A Macro Model of Heterogeneous Growth Ⅱ: An Existence Proof of Transitional Dynamics

In order to makes clear several important sources from which structural transitions occur, Dohtani (2018) constructed an endogenous growth model. However, per capita growth rates of the optimal paths obtained from the model is independent of the initial levels of macroeconomic variables. In other words, the model does not possesses transitional dynamics. This result is inconsistent with the well-known empirical evidence on convergence. See Barro and Sala-i-Martin (Chapters 11 and 12, 1995). In the present paper, by incorporating adjustment costs for investment into the model of Dohtani (2018), we will prove that a modified version yields transitional dynamics

Chaos resulting from nonlinear relations between different variables

In this study, we further develop the perturbation method of Marotto (1979) and investigate the general mechanisms responsible for nonlinear dynamics, which are typical of multidimensional systems. We focus on the composites of interdependent relations between different variables. First, we prove a general result on chaos, which shows that the cyclic composites of nonlinear interdependent relations are sources of chaotic dynamics in multidimensional systems. By considering several examples, we conclude that the cyclic composites play an important role in detecting chaotic dynamics

A Macro Model of Heterogeneous Growth

Developing the AK model, we construct an endogenous growth model with many industries. Unlike the original AK model, our model generates endogenous growth accompanied by the change of relative price. The growth rate of each industry is determined by such fundamental parameters as the rate of technological progress of the industry, the elasticity of marginal productivity of the industry and the elasticity of marginal utility of the goods produced by the industry. In our model, the persistent change of relative prices admits of consistent growths of heterogeneous industries. Therefore, our model gives a theoretical explanation of the persistent transition of industrial structure accompanied by a change of relative prices. The transition of industrial structure depends on the fundamental parameters. We derive an equation that relates the growth rate of relative price of an industry to the growth rates of capital stock and production of the industry. By using our model, we unifiedly explain many empirical facts that have been known so far. We also give a new theoretical viewpoint about the empirical fact that the relative price of investment is higher in poor countries relative to rich countries. We demonstrate that the empirical fact results from the myopia concerning consumption in the poor countries. Moreover, in the case where the number of consumption-goods industry is one, we incorporate population growth. In the modified model, we derive an equation which relates the growth rate of relative wage to the growth rates of capital stock and relative price

A Growth-Cycle Model of the Solow-Swan Type, Ⅰ

We construct an endogenous growth-cycle model of the Solow-Swan type. The equilibrium point of the growth-cycle model is the same as the steady state of the Solow-Swan growth model. Unlike in the Solow-Swan growth model, the representative household in the growth-cycle model, however, adaptively estimates his/her average income and determines his/her consumption in proportion to average income. We prove that if the steady state is unstable, any non-equilibrium path converges to a limit cycle. However, even if the steady state is stable, growth cycles can emerge. In fact, we prove that the growth-cycle model generates corridor stability. As a result, we prove that there exists an unstable cyclic path such that any path in the interior of the cyclic path converges to the steady state and any path in the exterior of the cyclic path tends toward a limit cycle (growth cycle). We also prove that a high economic growth rate is not compatible with a stable economy

An Endogenous Model of Heterogeneous Growth I : Additive Utility Function

Developing the AK model, we construct an endogenous growth model with many industries. Unlike the original AK model, our model generates endogenous growth accompanied by the change of relative price. The growth rate of each industry is determined by such fundamental parameters as the rate of technological progress of the industry, the elasticity of marginal productivity of the industry and the elasticity of marginal utility of the goods produced by the industry. These fundamental parameters are different among the industries. Therefore, our model gives a theoretical explanation of the persistent transition of industrial structure accompanied by a change of relative prices. Moreover, we derive an equation that relates the growth rate of relative price of an industry to the growth rates of capital stock and output of the industry. By using our model, we unifiedly explain several empirical facts that have been known so far. We also give a new theoretical viewpoint about the empirical fact that the relative price of investment is higher in poor countries relative to rich countries. Moreover, we incorporate population growth. In the modified model, we derive an equation which relates the growth rate of relative wage to the growth rates of output and relative price

An Endogenous Model of Heterogeneous Growth II : A Generalization to the CES Utility Function

