by jpfeifer » Fri Apr 08, 2016 4:14 pm
This is the last time I am going to post here regarding that matter, because I keep repeating myself. What SW do is that they loglinearize the nonlinear first order condition. While the marginal utilities appear in the nonlinear equation, they drop in the first order loglinear approximation, because the contribution of changes in marginal utility on the equation is 0 up to first order. This can be directly seen when conducting the loglinearization of the equation.
Loglinearization is conducted by taking a first order approximation around the deterministic steady state, i.e. conducting a multivariate Taylor expansion about the steady state. When doing so, the derivatives appearing in front of the actual variables are evaluated in steady state. As outlined above, in that process the first order partial derivatives of the equation with respect to marginal utility today and tomorrow as well as with respect to Tobin's q tomorrow are 0, because of the S' term evaluated in steady state appearing. This is standard knowledge in the literature and the reason why I recommended you to review the algebra behind this.
If you think this is not correct and have a different explanation, you are of course entitled to your opinion.