Does it matter in which currency goods are denominated in international trade? If prices are fully flexible, it should not matter. If nominal prices are rigid, it could matter. Models typically assume either producer currency pricing or local currency pricing, with the truth lying somewhere in between.
Michael Dotsey and Margarida Duarte build a model of international trade that allows to move between the two pricing regimes. They find that the economies are essentially equivalent, save for the behavior of the terms of trade. But empirically, that is not helpful either, seeing that real exchange rates are random walks. This implies that one should not bother with complex pricing mechanisms and focus on the real side of the economy, at least when studying the United States. Outcomes may be different in economies that are more involved in international trade.