We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker–Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power laws that describes how the number of cascades changes over time (cascades are consecutive transitions that end when the initial state is reached). The partial differential equation has a closed form solution which gives an explicit dependence of the number of cascades on their size and on time. Furthermore, the power law solution has a natural cut-off, a feature often seen in empirical data. This is due to the finite size a cascade can have in a finite time horizon. The derivation of the equation provides a justification for an exponent equal to 2, which agrees well with several empirical distributions, including Richardson’s Law on the size and frequency of deadly conflicts. Nevertheless, the equation can be solved for any exponent value. In addition, we propose an urn model where the number of consecutive ball extractions follows a power law. In all cases, the power law is manifest over the entire range of cascade sizes, as shown through log–log plots in the frequency and rank distributions.

The movie industry is a highly differentiated context where production studios compete in non-price product attributes, which influences the box office results of a motion picture. Because of the short life cycle and the constant entrance of new competitive products, temporal decisions play a crucial role. Time series of the number of movies on release and the sum of the box office results of the ten top motion pictures (ranked by box office result for that week) present a counterphased seasonality in the US movie market. We suggest that a possible reason is a risk sensitivity adaptation in the behaviour of the movie’s distributors. This paper provides a model supporting this hypothesis. We developed an agent-based model of a movie market, and we simulated it for 15 years. A comparable global behaviour exists when producers schedule the movies according to given risk-sensitive strategies. This research improves the knowledge of the US motion picture market, analyzing a real-world scenario and providing insight into the behaviour of existing firms in a complex environment.

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