# Price elasticity on comparison website. How to calculate price elasticity using the coefficient of the products position on site?

I am looking at the following paper. Glenn Ellison & Sara Fisher Ellison, 2009. "Search, Obfuscation, and Price Elasticities on the Internet," Econometrica, Econometric Society, vol. 77(2), pages 427-452, 03. Link: https://www.nber.org/papers/w10570

The model did not include price but did include rank ie. the position of a product on a comparison website sorted by price. They then used the model results to calculate price elasticity.

To summarise, the way they did this was: they treat log(1 + Rank) as a continuous variable and compute estimated elasticity when all variables are at their means by setting the derivative of Rank with respect to a change in Price equal to the inverse of the average distance between the twelve lowest prices. They concluded "the low-quality 128MB PC100 demand equation corresponds to an own-price elasticity of -25.0".

I can't understand this explanation, how is this price elasticity calculated?

• I understand where the $log(1+PLowRank)$ comes from. Essentially the steps are outlined in Section 2 of Chevalier and Goolsbee (2003). The authors have to determine the elasticity heuristically since the coefficient for $log(1+PLowRank)$ isn't helpful. So I think when they mention "setting the derivative of Rank with respect to a change in Price equal to the inverse of the average distance between the twelve lowest prices", they are referring to the derivative of the line interpolated among those 12 prices using the inverse distance weighing algorithm. – Kenneth Rios Oct 25 '18 at 14:55
• It is a standard theoretical procedure to go from rank to quantity by assuming a power law distribution over the rankings. Here is the Goolsbee paper. See section II that starts at the bottom of p. 7. I would work through it in an answer but I am hard-pressed for time this week. Apologies for that, but it is not too difficult though. – Kenneth Rios Oct 25 '18 at 15:05
• Thank you for your comments Kenneth. I read the Goolsbee paper and read into the inverse distance weighing algorithm but I must admit my understanding is still fuzzy. If you ever get the time I'd really like to see it worked through in an answer. – Sophie Oct 29 '18 at 3:59