| Abstract [eng] |
While studying the literature we observed a lack of applications of the Lambert W function in scientific papers. Even though in the last few decades stability of financial market has been studied extensively, when dealing with more complicated models, for example with models that factor in the delay between supply and demand, the researchers had to used numerical methods, which can only give approximate results. When then mathematic model is described as delay differential equation we can apply Lambert W function ant obtain exact results without needing many computational resources. We have applied Lambert W function to analysis of market price stability and the described methodology was used to solve two real world solutions: 1. Mathematical model of poultry meat market price – linear delay differential equation. To obtain the solution of models characteristic equation Lambert W function was used. We analyzed the position of the solutions of characteristic equation on complex plane and by doing so determined whether the system is stable or not. The dynamics of the solution of the original equation was also studied and parameter values, at which system is stable where obtained. 2. Linearized Kalecki model – nonhomogenous delay differential equation. We analyzed the values of the solution in terms of time, determining the stability with different values of the delay argument. The main result of this project is the application of Lambet function to the analysis of market price stability. We analyzed the poultry price model with constant value of the delay argument τ = 0.9 and different values of market stability coefficient. We observed, that the system is stable, when values of market stability coefficient are equal to 0.9 and 1.4, but when the value of the coefficient is 1.9 system becomes unstable. Kalecki model was analyzed with different values of the delay argument. Results showed, that system is stable, when values of delay argument are between 0.1 and 0.6. When the delay is equal to 0.61 the system becomes unstable. From the obtained results we can conclude that Lambert W function can be applied to analysis of market price stability, when they are described by delay differential equations. |
