How to properly interpret logged interaction variables

Question

I have the following regression to interpret elasticity of demand:
$$ln(demand) = const - 0.6*ln(fare)$$
I understand that a 1% increase in fare results in a 0.6% decrease in demand
I want to add dummy variables for days of week (excluding one of course) and want to add day-of-week $* ln(fare)$ interactions so that I can determine the elasticity of demand by day of week (which I already know differs)
For simplicity in writing this out let’s pretend there are 3 days: mon, tue, wed and I will leave Wednesday out as my base
So now my regression is:
$$ln(demand) = const - 0.45*ln(fare) + 10*(mon) + 15*(tue) + 0.05*ln(fare)*(mon) - 0.15*ln(fare)*(tue)$$
Here, my elasticity of demand for Wednesday is interpreted as a 1% increase in fare results in a 0.45% decrease in demand
But my question is how to interpret elasticity of demand for Tuesday. I know that the interaction and even the dummies are relative to the base.
So the question is for Tuesday should The coefficients be added? Or multiplied since it’s logged?

1muflon1 · Answer

You have to add them even though they are in $ln$ the model is still linear in the parameters. To see this more clearly you can rewrite your equation as (I use $D$ for demand, $c$ for constant, $F$ for fare and $M$ and $T$ for Monday and Tuesday):
$$ln(D) = c  + left(- 0.45  - 0.15(T)  + 0.05(M) right)ln(F) + 10(M) + 15(T)$$
Hence when both $T=0$ and $M=0$ the elasticity for Wednesday is -0.45, when $T=1$ and $M=0$ (i.e. its Tuesday) the elasticity is $-0.6$, and finally when $T=0$ and $M=1$ (i.e. its Monday) the elasticity is $-0.4$.
If you want to learn more Wooldridge discusses this in his introduction to econometrics in chapter 7.

How to properly interpret logged interaction variables

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