Question

Suppose that there are two products: clothing and soda. Both Brazil and the United States produce each product. Brazil can produce 100 units of clothing per year and 50 units of soda (where units are thousands of cans). The United States can produce 65 units of clothing per year and 250 units of soda. Assume that costs remain constant. For this example, assume that the production possibility frontier (PPF) is a straight line for each country because no other data points are available or provided. 
What would be the production possibility frontiers for Brazil and the United States? (Without trade, the United States produces AND CONSUMES 32.5 units of clothing and 125 units of soda. Without trade, Brazil produces AND CONSUMES 50 units of clothing and 25 units of soda.) Which product should each country specialize in, and why? What is the labor-intensive good? What is the Marginal Rate of Transformation impact? What is the labor-abundant country? What is the capital-abundant country? Could trade help reduce poverty in Brazil and other developing countries?

Accepted Answer

In this example the United States should specialize in soda and Brazil should specialize in clothing, as they have larger capacities for manufacturing these products. The most labor- or capital-abundant country would depend on the amount of labor and costs necessary to create each specific product. Brazil could theoretically reduce poverty through trade if it had a surplus of specialized product.

Answer

The question is one of production specialization and illustrates what Adam Smith advocated: that by specializing in products where there is a competitive advantage, each country can improve its own productivity to the benefit of both. Telling us that units of soda equal thousands of cans does not add much to our understanding of productivity so for the purposes of this question, we should disregard that detail.
Brazil can produce 100 units of clothing per year and 50 units of soda. The U.S. can produce 65 units of clothing per year and 250 units of soda. This means that the U.S. produces only 65% as much clothing as Brazil produces each year, but five times as much soda. Conversely, Brazil produces 154% as much clothing, but only 20% as much soda.
This implies that the U.S. should forego the manufacture of clothing and let Brazil take care of that, while it (the U.S.) specializes in the more capital intensive manufacture of soda, which it can complete at five times the rate of Brazil.
The PPF assumes that the production of one product can only increase if the production of the other product falls so that resources can move from one to the other. In terms of the marginal rate of transformation, the U.S. produces 3.85x as much soda, the capital intensive product, as it does clothing. Thus, if it produces only soda and shifts all resources to that product, assuming a linear substitution between the two, the nation can produce 500 units of soda. If Brazil, which can produce two units of clothing for each unit of soda, produces only clothing, it can manufacture 250 units of clothing when it produces no soda.
We know in the real world the surplus that each country manufactures can be exported to trade partners. Thus, even though in this particular example, both countries can fully satisfy demand from their own internal resources, it still is more efficient to specialize. They can export their surplus and increase their revenue, which would help reduce poverty in Brazil.

Answer

The PPF (production possibility frontier) of an economy describes the maximum output that the economy can produce. Here, only two goods, clothing (C) and soda (S), are considered. The PPF's for the two countries, Brazil and the US, are assumed to be straight lines to keep the model simple. The PPF line describes the possible combinations of output of the two goods, C and S. Points below the PPF represent the situation where the economy is producing goods at less than full capacity. Any point above the PPF represents a combined output that isn't achievable under current production possibilities (according to resources and available labour), but might be achievable in different conditions.
1) In the case of Brazil, we're told it can produce 100 units of clothing per year, or 50 units of soda. Therefore, its PPF is given by
2S + C = 100
because when C = 0, S = 50 (so that 2S = 100), and when S = 0, C = 100. The coefficients of S and C arise from the fact that production of C (clothing) to S (soda) is in the ratio of 2 to 1.
Similarly, the PPF for the US is given by
(65/250)S + C = 65
because when C = 0, S = 250 and when S = 0, C = 65.
2) We are told that, without trade, the US produces 32.5 units of clothing and 125 units of soda. This is the point on the PPF for the US where C = 32.5 (horizontal axis) and S = 125 (vertical axis). This is halfway along the PPF line, as the value of C is half of what the US can produce in total of C (65 units), and similarly the value of S is half what it can produce in total of S (250 units).
   We are also told that, without trade, Brazil produces 50 units of clothing and 25 units of soda. On Brazil's PPF line this would be the point where C = 50 (horizontal axis) and S =25 (vertical axis). Again, this is halfway along the PPF line as C is half the value that Brazil can produce (100), and S is half the value that it can produce (50).
3)  Assuming the Ricardian model of trade, that is, perfect competition, we can compare the opportunity costs for not producing each good in turn for each country to work out which product which country should export and which product which country should import.
The opportunity cost of a particular good produced is expressed in terms of how many units of another good could be produced in the same unit of time (here 1 year). This assumes that there is a limit to the availability of labour.
For each of the goods (C and S here), the opportunity cost is the ratio of possible production of one to the other. That is, it is the slope of the PPF line. If the line is drawn with C as the horiztonal or x axis and S as the vertical or y axis then the PPF is described as
S = 50 -(1/2)C   for Brazil,  and
S = 250 - (250/65)C  for the US.
The opportunity costs of making clothing (C) in terms of soda (S) are then, respectively, OC(Brazil) = 1/2 and OC(US) = 250/65. That is, labour devoted by Brazil to making one unit of C could have been used to make only 1/2 a unit of S. The opportunity costs of making S in terms of C are the reciprocal of those of C in terms of S, that is, OC(Brazil) = 2 and OC(US) = 65/250.
Because Brazil can make relatively more of C (clothing) than S (soda), whereas the US can make relatively more of S than C, Brazil has the comparative advantage in making clothing and the US the comparative advantage in making soda. Brazil can make OC = 2 times the amount of clothing that it can make of soda. In contrast, the US can make OC = 250/65 times the amount of soda that it can make of clothing. Therefore there are gains to be made from trade for each country in a trade agreement between the two where Brazil specializes in making clothing and the US specializes in making soda.
Because the US has a larger capacity for production, despite having a larger capacity for consumption also, it would either need to under-produce soda (it can produce 250 units of soda, but Brazil only requires 25 units and the US itself only require 125, meaning they can produce 100 units more than is required for these two countries alone) or enter a trade agreement with other (developing) countries such as Brazil that have relatively low production and consumption in comparison to the US (and similar developed countries).

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