Price elasticity in logistic regression with log price

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?

• I am not sure if your derivation is correct. See here for the log price case. The levels elasticity derivation is pretty similar. – Dimitriy V. Masterov Aug 6 '18 at 20:39
• My derivation is correct, elasticity is dQ/dp * p/Q, not just dQ/dp. – Daria Aug 7 '18 at 21:20