Log-linearization of adjustment cost

I am trying to log-linearize an FOC. I am doing this because I have already log-linearized rest of the model except this FOC. It contains the following adjustment cost (as in Gerali et al, 2010)

- \frac{\kappa_{B}}{2}\left(\frac{K_{t}^{B}}{L_{t}} - \nu_{B}\right)\left(\frac{K_{t}^{B}}{L_{t}}\right)^{2}

where K_{t}^{B} is bank capital, L_{t} is bank credit and \kappa_{B} and \nu_{B} are constants. I wonder how to correctly log-linearize this term. I took a stab at this and I log-linearized it like this:

- \kappa_{B}\left( \frac{K^{B}}{L} - \nu_{B} \right)\left( \frac{K^{B}}{L} \right)\left( \widehat{K_{t}^{B}} - \widehat{L}_{t}\right) - \frac{\kappa_{B}}{2}\left( \frac{K^{B}}{L} \right)^{2}\left( \frac{K^{B}}{L} - \nu_{B}\right)\left( \widehat{K_{t}^{B}} - \widehat{L_{t}}\right)

Can I please ask whether it looks correct and if there’s a way to check this? Will appreciate any advice and pointers.

Hi VS19,

Your third term from the left in the second line should be squared (see in red).
Indeed, if you account for this, you can re-write your whole second line more compactly as follows:

- \frac{3}{2}\kappa_{B}\left(\frac{K^B}{L}\right)^{\textcolor{red}{2}}\left(\frac{K^B}{L} - \nu_B\right)\left(\widehat{K_t^B}-\hat{L_t}\right)

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