The Challenges of Using Ranks to Estimate Sales

Researchers have frequently used publicly available data on product ranks to estimate nonpublic sales quantities under the assumption that the distribution of sales follows a power law. Using population sales data for a product frequently thought to follow a power law—books—we find the (double logged) rank-sales relationship, contrary to assumption, is not linear but is instead concave. We demonstrate that this concavity is sufficiently strong to require a reevaluation of hundreds of results from analyses that had assumed linearity and we go on to demonstrate that the nonlinearity can cause poor predictions of sales for either poor selling or high selling titles, and sometimes both. We illustrate some of the problems of applying a linear technique to a nonlinear relationship by examining the claim that the greater product variety made available to shoppers on the Internet has a large positive impact on social welfare and also claims about sales levels in top 20 and top 50 “charts.” In spite of these difficulties, the concavity appears to be sufficiently similar across time and book categories to allow the use of simple nonlinear specifications that provide reasonable predictions of sales from ranks using samples with only a modest number of observations.