Question

What would the production possibility frontiers be for Brazil and the United States?
Without trade, the United States produces AND CONSUMES 32,500 units of clothing and 125,000 cans of soda. Without trade, Brazil produces AND CONSUMES 50,000 units of clothing and 25,000 cans of soda.
Denote these points on each COUNTRY’s production possibility frontier. Using what you have learned and any independent research you may conduct, which product should each country specialize in, and why?

Accepted Answer

You do not have enough information here to answer any of these questions. 
In order to find the production possibility frontiers (PPFs) for the two countries, you would need to have at least two ordered pairs for each country.  Right now, we have only one ordered pair for each.  We know that the US can produce and consume 32,500 units of clothing and 125,000 cans of soda.  We know that Brazil can produce and consume 50,000 units of clothing and 25,000 cans of soda.  The problem is that we do not have any other points.  For example, we do not know how many cans of soda Brazil could produce if it only produced 40,000 units of clothing.  We do not know how many units of clothing the US could produce if they only produced 100,000 cans of soda.  We need more points so that we can draw PPFs through those points.  As it is, you could denote those points, but you would not be able to draw any curves to show the PPFs.
As far as what product each country should specialize in, you do not know enough because you cannot determine opportunity cost.  The idea of comparative advantage says that a country should specialize in the good for which their opportunity cost is lower than that of their trading partner.  The problem here is that you do not know the opportunity costs for either country because you only have one data point for each.  I would guess that the US should specialize in soda because rich countries rarely have a comparative advantage in something as labor-intensive as the garment industry, but I have no numbers to back this up.
Please check your assignment to see if you have been given more information.

Answer

Each country ought to specialize in making whichever product costs its economy relatively the least. We can define such a cost in many terms; popular definitions include the lost opportunity to make other things of value to the economy or an economy's allocation and production efficiencies.

The United States, at its most efficient economic state, is utilizing soda at a rate nearly four times greater it does clothing: 125,000 sodas : 32,500 units of clothing = ~3.846:1.

Brazil, on the other hand, is said to utilize soda only half as much as clothes at its own optimal PPF state: 25,000 sodas : 50,000 clothes = 1:2.

Sometimes these ratios are expressed as the "slope" of the respective nations' PPF functions, but more generally, they represent the relative value each commodity has to one another within said country. If each nation ought to specialize in producing that which it...

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