# NK Wage Phillips Curve a la SGU or EHL or doesn't really matter

Hi there,
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.

Best,
Peter

That question is really hard to answer. The wage PC gives you a relation between the aggregate MRS gap and inflation. In the SGU setup the functional form of the felicity function does not matter by construction. In the EHL framework, you have to trace out how aggregate wage changes translate into changes in idiosyncratic hours worked (the \varepsilon_w) and how the idiosyncratic hours worked affect the idiosyncratic utility (the \phi ternm that measures the total elasticity of the MRS). The last term is sometimes hard to derive as you correctly point out.

Now you are introducing a multi-sector structure. In that case it is hard to predict what will happen as the Phillips curve and the terms appearing in it might take a different form than in the standard NK model. But nevertheless, the SGU case will by construction be easier to implement as it simplifies aggregation considerably. And in terms of restrictiveness of assumptions, there is not that much of a difference. Either you need complete markets (SGU) or pooling via unions (SGU).

Many thanks Johannes, I will definitely go for the SGU case, no need to make things too complicated.

Best,
Peter