Error term in log-level models?

I am running a decomposition of log wages for two time periods and want to explain the variance of the error term.

My question is: If there is a wage growth trend, will this automatically increase the variance of my error term over time (keeping the explanatory power of other variables in the wage decomposition constant)?

In a model of level wages, a doubling of all wages would increase the error variance by a factor of 2^2=4 I believe.
How are things in a model of log-wages, and -if necessary- how do I control for the influence of wage growth on error variance?

Ultimately, I am interested in whether the variance of the error term declines over time due to a (hypothesized) increasing explanatory power of the variables in my wage regression.