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[jpfeifer]

Take a look at Fernandez-Villaverde Rubio-Ramirez (2006) - A Baseline DSGE Model. The lambdas in that equation multiply with S' which is 0 in steady state. Because of this, they drop out at first order in the first equation. The same does not happen in the second equation.

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[jpfeifer]

The S'' is a typo in this line as is the exp(q_t) after the expected value, which should obviously be q_{t+1}. The FOC is just rewritten so the S' from the previous line should stay S'. It only becomes S'' during linearization (which is why the lambdas and the q_{t+1} drop).

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[jpfeifer]

Please check your computations. As I said, in the rewritten and not yet linearized equation, it should be S' not S'' as there has been no transformation performed, only a replacement of the variables x by exp(log y). In the next line (linearized equation), when taking the derivative the treatment of investment I is different than the one of q and lambda. The S'' appears when taking the derivative with respect to investment (it is the inner derivative). This explains why the investment terms do not drop as S'' >0 in steady state. In contrast, when taking the derivative with respect to q and lambda, the S' remains, which is 0 in steady state and explains why these variables drop.

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[jpfeifer]

It''s simple derivative rules. When taking a multivariate first order Taylor approximation of lambda*S' you get

lambda'(SS)*S'(SS)*dlambda+lambda'(SS)*S''(SS)*dS'

where the first term drops because S'(SS)=0.

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[jpfeifer]

They do disappear in the linearized equation. This is a mathematical fact, not an assumption of SW. In the second term, the elements belonging to lambda an Q are evaluated in steady state where lambda(+1)/lambda is 1. Similarly, the steady state of Q(+1) is 1. They are NOT variables, but constant terms in the partial derivative with respect to investment. I think you should read up on linearization and multivariate Taylor approximations.

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[jpfeifer]

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.