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.