Power Law of Customers’ Expenditures in Convenience Stores

We can observe a power law in critical phenomena in natural sciences. Such a power law is also empirically exhibited in many economic phenomena. In one of the older studies conducted approximately 120 years ago, Pareto first reported that the probability density function of personal wealth in Italy follows a power law [1]. Following this research, the power law was also found to hold for city size [2, 3], the circulation of magazines and newspapers [4], book sales [5], and the number of incoming links to a webpage from another webpage [6, 7]. For the last ten years, investigations on this subject particularly focused on the fact that the probability density functions of market price fluctuations [8, 9] and company size both follow power laws [10–17], based on the analysis of high-frequency data. Recently, point-of-sale (POS) databases containing detailed records of all customer purchases in shops were published and attracted the attention of both physicists and economists. For example, an accelerating power law growth for book sales was found in the time series of the book sales on Amazon.com [18]. Further, it was reported that approximately 20% of a store's products contribute to 80% of its sales [19, 20]. In this paper, we focus on a POS database for convenience stores. First, we provide a description of the POS database. Next, we investigate that the probability density function of the expenditure of a person during a single shopping trip. The top 10 % of the function follows a power law with an exponent of –2.5. Moreover, the exponent is independent of the store's location, the shopper's age, the day of week, and the time of day. In this study, we analyze a POS database containing detailed records of all purchases at a convenience store chain. The database comprises " NEEDS-SCAN/CVC receipt data " and is published by Nikkei Media Marketing, Inc. The convenience store chain sells many foods manufactured by Yamazaki Baking Co., Ltd. Therefore, we believe that the name of this chain is Daily Yamazaki Co., Ltd.; however, the Nikkei has not officially announced the chain's name. The database recorded the receipt number, the store location, the purchase date and time, the customer's age and sex, and the name of the item for each item purchased. The general location of a store is chosen from among the following four categories: in the vicinity of a railroad station, a business …