12.1 Sales from Rank
use an experiment to infer demand
From low-selling books (where demand was either known or very small) and purchased large quantities (relative to their low demand)
Data from Amazon
As the ranks for the books changed, the relationship between the sales rank and demand can be inferred.
However, this is practical only to very low-selling books.
(Brynjolfsson, Hu, and Smith 2003b)
- With additional demand data from a book publisher, they can derive the relationship between the sales and rank.
- Without demand data, they can infer the demand data
On Apple’s App Store, there are three rank list:
- Top-free apps
- Top-paid apps
- Top-grossing apps: based on revenue generation
Assuming a Pareto distribution
\[ d_{r_p} = b_p \times r_p^{-a_p}|1 \le 200 \]
where
\(d_{r_p}\) = downloads at rank \(r_p\) in the top-paid
\(b_p\) = scale factor that is dependent on the total market size for iPad or iPhone apps
\(a_p\) = shape of the Pareto curve
For apps in the top-grossing list, we assume a Pareto distribution for each app (to estimate use a simple truncated OLS regression)
\[ \begin{aligned} pd_{r_g} &= p \times d_{r_p} = b_g \times r_g^{-a_g} \\ \log(r_g) &= \frac{1}{a_g} \times \log\left( \frac{b_g}{b_p} \right) + \frac{a_p}{a_g} \times \log (r_p) - \frac{1}{a_g} \times \log(p) \\ \log(r_g) &= \beta_0 + \beta_1 \times \log(r_p) + \beta_2 \log(p) \end{aligned} \]
where
\(p\) = product price
\(b_g\) = scale factor that is dependent on the total market size for iPad or iPhone apps
\(d_{r_p}\) = downloads of the same app in the top-paid list
Assumption: top-grossing apps generate revenue from upfront pricing only
Free and paid apps may have additional purchasable features inside the app, but these purchases will not be included in the paid apps (which is reasonable since paid apps make most of their money from upfront prices) (p. 1256).
It’s reasonable to assume that in-app features are most common in free apps (p. 1256)
Apps’ rank and price are assumed to be independent across sectional data (even when an app appears multiple times).
\(a_g = - 1/\beta_2\)
\(a_p = -\beta_1/\beta_2\)
\(\frac{b_g}{b_p} = \exp(-\beta_1/\beta_2)\)
To recover individual values of the scale parameters, we aggregate downloads across apps in a day (since actual downloads for an app is not available) \(D_t = \sum d_{r_p} = b_p \sum_{r=1}^N r_p^{-a_p}\)
Using the total number of downloads of top ranked apps, the two parameters are
\(b_p = (\sum_{r_p=1}^N d_{r_p})/(\sum_{r_p = 1}^N r_p^{-a_p})\)
\(b_g = \exp(-\beta_0 / \beta_1)\times (\sum_{r_p=1}^N d_{r_p})/(\sum_{r_p = 1}^N r_p^{-a_p})\)
Data:
Apple, Appshopper (shutdown in 2021), AppAnnie (now is data.ai), Mobilewalla
Period: April 2011 - May 2011
Data: 200 paid apps, 200 grossing apps and their price data.
(He and Hollenbeck 2020) On Amazon, which is a generalized results from (Chevalier and Goolsbee 2003)