How to Calculate the Marginal Effect When a Variable is Zero?

[ad4stra]

Hi everyone! I have (I think a major) problem so I'm hoping you could give me some useful advice.
I am working on my thesis where I am statistically estimating this type of equation:

dlny=α+β₁*dlnx+β₂*dlnz+β₃*dlnx*dlnz

where dln stands for differencing a natural logarithm of the variable, eg

dlnz=ln(z_t)-ln(z_t-1)

so basically the approximation of the growth rate of x.

After I estimate this equation, I get certain values for α and for the βs and I have this nice equation.

My research question is:
What will be the marginal effect of an increase in dlnx on dlny if z doesn't exist (ie it is zero).

Obviously, the derivative of dlny with respect to dlnx is

∂lny/∂lnx=β₁+β₃*dlnz

and now I need to calculate how much that is when z (not dlnz) doesn't exist.

Obviously, I have problem since dlnz in this case isn't defined:

dlnz=ln(0)-ln(0)

Is there any way I can manipulate mathematically my problem in order to get an answer to my question??

Thank you very much in advance!

 

[ad4stra]

* so basically the approximation of the growth rate of z