# Loglinearization of equation governing evolution of capital

I have a model in which bank capital evolves according to
K_{t}^{B} = (1 - \delta^{B})\psi_{t}K_{t-1}^{B} + \Pi_{t}
where \psi_{t} is an AR(1) shock in logs and \delta^{B} is a constant. I need to loglinearize the above equation. Is the following loglinearization correct? Will apreciate any help.
\widehat{K_{t}}^{B} = (1 - \delta^{B})(\widehat{\psi_{t}} + \widehat{K_{t-1}}^{B}) + \widehat{\Pi_{t}}

\hat{K_{t}^{B}} = (1- \delta^{B})* \psi_{t} * \hat{K_{t-1}^{B}} + \frac{\overline{\Pi}}{\overline {K^{B}}} \hat{\Pi_{t}}

@eisamabodian , thank you so much. Can I kindly ask, won’t \psi_{t} be loglinearized in the first term on RHS? And the first term on the RHS contains multiplication of two variables, won’t it make it non-linear?

If \psi_{t} is a fixed parameter in your model my
Log-linearized equation is true but if this is a variabe in your model therefore my linearized equation is not true and you can use the following equation :

\hat{K_{t}^{B}} = (1- \delta^{B}) \star \overline{\psi} \star ( \hat{\psi_{t}} + \hat{K_{t-1}^{B}} ) + \left( \frac{\overline{\Pi}}{\overline {K^{B}}} \right) \star \hat{\Pi_{t}}

In this new equation \psi_{t} is a variable in the model.

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Yes, the last one looks correct.

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