I build a model who contains two production sectors. Then I want to use OSR to find some optimal monetary policy. I have to derive the welfare function which contains output gap and inflation. Because I have two production sectors. It is so complex to derive the output gaps and inflation dispersion. But I set log consumption in utility function. So in second order approximation the welfare function contains only variance of labors. I wonder if I could just put variance of labor in the welfare loss function in OSR instead of deriving deriving complex output gaps and inflation dispersion. Thank you!
Your welfare function just needs to be correct. It does not need to be in terms of output gap and inflation. This is often just a convenient and intuitive representation.