Analyzing market price equilibrium dynamics with differential equations: incorporating government intervention and market forces

Abstract

This study seeks to investigate price stability in a dynamic market, where prices are subject to sudden impacts akin to those observed during the Covid-19 lockdown in 2020, as well as other influences introduced naturally or by price regulatory agencies. By examining functions derived from price observations, changes in prices, and changes in the rate of price changes, the study analyzes their stability amidst various influences. These influences are incorporated by examining factors affecting supply and demand quantities, which are modelled using a second-order linear differential equation; P′′(t)+a₁P′(t)+a₀P(t)=f(t). This study builds upon the research of Espinoza and Bob Foster, who analyzed a second-order differential equation with a constant inhomogeneity. It employs matrix methods to assess the stability of systems of differential equations. To analyze impulsive price changes modelled using the Dirac delta function and persistent price changes modelled with Heaviside's unit step function, the Laplace technique and its general inversion formula are applied. The study identifies conditions under which stability in the system can be maintained.