# Log linearization of Tobin Q equation (SW, 2003) **URL:** \ **Category:** General DSGE Modeling\ **Created:** [9 December 2023 17:42 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730 "2023-12-09T17:42:06Z")\ **Posts on this page:** 9\ **Page:** 1
**Author:** ![MarcoUrbani](https://forum.dynare.org/letter_avatar_proxy/v4/letter/m/f6c823/32.png) [@MarcoUrbani](https://forum.dynare.org/u/MarcoUrbani)\ **Post date:** [9 December 2023 17:42 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/1 "2023-12-09T17:42:06Z")
Hi everybody, I am struggling with the following log linearization (it is the FOC for K(t) in SW, 2003); in particular, given that functional form for the capital utilization function, I was wondering whether I have to take first order taylor expansion of the entire function when log linearizing(in that case, I use the entire function evaluated at utilization rate=1 which is the stady steady state value of capital utilization as a pivotal point) or just of the exponential term (in that case, I just use utilization rate = 1 as a pivotal point). I’ve tried both ways, but I am not able to obtain equation 30. Would anyone be so helpful to give me some insights? Thank everybody for the attention [tobin q log linearization.pdf](https://forum.dynare.org/uploads/short-url/4dGBz29D5Gto0cFwqABGrsNTTjR.pdf) (443.3 KB) ---
**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\ **Post date:** [11 December 2023 13:12 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/2 "2023-12-11T13:12:45Z")
You need to do a full Taylor of the whole equation. ---
**Author:** ![MarcoUrbani](https://forum.dynare.org/letter_avatar_proxy/v4/letter/m/f6c823/32.png) [@MarcoUrbani](https://forum.dynare.org/u/MarcoUrbani)\ **Post date:** [12 December 2023 00:50 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/3 "2023-12-12T00:50:53Z")
What do you mean exactly by doing a full Taylor expansion of the whole equation? I have always done log linearization on the single terms (ie. x’=log(x(t)) - log(x) ==\>x(t)=x_exp(x’)==\>x(t)=x_(1+x’)) and I have obtained the rest of the equations in the SW paper this way. In this particular case, I do not kwon how to treat the exponential term in the variable capital utilization function, as if I applied the aforementioned method, I wouldn’t get rid of the exponential. Thanks for being helpful. ---
**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\ **Post date:** [12 December 2023 12:24 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/4 "2023-12-12T12:24:56Z")
I case like this it’s easier to do a Taylor approximation than the Uhlig trick. {e^{\psi \left( {{u\_{t + 1}} - 1} \right)}} \approx \psi {e^{\psi \left( {u - 1} \right)}}\left( {{u\_{t + 1}} - u} \right) = \psi {e^{\psi \left( {u - 1} \right)}}u\frac{{\left( {{u\_{t + 1}} - u} \right)}}{u} = u\psi {e^{\psi \left( {u - 1} \right)}}{\hat u\_{t + 1}} ---
**Author:** ![MarcoUrbani](https://forum.dynare.org/letter_avatar_proxy/v4/letter/m/f6c823/32.png) [@MarcoUrbani](https://forum.dynare.org/u/MarcoUrbani)\ **Post date:** [14 December 2023 10:38 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/5 "2023-12-14T10:38:51Z")
Since the steady state value of capital utilization rate is 1 by assumption, is it correct to claim that the resulting value for that expression is u(hat)(t+1)? If so, can I procceed with the Uhlig trick in the rest of equation? I have tried so but I get something very different from the SW paper. Thank you in advance ---
**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\ **Post date:** [14 December 2023 12:04 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/6 "2023-12-14T12:04:21Z")
Given that the steady state is 1, percentage deviations and absolute deviations are the same. Thus, the answer should be yes. ---
**Author:** ![Dean](https://forum.dynare.org/letter_avatar_proxy/v4/letter/d/3be4f8/32.png) [@Dean](https://forum.dynare.org/u/Dean)\ **Post date:** [12 February 2026 09:34 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/7 "2026-02-12T09:34:29Z")
Dear @jpfeifer I am a little bit confused with log linearization of time-varying variables in exponent, what if the function is {\psi(u\_{t+1}^{x\_{t+1}}-1)} ---
**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\ **Post date:** [3 March 2026 11:13 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/8 "2026-03-03T11:13:21Z")
It should be something like \begin{gathered} \psi \left( {u\_{t + 1}^{{x\_{t + 1}}} - 1} \right) = \psi \left( {{e^{{x\_{t + 1}}\ln {u\_{t + 1}}}} - 1} \right) \hfill \\ \approx \psi \left( {{u^x} - 1} \right) + \psi \ln u{e^{x\ln u}}\left( {{x\_t} - x} \right) + \psi x\frac{1}{u}{e^{x\ln u}}\left( {{u\_t} - x} \right) \hfill \\ \approx \psi \left( {{u^x} - 1} \right) + \psi \frac{{\ln u}}{x}{u^x}\frac{{\left( {{x\_t} - x} \right)}}{x} + \psi x{u^x}\frac{{\left( {{u\_t} - x} \right)}}{u} \hfill \\ \approx \psi \left( {{u^x} - 1} \right) + \psi \frac{{\ln u}}{x}{u^x}{{\hat x}\_t} + \psi x{u^x}{{\hat u}\_t} \hfill \\ \end{gathered} ---
**Author:** ![Dean](https://forum.dynare.org/letter_avatar_proxy/v4/letter/d/3be4f8/32.png) [@Dean](https://forum.dynare.org/u/Dean)\ **Post date:** [5 March 2026 03:42 UTC](https://forum.dynare.org/t/log-linearization-of-tobin-q-equation-sw-2003/24730/9 "2026-03-05T03:42:59Z")
Dear @jpfeifer, thanks a lot. I believe it should be: {\approx\psi(u^x-1)+\psi xu^x\ln u \hat{x}\_{t+1}+\psi xu^{x}\hat{u}\_{t+1}}