Dohtani (2019) constructed an endogenous growth model with heterogeneous industries. In the growth model, the growth rate of each industry is determined by such fundamental parameters as the rate of technological progress of the industry. Through the growth model, the paper derived a fundamental equation that gives an analytical relation among the growth rates of relative price, consumption and capital stock of each industry. However, the fundamental equation is derived under the assumption that the household possesses an additive utility function. In the present paper, we generalize the model to a growth model with the CES utility function and introduce a new notion that gives a generalization of the growth rate in the usual sense. By using such a notion, we derive the same results derived by Dohtani (2019

A Business Cycle Model of Speculation from a Viewpoint of Minsky and Shiller Ⅱ: Global Dynamic Analysis

We construct a 3-dimensional extension of the dynamic IS-LM model, in which the money demand function depends not only on income but also on a rate of change in expected income (RCEI). We demonstrate the occurrence of limit cycles in the extended IS-LM model. Our arguments are essentially derived from the remarkable viewpoint of H. P. Minsky and J. R. Shiller concerning financial markets. We assume that the money demand negatively correlates with RCEI. Such a negative correlation results from a speculative behavior. We demonstrate that the negative correlation is an important source of unstable equilibrium and therefore, business cycles. Firstly, we transform the extended IS-LM model into a 2-dimensional Lienard system and prove the occurrence of a stable limit cycle in the Lienard system. Secondly,by using a Hopf bifurcation theorem, we demonstrate the occurrence of a Hopf cycle in the extended 3-dimensionl IS-LM model. Our model possesses two types of self-fulfilling prophecy

Equilibrium Pigou Cycle

We construct a simple endogenous model that describes business cycles resulting from self-fulfilling prophecies. In our model, the goods market is assumed to be continuously cleared. Being guided by Pigou\u27s 1926 insight, we try to show that the expectation about future economy plays an important source of business cycles. We assume that firms expect the occurrence of business fluctuations. Then, the firms will try to calculate expected income. Under such an assumption, we demonstrate the occurrence of business cycles. Since the expectation of business fluctuations yields business fluctuations, the results of our model are based on self-fulfilling prophecies. Using the Hopf bifurcation theorem, we detect a limit cycle in our model. Moreover, by executing numerical simulations, we describe the possible occurrence of stable limit cycles and show that the width of such a limit cycle depends on the degree of intensity of animal spirits

A Business Cycle Model of Speculation from a Viewpoint of Minsky and Shiller Ⅰ: Construction of Model and Its Local Analysis

We construct a 3-dimensional extension of the dynamic IS-LM model, in which the money demand function depends not only on income but also on a rate of change in expected income (RCEI). We demonstrate the occurrence of limit cycles in the extended IS-LM model. Our arguments are essentially derived from the remarkable viewpoint of H. P. Minsky and J. R. Shiller concerning financial markets. We assume that the money demand negatively correlates with RCEI. Such a negative correlation results from a speculative behavior. We demonstrate that the negative correlation is an important source of unstable equilibrium and therefore, business cycles. Firstly, we transform the extended IS-LM model into a 2-dimensional Lienard system and prove the occurrence of a stable limit cycle in the Lienard system. Secondly, by using a Hopf bifurcation theorem, we demonstrate the occurrence of a Hopf cycle in the extended 3-dimensionl IS-LM model. Our model possesses two types of self-fulfilling prophecy

Animal Spirits and Business CyclesⅠ: A Dynamic Model and Its Nonlinear Analysis

We construct a simple Keynesian business cycles model in which animal spirits is incorporated into the model. We assume that each firm possesses the Keynesian investment function depending on demand. Like the well-known Kaldor model, we connect the animal spirits with the degree of response of investment to demand. However, for each firm we assume that even if the level of demand is the same, the strength of it is different between upswing and downswing of income. That is, we assume that the animal spirits of each firm is strong (resp. weak) in the case where demand increases (resp. decreases). This assumption implies that the Keynesian investment function of each firm is asymmetry. Unlike many Keynesian business cycles models, we introduce a certain kind of homogeneity among such a type of investment function. Moreover, we construct a statistical model that builds a bridge between microinvestment and macro-investment and derive a nonlinear macro-investment function. By using the investment function, we construct a simple business cycles model. We demonstrate that the nonlinearity yields a limit cycle in our model and detect the occurrence of a generalized Hopf bifurcation