Log-linearization of current account equation

Hello everyone,

I am working on my thesis, doing a two-country model with traded and non-traded goods considering incomplete asset markets and imperfect pass-through. In a sense, I am following Benigno & Thoenissen (2008).

I have been having troubles in log-linearizing my current account equation, which is given by

\begin{gather} \frac{S_tB_{F,t}}{P_t(1+i^*_t)} \frac{1}{\Theta\Big(\frac{S_tB_{F,t}}{P_t}\Big)} - \frac{S_tB_{F,t-1}}{P_t} + \frac{P_{H,t}}{P_t}Y_{H,t} + \frac{P_{N,t}}{P_t}Y_{N,t} - Z_t \end{gather}

where \Theta(\cdot) is the cost function that drives a wedge between the return on foreign currency denominated bonds and Z_t = C_t + I_t.

I have seen in different papers that it is standard to assume that b_t = \frac{S_tB_{F,t}}{P_t} and \Theta(b)=1, where b is the steady state value, and that the steady-state ratio of net foreign assets to GDP, which would be in my case \bar{b}/\bar{Z} is assumed to be zero.Let \varepsilon = -\Theta '(b)Z.

Moreover, I have seen that it is standard to say that \hat{b}_t is the deviation of foreign currency denominated bond holdings from their steady state, relative to domestic GDP. Hence, the log-linearized expression of the current account takes the form of

\begin{gather} \beta \hat{b}_t = \hat{b}_{t-1} + ... \end{gather}

My question is how can I log linearize the left hand side of my current account expression? Because from what I have read, a normalization with the GDP is used, so \hat{b}_t is not a typical log-linearization per se.

I would appreciate very much any help.

Alejandro

Simply divide the equation by GDP, define the resulting ratio X_t\equiv\frac{B_t}{Y_t} as a new variable and then loglinearize in X_t. Of course, you will need to add the linearization of X_t\equiv\frac{B_t}{Y_t} as a new equation to your model.

I understand that part. But I have read that papers assume that the steady state value of the X you defined is zero. Given my understanding of the log-linear method and Uhlig’s method, doing that under that assumption would lead to zero the first term of the left hand side equation and the right hand side. And that is what I do no understand on how to proceed with that, because I know I need to have that term in the final log-linear equation.

I just noticed I missed an “=” sign in the current account equation I wrote. It should be like this:
\begin{gather} \frac{S_tB_{F,t}}{P_t(1+i^*_t)} \frac{1}{\Theta\Big(\frac{S_tB_{F,t}}{P_t}\Big)} = \frac{S_tB_{F,t-1}}{P_t} + \frac{P_{H,t}}{P_t}Y_{H,t} + \frac{P_{N,t}}{P_t}Y_{N,t} - (C_t+I_t) \end{gather}

If the steady state of that term is 0, you typically linearize instead of loglinearizing. The expression is already in percent, though in percent of GDP.

Then I am a little confused. Would that imply that the whole current equation should be linearized instead of log-linearized, or can I mix a linearization for what is needed and a log-linearization for the rest of the terms?

No, you can easily mix them.