Second Order Approximation Welfare Loss Function

I’m looking for the second order approximation of a two-agent economy’s welfare loss function, following the F. Bilbiie’s (2008) approach.
My question is the following: as I’m trying to express the welfare loss in terms of inflation and output, after I derive the second order approximation of households’ utility, can I do substitutions employing log-linearised optimality conditions approximated up to first order?

Thanks in advance,
Francesco Ricciutelli

Generally, the answer is no, unless there is an undistorted steady state.

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