How to model commodity price when the economy is a 'large' economy

In SOE DSGE models for commodity-exporting economies, commodity price is assumed to follow an exogenous process. The most basic one is P_t = \rho P_{t-1} + e_t, where e_t is the commodity price shock. But the source of e_t is often not discussed. I guess it doesn’t matter much here whether e_t is associated with a global supply or a global demand shock.

But if the commodity-exporting economy is ‘large’ and can affect world prices. Do the following statements sound correct?

  1. If we still use P_t = \rho P_{t-1} + e_t when the economy is ‘large’, then e_t essentially emanates from a global demand shock or supply shock of other producers.
  2. If the commodity-exporting economy is ‘large’, can we assumed the following equation for how commodity price evolves: P_t = \rho_0 C_t + \rho P_{t-1} + e_t,\;\; \rho_0<0. Where C_t is the commodity output of the commodity-exporting economy, and \rho_0 captures, for example, the openness of the economy and how strongly it affects world prices (P_t). So here, P_t is determined both endogenously via C_t and exogenously via e_t.

I have tried to find if there exists a more conventional way of modeling commodity prices when the economy is large, but it appears there are no conventions unlike in SOE models. Thanks for any comment!!