METHODS OF COMPARATIVE STATICS AND DYNAMICS IN THE THEORY OF ECONOMIC CYCLES

The subject of this work is the problem of describing the dynamic behavior of the price in the market of one product. The typical balance of interaction of supply and demand functions depending on the price is considered. The dynamic model of price evolution is based on the assumption that the demand function at a given time depends on the supply function at all previous points in time, i.e. there is a process with an aftereffect. The core of the integral transformation is a characteristic of the second order, which can initiate periodic regimes in price variables. The aim of the work is to synthesize a mathematical model of price changes in the market of one product and study the stability of its equilibrium states with the manifestation of the structure of marginal cycles. The task of the study is to demonstrate the degree of connection between the problem of stability of equilibrium and the problem of obtaining fruitful results in comparative statics. This duality is the principle of conformity of P. Samuelson. The basic mathematical model of the studied process of price dynamics is a system of two nonlinear differential equations of the first order. The research methods are the nonlinear theory of analysis of dynamical systems, the mathematical theory of stability of systems of differential equations, the conceptual apparatus of analysis of typical bifurcations of birth (death) of the boundary cycle, known as the Andronov-Hopf bifurcation. As a result of a detailed analysis of the properties and parameters of self-oscillating modes, a double cycle is revealed, i.e. there is a fact of coexistence around the equilibrium state of stable and unstable limit cycles. Subsequent mathematical transformations prove that the line of demarcation of these two cycles is completely determined by the static parameters of the studied system, which illustrates the mechanism of action of the principle of conformity of P. Samuelson. Conclusions: on the example of the functioning of the labor market, a comprehensive analysis of the stability of two positive equilibrium states, which characterize the effects of substitution and income. Computer simulations are used to perform computational experiments that demonstrate self-oscillating modes of price changes. As a result of the analysis of the obtained numerical results, it is possible to draw a conclusion about the stability of the limit cycle in the vicinity of the equilibrium state, which corresponds to the substitution effect.

Acoustic Identification of the Voicing Boundary during Intervocalic Offsets and Onsets Based on Vocal Fold Vibratory Measures

Methods for automating relative fundamental frequency (RFF)—an acoustic estimate of laryngeal tension—rely on manual identification of voiced/unvoiced boundaries from acoustic signals. This study determined the effect of incorporating features derived from vocal fold vibratory transitions for acoustic boundary detection. Simultaneous microphone and flexible nasendoscope recordings were collected from adults with typical voices (N = 69) and with voices characterized by excessive laryngeal tension (N = 53) producing voiced–unvoiced–voiced utterances. Acoustic features that coincided with vocal fold vibratory transitions were identified and incorporated into an automated RFF algorithm (“aRFF-APH”). Voiced/unvoiced boundary detection accuracy was compared between the aRFF-APH algorithm, a recently published version of the automated RFF algorithm (“aRFF-AP”), and gold-standard, manual RFF estimation. Chi-square tests were performed to characterize differences in boundary cycle identification accuracy among the three RFF estimation methods. Voiced/unvoiced boundary detection accuracy significantly differed by RFF estimation method for voicing offsets and onsets. Of 7721 productions, 76.0% of boundaries were accurately identified via the aRFF-APH algorithm, compared to 70.3% with the aRFF-AP algorithm and 20.4% with manual estimation. Incorporating acoustic features that corresponded with voiced/unvoiced boundaries led to improvements in boundary detection accuracy that surpassed the gold-standard method for calculating RFF.

The dynamics of export-import interaction

This work is devoted to the dynamic interpretation of the basic provisions of the quantitative theory of money. Namely, the construction of models of price changes for marketable products in the performance of export-import activities both in discrete and continuous time. A number of hypotheses are used to determine the conditions for the violation of the equilibrium states of the trade balance using the classical macroeconomic Fisher equation. An overview of scientific works is presented, which highlights the main factors for the implementation of foreign economic activity: exchange rates and devaluation. The analysis of the considered sources testifies to the presence of the problem of stability of the observed dynamic processes in the vicinity of the state of equilibrium, which have the traditional name of the Marshall-Lerner condition. A detailed study of the stability criteria of a discrete dynamic model with quadratic nonlinearity demonstrates a significant variety of trajectories of the studied process. These are, in particular, increasing or decreasing aperiodic behavior, oscillating processes of a fixed period, bifurcations of doubling the period and chaotic trajectories. The limits of distribution of various types of evolutionary changes in terms of elasticities as important indicators of export-import operations are indicated. For models of pricing dynamics in continuous time, a detailed analysis of the structural instability of equilibrium states was performed. Saddle-nodal bifurcation and no less important Andronov-Hopf bifurcation, which is associated with the formation of a boundary cycle around the equilibrium state, are distinguished. It is proved that the found cycle is unique and stable. For a discrete model of price formation, appropriate calculations are performed to demonstrate different types of evolutionary behavior. The given discrete model based on the Fisher equation can be used for qualitative forecasting (by trajectories) of the dynamics of internal pricing without the use of traditional methodology of econometric analysis of time series. This model is characterized by the fact that it is reduced to a single complex parameter and this greatly simplifies the definition of the corresponding types of dynamical regimes in the vicinity of equilibrium states.