There are two "that" in your sentence. Which one are you referring to?
understand why many economists believe THAT unrestricted free trade between nations will raise the economic welfare of countries THAT participate in a free trade system.
The first one refers to whole subordinate clause that follows:
Economists believe: Unrestricted free trade between nations will raise the economic welfare of countries...
The second one could be replaced by "which":
The economic welfare of countries WHICH participate in a free trade system