Sorry for such a long question…but I am confused about this.
Let’s say we have the following budget constraint for the household (where B_t is foreign bond, \Pi_t is firm profit):
P_t C_t + B_t = W_tN_t + R_{t1} B_{t1} + P_t\Pi_t
Using the firm’s profit function,
P_t\Pi_t = P_tY_t  W_tN_t
We can derive resource constraint,
C_t + \frac{B_t}{P_t}  \frac{R_{t1} B_{t1}}{P_t} = Y_t

In GC Lim, PD McNelis  2008, there is a definition for export (X_t = f(\cdot)) in their models, so the authors define the following evolution of foreign debt \frac{B_t}{P_t} = \frac{R_{t1} B_{t1}}{P_t} + X_t so that goods market clearing condition is Y_t = C_t + X_t.

In J Galí  2015, they do a similar thing in chapter 8, where export (X_t = f(\cdot)) is defined, and goods marketclearing condition is Y_t = C_t + X_t (Assuming households does not consume foreign goods). However, there is no mention of foreign debt evolution ( \frac{B_t}{P_t} = \frac{R_{t1} B_{t1}}{P_t} + X_t). They just state that goods market clearing condition requires that Y_t = C_t + X_t. I guess this requirement would mean that \frac{B_t}{P_t} = \frac{R_{t1} B_{t1}}{P_t} + X_t (in incomplete market case), which can be derived by matching resource constraint (C_t + \frac{B_t}{P_t}  \frac{R_{t1} B_{t1}}{P_t} = Y_t) to the goods market clearing condition (Y_t = C_t + X_t)

In Valerio Nispi Landi notes, there is no export, so he defines trade balance as TB_t = \frac{B_t}{P_t}  \frac{R_{t1} B_{t1}}{P_t}. And goods market clearing condition is Y_t = C_t + TB_t.
So it seems to me one cannot have both export and trade balance variables in the model, right? For example, if you have export defined in the model, resource constraint suggests that \frac{B_t}{P_t} = \frac{R_{t1} B_{t1}}{P_t} + X_t. So you cannot also define TB_t = \frac{B_t}{P_t}  \frac{R_{t1} B_{t1}}{P_t}, which would just results in TB_t = X_t. I understand this makes sense since TB_t = X_t  M_t, and M_t = 0 here in the model.
But even when imports are not zero,
 Foreign debt evolution equation is defined as \frac{B_t}{P_t} = \frac{R_{t1} B_{t1}}{P_t}  M_t + X_t in GC Lim, PD McNelis  2008, so that domestic goods market clearing is Y^h_t = C^h_t + X_t
 Following Valerio Nispi Landi notes, trade balance would be TB_t = \frac{B_t}{P_t}  \frac{R_{t1} B_{t1}}{P_t} + M_t, so that domestic goods market clearing is Y^h_t = C^h_t + TB_t.
To me, it seems TB_t = X_t. However, if you put export demand (X_t = f(\cdot)) in your model, and you define foreign debt evolution \frac{B_t}{P_t} = \frac{R_{t1} B_{t1}}{P_t}  M_t + X_t, so that goods market clearing condition is Y^h_t = C^h_t + X_t (as in GC Lim, PD McNelis  2008 and Gali  2015), you cannot also define TB_t = \frac{B_t}{P_t}  \frac{R_{t1} B_{t1}}{P_t} + M_t in same model, right? Because it would just mean TB_t = X_t. Or maybe I am missing something here…
Or if we have export in the model, then we need to change the definition of trade balance from say TB_t = \frac{B_t}{P_t}  \frac{R_{t1} B_{t1}}{P_t} + M_t to TB_t = X_t  M_t. Thus, definition of trade balance is discretional and we can choose how we define it…
Thanks for any reply…