Path dependence is a key feature of complex economic systems. It implies that history matters in the long-term evolution of markets and economies. Path dependence can be viewed as the dynamic version of positive feedback effects. This paper focuses on the nonlinear neoclassical economic growth model with the Cobb-Douglas production function, which accounts for problems related to pollutant emissions. It was found that only selected initial forms have a chance to develop. Present states depend on past states, even though the historical circumstances that had affected the past states may no longer be relevant. The choice among different histories may be a stochastic process. The understanding of economic growth suggested in this study stands in opposition to the neoclassical tradition based on equilibrium states or paths independent of the system's history. In contemporary economics, the idea of path dependence is most often used in studies on the high-tech industry, where the researchers are focused on such phenomena as innovation processes, monopolization, or the causes of ineffective technical solutions. The analysis of historical conditions is almost entirely carried out with the use of qualitative methods, since the subject of the research is non-formalized. In addition, the theoretical basis for conducting relevant empirical research is still missing. As a result of the development of complexity economics in recent years, numerous dynamic features of complex economic systems can be examined with the application of quantitative methods which, in effect, strengthens the bonds between theory and practice. Rare exceptions include path dependence relations. The aim of this article is to fill this gap and to create a theoretical basis for quantitative research on historical conditions in economics. This is a necessary condition for undertaking empirical research. The theoretical search started with the Keynesian model of the Samuelson-Hicks trade cycle, to demonstrate that conventional economics completely omits the most interesting path-dependence cases. It turns out that only the neoclassical model of economic growth, taking into account two power laws, provides appropriate dynamic characteristics for a full description of path dependence relations. Therefore, appropriate theoretical bases can be provided only by complexity economics. It may seem that, in this work, the dependence on history is restricted to two successive time steps in the case of the Samuelson-Hicks model and a single step in the neoclassical model of economic growth by Day. However, it examines an ordered path dependence, where events are chronologically ordered and the impact of earlier events on the later ones occurs through intermediary events. It should be remembered that events are constantly affected by environmental stimuli that are reflected not only in initial conditions, but also in the values of the parameters for all periods. Thus, it is not a case of short-term memory.