence with respect to energy density. These exact solutions are given in the form of hypergeometrical functions. We proved simple theorem which illustrated as viscosity effects can solved the initial singularity problem present in standard cosmological model.

In this paper, we consider a model of economic growth with a distributed time-delay investment function, where the time-delay parameter is a mean time delay of the gamma distribution. Using the linear chain trick technique, we transform the delay differential equation system into an equivalent one of ordinary differential equations (ODEs). Since we are dealing with weak and strong kernels, our system will be reduced to a three- and four-dimensional ODE system, respectively. The occurrence of Hopf bifurcation is investigated with respect to the following two parameters: time-delay parameter and rate of growth parameter. Sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation are presented in case of time-delay parameter. Numerical studies with the Dana and Malgrange investment function show the emergence of two Hopf bifurcations with respect to the rate growth parameter. In this case, we have been able to detect the existence of stable long-period cycles in the economy. According to the time-delay and adjustment speed parameters, the range of admissible values of the rate of growth parameter breaks down into three intervals. First, we have stable focus, then the limit cycle and finally again the stable solution with two Hopf bifurcations. Such behavior appears for some middle interval of the admissible range of values of the rate of growth parameter.

We consider the model of economic growth with time delayed investment function. Assuming the investment is time distributed we can use the linear chain trick technique to transform delay differential equation system to equivalent system of ordinary differential system (ODE). The time delay parameter is a mean time delay of gamma distribution. We reduce the system with distribution delay to both three and four-dimensional ODEs. We study the Hopf bifurcation in these systems with respect to two parameters: the time delay parameter and the rate of growth parameter. We derive the results from the analytical as well as numerical investigations. From the former we obtain the sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation. In numerical studies with the Dana and Malgrange investment function we found two Hopf bifurcations with respect to the rate growth parameter and detect the existence of stable long-period cycles in the economy. We find that depending on the time delay and adjustment speed parameters the range of admissible values of the rate of growth parameter breaks down into three intervals. First we have stable focus, then the limit cycle and again the stable solution with two Hopf bifurcations. Such behaviour appears for some middle interval of admissible range of values of the rate of growth parameter.

We study an impact of the financial intermediation on economic growth. We assume the simple model of the economic growth in the form of an autonomous dynamical system with a financial sector represented by banks and real sector represented by households and firms. We assume that financial intermediation services are described by financial intermediation technology which is a function depending on the share of labor employed by banks. Investments realized by firms depend not only on savings accumulated by banks but also on financial intermediation technology. We obtain a three-dimensional dynamical system and analyze the existence of a saddle equilibrium in the growth process associated with financial intermediation. Using mathematical methods of dynamical systems, we analyze growth paths, and we study the stationary states of the system and their stability. We found that equilibrium is reached only by trajectories located on two submanifolds. The resulting analysis provides an insight into the saddle solution with a stable incoming separatrix lying on one of the invariant manifolds.

The article proposes nonlinear optimal control for a multi-agent Kaldor-Kalecki macroeconomic model of linked economies. The Kaldor-Kalecki model has been analysed mostly by considering closed economies in which interactions with other economies or financial institutions are overlooked. In the present article, a system of linked economies is considered in which the Kaldor-Kalecki model of each individual economy is retrofitted with a fraction of the GDP produced by the other economies. In this context a nonlinear optimal control scheme is developed for the entire financial system consisting of the interacting economies. The global stability properties of the control method are proven through Lyapunov analysis.