I'm estimating demand and calculating price elasticity using logistic regression.
In logistic regression with level price, elasticity is $$ \alpha*price*(1-share)$$ while if one uses log of price, elasticity is $$ \alpha * (1-share) $$ I've noticed that if I estimate regression using level price, my elasticities vary highly within products. However, when I use log of price, elasticities for different products are very close to each other. I understand that it is because I have 22 products, and thus shares are very small and elasticities are mostly driven by $\alpha$.
Could someone please explain me on a high level why this happens. And should I use the regression with level price if I do not trust the elasticity estimates of the log price regression?