A Question on shock equation

Dear Professor Pfeifer,

As to a shock like housing demand shock {j}_{t} in utility function {{E}_{0}}\sum\limits_{t=0}^{\infty }{{{\beta }^{t}}\left( \log {{c}_{t}}+{{j}_{t}}\log {{h}_{t}}-{{\varphi }_{t}}{{n}_{t}} \right)}, the shock equation is written as \ln {{j}_{t}}=\left( 1-\rho \right)\ln {j}+\rho \ln {{j}_{t-1}}+{{e}_{t}}.

I wonder if I set utility function as {{E}_{0}}\sum\limits_{t=0}^{\infty }{{{\beta }^{t}}\left( \log {{c}_{t}}+{{\varepsilon }_{t}}j\log {{h}_{t}}-{{\varphi }_{t}}{{n}_{t}} \right)}, and use \ln {{\varepsilon}_{t}}=\rho \ln {{\varepsilon}_{t-1}}+{{e}_{t}}. Do they have same effect?

Thank you for reading this post.

Yes, that should be equivalent (except for the scaling of the standard deviation of e_t)