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For a regression model: Y = B0 + B1.X + B2.X2 + U, B1 and B2 is the marginal effect on Y wrt X and X2.

Now, for a log - log regression model, log(π‘Œ) = 𝐡0+ 𝐡1.log(𝑋) + 𝐡2.log(𝑋2) + π‘ˆ, B1 and B2 is the elasticity Y wrt X and X2.

In my case, I'd like to estimate elasticity Y wrt X, when X2 is not constant. i.e cross elasticity effect.

In my case, Y is units sold and X is price and X2 is available substitutes. The standard interpretation doesn't apply here because there is cross-elasticity effect of X2, number of available substitutes on Y.

How would I interpret elasticities and cross-elasticities in this case?

ps. I realize price is endogenous and need to add an instrument to deal with problem.

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