Interpreting log in regression

[sarahgi]

Good morning,
I am new to this forum and would like some help with the interpretation of the following regression. I hope I am in the right place.

logWaste = -1.85(logGDP) +0.1(logGDP)^2

Now I am trying to give an interpretation of what would happen to Waste if GDP increases by x%.
I usually report the elasticity easily but I am not sure how to work with this since I have two different values of GDP that I should take into account when differentiating. I hope this makes sense and someone can give me a hand.

Many thanks to all!

 

[ksdhart2]

Letting \(w\) denote the Waste and \(g\) denote GDP, and assuming that we're using the natural log, then the problem can be restated as:

\(\displaystyle \ln(w) = -1.85 \ln(g) + 0.1 \ln^2(g)\)

If we raise \(e\) to both sides, to counteract the natural log, we get:

\(\displaystyle e^{\ln(w)} = e^{-1.85 \ln(g) + 0.1 \ln^2(g)}\)

\(\displaystyle w = e^{-1.85 \ln(g) + 0.1 \ln^2(g)}\)

By continuing to simplify this down into a "nicer" form, you can get to the point where taking the derivative to find \(\displaystyle \frac{dw}{dg}\) will be a matter of applying the quotient rule, which shouldn't be that bad.