Econometrics: Is elasticity meaningful in my, or any, regression?
The answer to the question is yes, it is indeed meaningful (at least mathematically speaking). If you estimate the linear equation

$$ W = \beta_0 + \beta_1 PTR, $$

then $\beta_1=\frac{\partial W }{\partial PTR}$, meaning that $\beta_1$ represents the marginal change of $PTR$ over $W$. Now, if you estimate

$$ log(W) = \beta_0 + \beta_1 log(PTR), $$

then $\beta_1=\frac{\partial W}{\partial PTR}\cdot\frac{PTR}{W}$, which is the very definition of elasticity.

Generally speaking, linear transformations only affect the interpretation given to the coefficients, but the validity of the regression itself (in broad economic terms) is given by the model's assumptions and the economic phenomena being analyzed.