The κ-exponential function, representing a generalization of the exponential function, has been firstly introduced in physics, and, then, it has been considered in a noteworthy number of fields because of its ability to take rare events into account. Among the possible applications of this function, one of particular interest is in economics in which rare events may consist in natural disasters, such as earthquakes that reduce the supply of capital, or epidemics or other external shocks influencing the supply of intermediate inputs, human or physical capital. Starting from the κ-exponential function, the κ-logistic function, which is a generalization of the sigmoidal function, can be obtained and used to describe production functions in a unique setting to take into account (1) several shapes usually considered in economics (i.e. concave and non-concave production functions), (2) economies at different development levels, and, (3) the possible occurrence of rare events. In this paper, we investigate the economic growth model as proposed by Böhm and Kaas (2000), wherein the production function utilizes the κ-logistic function. We provide theoretical results confirmed by extensive computational experiments and in line with economic literature showing that a poverty trap may emerge together with fluctuations, multistability and complex dynamics. •The κ-logistic production function is a generalization of the sigmoidal production function.•Economic growth is considered across various levels of development.•Non-concave production functions can lead to the occurrence of multistability and complex dynamics.•The poverty trap can be exhibited.