Adam Krawiec, Jagiellonian University, Kraków, Poland
Akio Matsumoto, Chuo University, Tokyo, Japan
Anna Połeć, Jagiellonian University, Kraków, Poland

Abstract: This paper is a study of applications of the Lambert W function to solve delay differential equation models that arise in economics and finance. The Lambert W function, one of the mathematical special functions, is a basis of an effective technique to examine the stability and the dynamical regimes, especially cyclical one. The introduction to the Lambert W function is presented, showing its analytical properties. To demonstrate the method at work two economic models are chosen. First, the model of economic growth with a production delay and second, the market price model with supply dependent on a delayed price. In the Solow model of economic growth with a delay, the conditions are derived to determine how the interaction of the delay and the population growth rate can lead to the appearance of cyclical behavior in the economy and possibly the switch in stability near the steady state, and emergence of constant fluctuations and growth cycles, in consequence. Second, in the model of market price, similarly to the Solow model with a delay, exact conditions for the emergence of fluctuations and possible stability switch are constructed.

Keywords: delay differential equations, Lambert W function, Solow model, market price model, stability analysis