Can we do welfare analysis after log_linearizing?

Dear all ,I have learned that we can do welfare analysis using a recursive definition of welfare like Welfare = Util + beta*Welfare(+1).
But this is based on the first order condition of the model, without log-linearizing it.
But it is often hard to calculate steady-state values of all the endogenous variables and log-linearizing it will make it easy.
Here comes the question, Can we do welfare analysis after log_linearizing?
If could, would you give us some example papers?
Thanks

The general answer is no, you cannot do this. Due to certainty equivalence, stochastics will not affect the welfare ranking. In some cases, you can do a naive linear-quadratic approach, but generally, full second order is required. See Woodford (2002): “Inflation Stabilization and Welfare” for a thorough discussion