Physica A ( ) – Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Collective purchase behavior toward retail price changes Hiromichi Ueno a,∗ , Tsutomu Watanabe b , Hideki Takayasu c , Misako Takayasu a a Department of Computational Intelligence & Systems Science, Interdisciplinary Graduate School of Science & Engineering, Tokyo Institute of Technology, 4259-G3-52 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan b Institute of Economic Research, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8603, Japan c Sony Computer Science Laboratories Inc., 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan article info Article history: Received 5 September 2010 Available online xxxx Keywords: Collective behavior Power law POS data Log-normal distribution abstract By analyzing a huge amount of point-of-sale data collected from Japanese supermarkets, we find power law relationships between price and sales numbers. The estimated values of the exponents of these power laws depend on the category of products; however, they are independent of the stores, thereby implying the existence of universal human purchase behavior. The rate of sales numbers around these power laws are generally approximated by log-normal distributions implying that there are hidden random parameters, which might proportionally affect the purchase activity. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Developments in information technology have enabled the storage of large volumes of high-frequency data of human activities, and soon, scientists began paying attention to such data [1]. People act intentionally based on their own will; in this sense, human behavior should be very different from the motion of materials. It will be very difficult to find a universal law for individual behavior, which may be based on private preferences or habits; however, there is a possibility that universal statistical laws can be found in collective human behavior. Several pioneering studies have reported possible universal laws in the mass of human activity. At the end of the nineteenth century, Pareto investigated individual income distribution in many countries and found that power laws were dominant in the case of the high-income group of people [2]. In 1949, Zipf listed power law distributions in various types of human behavior from word-frequency to city population [3]. Shockley pointed out that the distribution of the productivity of scientists followed a log-normal law in 1957 [4]. In 1963, Mandelbrot found scale-invariance and a power law distribution in the market prices of cotton [5], and in 1981, Montroll analyzed the price distribution of products, and found that the distribution follows a log-normal law [6]. Electronic databases became available from the end of the last century, and the quality of data analysis rose considerably. In 1995, Mantegna and Stanley confirmed power law distributions of market price changes [7]. M.H.R. Stanley et al. surveyed business firm databases and discovered that the variance of the growth rate of annual sales of a firm decreases following an inverse power law of its sale in 1997 [8], and Redner found a power law distribution in scientific citations in 1998 [9]. Recently, sales data such as point-of-sale (in short POS) data are studied from the viewpoint of physics. Sornette et al. analyzed a time series data of book sales obtained from Amazon.com and found that a functional form of increase and decrease in bestsellers can be approximated by power laws [10,11]. Groot observed fluctuations in sales using sales data collected from Dutch supermarkets and observed that these fluctuations exhibit properties similar to those of the stock market [12]. Fu et al. reported a universal growth rate distribution through an exhaustive investigation of various economic activity data such as the POS of products, business firm’s sales, and even GDP [13]. Mizuno et al. focused on the amount ∗ Corresponding author. E-mail address: ueno@smp.dis.titech.ac.jp (H. Ueno). 0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.09.032