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A production possibilities curve or production possibility frontier can be used to determine whether a country is producing to its full potential. So it could tell you whether a country has economic problems. However, the PPC is really not well suited for this because it is more of a theoretical tool that is used to teach about opportunity costs and such.
A PPC shows what combinations of two products or types of products can be made given available resources. You can see, for example, what combinations of cars and bicycles can be made or what combination of military goods and consumer goods. It shows what combinations are efficient, inefficient, and impossible. By looking at what combination a country is producing, you can see whether its output is efficient or not.
The problem is that a PPC only measures combinations of two things. Countries produce more than two kinds of goods and therefore it is hard to use a PPC to really look at the whole of a country's output.
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