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So are there any circumstances where two countries trade with each other and they both do not gain anything? is it possible? and if you could also explain why that is the case? I have looked around and haven't really been able to find any answers.

Thanks in advance

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If I understand well your point, you want to know whether 2 agents (say countries) can make transaction if they don't each gain from that trade.

In general, agents optimize some function, called the objective, with respect to some constraints, and 2 agents trading with each other will keep trading until both marginal utilities are equal (first-order conditions of the maximization problem). So to answer your question, an agent won't trade if that trade does not increase her utility. Having said that, in a world where there are some additional constraints (in addition of the budget constraints), or some externalities you can end-up in situations where agents trade while it is not strictly speaking beneficial for them. Specifically, there may be some rules that require some externalities to be strictly lower than some values, and that would yield to trade that indeed reduce the utility of the agent.

As an example, let's think about a country that is endowed with a technology which produce some output and some externalities, say pollution. Then without going through the specifics of such a model, if (once the equilibrium is reached) a new rule says that the amount of pollution that a country can produce needs to be lower than some value (which is assumed to be higher than what was originally produced), the country may need to trade with some other country (with higher cost, but lower pollution) to fulfill the requirement, then resulting in less utility than previously but the require amount of pollution.

To sum-up, in general (free market, with perfect competition, etc.) this won't be the case, but such situations could arise in the presence of additional rules (or externalities that need to be internalized or additional constraints to the maximization program).

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The reason you probably find it hard to think of a case where this happens is because rarely do countries go around trying to make poor trades (read: never). If you trade real goods, there will almost always be a comparative advantage of trade to be had as long as countries have different production possibility frontiers in different ratios. Different opportunity costs of making more of one good means you might want another countries' labor to offer that service.

I can't give any specific examples of trades that were bad for each party where each party also was trying to act in there own self interest. Maybe if two countries were trading public debt with each other before a recession, they might end up with riskier assets than they bargained for, so even though the expected value of the debt was worth enough before a negative growth shock, you might get something less favorable in practice. Even then I don't think a situation like that is very likely to happen.

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There are two circumstances where there are no gains from trade that I am familiar with. However, there are always non-negative gains from trade in the standard model. If a trade was bad, the countries simply reject it, it is a consensual trade.

First, if the opportunity costs are equal between the two countries, there is nothing to gain from specialization, the countries are identical and there is no benefit from producing the good abroad rather than at home.

Second, if both countries are entirely interested in only one of the goods, there can be no appreciable gains from trade. Suppose they produce nothing but gold or dung. Then both countries only produce gold, and regardless of their comparative opportunity costs, they cannot be incentivized to trade that gold for any amount of dung.

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