A simple view of the heavy-tailed sales distributions and application to the box-office grosses of U.S. movies

TL;DR

This paper introduces a simple stochastic model explaining the power-law distribution of sales, specifically applied to U.S. movie box-office grosses, and confirms its predictions with real data analysis.

Contribution

It presents a new, analytically derived model for sales distribution that accurately describes the power-law behavior observed in movie income data.

Findings

The model predicts the power-law exponent of ROI distribution from weekly income ratios.

Empirical data confirms the exponential decay of weekly income as predicted by the model.

The model effectively captures the heavy-tailed nature of sales distributions.

Abstract

This letter treats of the power-law distribution of the sales of items. We propose a simple stochastic model which expresses a selling process of an item. This model produces a stationary power-law distribution, whose power-law exponent is analytically derived. Next we compare the model with an actual data set of movie income. We focus on the return on investment (ROI), defined as the gross income divided by the production budget. We confirm that the power-law exponent of ROI distribution can be estimated from the ratios of income between two adjoining weeks, as predicted by the model analysis. Moreover, exponential decay of weekly income is observed both in the model and actual income. Therefore, the proposed model is simple enough, but it can quantitatively describe the power-law sales distribution.