Dynamics of Formation of Transitional Prices on the Chain of Sequential Markets: Analytical Model

Abstract

Although the problem of formation of market prices, determination of equilibrium prices within the model “Demand – Supply” is quite known and a great number of both theoretical works and works that summarize the results of observations are devoted to its research, this problem remains relevant, especially as to the dynamics of pricing processes and the stability of equilibrium prices in relation to changes in parameters that characterize the state of the system. Most studies addressing these issues focus on either a particular local market or the global market for some products in general. The purpose of this work is to build a mathematical model that would allow us to analyze general issues related to the formation of transitional prices in the finite N-dimensional chain of sequential markets in accordance with the scheme of market equilibrium. An analytical model is proposed that makes it possible to study the dynamics of prices in adjacent markets. Within this model, which is based on the determination of processes using a system of integral equations, it was assumed that the impact on the chain of sequential markets and the response to this impact are continuous over time. The dynamic aspect of the proposed pricing model in the vertical sequence of markets is the existence of an “after-effect”, which is described in an integral form by the delay distributed over time. The issues of adequacy of the model were examined, its internal coherence was studied, the correctness of the transition from the mathematical model of dynamics as a system of integral equations to the model in the form of a system of linear algebraic equations was substantiated. The conditions for the existence of the solution for this system of equations and the area of its stability are formulated. The mathematical model proposed in this paper allows for a qualitative analysis of the system states (by phase trajectories). Examples of numerical implementation of our analytical model for two and three sequential markets are given, equilibrium prices for each link of the chain of sequential markets are determined. Applying simulation modelling, the stability of the solution in relation to changes in such parameters of the model as the elasticity of demand and supply in the market under study and cross-elasticities in adjacent markets as well as the impact of these parameters on such dynamic indicators of the market system as the rate of attainment of equilibrium was examined.