Log-Linearization of adjustment costs

Hi everybody,
I have a problem in log-linearizing the typical quadratic adjustment cost function. The entire expression (it is the FOC for I_{t} in SW’03) reduces to zero. How can I deal with it?
DYNARE TEST.pdf (72.4 KB)

Your equation (2) will have S'' appear during linearization due to the Taylor approximation introducing another derivative. That term will not be 0.

Are you meaning that I should treat S() and S'() as variables rather then expliciting functional forms, and then find out the expression for \hat{S} and \hat{S}' by Taylor approximation?

No, I am saying that in the FOC, you should have S' appearing, which will be I_t/I_{t-1}-1. When you linearly approximate this expression, you should obtain something like \hat i_t-\hat i_{t-1}, which is not 0.

In the FOC I have I_{t}S' with S'=\frac{\gamma}{I_{t-1}}\left[\frac{I_{t}}{I_{t-1}}-1\right]; can’t get why you have depreciation rate of capital. But the point is the following. When I perform log linearization by Uhlig trick, I get:
\frac{I}{I}\gamma\left[\frac{I}{I}-1\right]\left(1+\widehat{\frac{I_{t}}{I_{t-1}}}+\widehat{\frac{I_{t}}{I_{t-1}}-1}\right) which results in zero.
The same reasoning applies for the forward part of thr FOC, isn’t it?
Thanks for the support professor

The delta should have been a 1. I fixed it. I gave up on the Uhlig trick long ago due to messy issues like this. Try a Taylor approximation. Also, you are approximating in the individual variables, not full expressions.

If I perform a Taylor approzimation on the whole equation, I will do a Taylor approximation in 4 variables right? I can’t arrive to the log linear form proposed by SW.
(ie.x_{t}=\frac{I_{t}}{I_{t-1}}, x_{t+1}=\frac{I_{t+1}}{I_{t}}, Q_{t}, Q_{t+1})
I can’t arrive to the log linear form proposed by SW by doing so.

You should have 6 variables. You can try