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 Uhligâ€™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 â€ś=â€ť 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?

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No, you can easily mix them.

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