I found a paper of yours Johannes comparing the derivation of the NK wage Phillips curve in the spirit of Erceg, Henderson, and Levin (2000) (and in Gali’s book) and Schmitt-Grohé and Uribe (2006b). I know pretty much that up to a first order approximation, the only difference is the (1-epsolon_w*phi)^-1 in the shope of the curve captured by, say, a parameter called kappa_w. It comes from turning the idiosyncratic marginal rate of substitution into an aggregate one in the EHL case sthe economy is populated by monopolistically competitive household members whereas SGU make the assumption of labour unions.
So my question is rather methodological.
When adhering to the EHL framework, we need to know the exact functional form of the linearized MRS. In the case of SGU that is never the case and we always end up with the standard slope ( (1-betatheta)(1-theta))/theta. Am I right?
I am asking since in my case of a multisector econoy (durable and non-durable sectors) the MRS won’t be the same as in the basic NK model. It isn’t really difficult to derive the NK Wage Phillips curve a la EHL in my case, but I was wondering whether it would really make sense since the SGU one seems to be rather general and, up to a 1st order approximation, we always end up with the same expression.