# Composite consumption assumptions of SOE models comes from how we model exports?

From reading the literature, I have found different ways authors model consumption in SOE models. While I have not found a detailed discussion in this regard, I have observed the following…not sure though if my observation is entirely correct? Whenever you have the time you can kindly leave a comment…thanks!.

So on the household side of SOE models, the household can trade

1. Only foreign assets (B^*_t)
2. Or foreign assets and consumer goods. So we get composite consumption C_t = CES(C^h_t, C^f_t) as in Gali. And then we can do things like S_t = \frac{P^f_t}{P^h_t}, P^f_t = {\epsilon_t}{P^*_t} (LOOP), and so on.

The production side of SOE models should not necessarily have exports, but if we want to include exports, then there are two ways:

1. One sector model: Y_t = C^h_t + X_t, where X_t is exports.
2. Two sector model: Y_t = Y^N_t + Y^T_t, where Y^T_t is sometimes the exportable good.

So the above gives 4 possible combinations of modeling consumption at the household side of SOE modelS, right? Thus

1. Only foreign assets, and One sector model (C_t).
2. Only foreign assets, and Two sector model (C_t = C^N_t + C^T_t).
3. Foreign assets, trade in consumer goods, and One sector model (C_t = C^h_t + C^f_t)
4. Foreign assets, trade in consumer goods, and Two sector model (C_t = C^h_t + C^f_t, where C^h_t = C^N_t + C^T_t). Sometimes other people also do (i.e., C_t = C^T_t + C^N_t, where C^T_t = C^h_t + C^f_t)…I guess they are similar.

So these are the four ways of modeling consumption in SOE models, right? So for example, if I am using a 2-sector (SOE) model and I do not care about S_t = \frac{P^f_t}{P^h_t}, P^f_t = {\epsilon_t}{P^*_t} (LOOP), and so on as in Gali, then I would go for number 2, right? Thus, C_t = C^N_t + C^T_t, and I do not need C^f_t in the model, I guess. Or maybe these observations are wrong? Thank you.