Why would trade lead to allocative efficiency?

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Allocative efficiency (AE) is the economic principle where production is allocated to the most efficient use, also referred to as the optimal distribution of goods and services.

The concept of AE is simple.  If Canada produces maple syrup more cheaply than the US, then the US should trade something of value it produces to Canada for the maple syrup. 

Allocative efficiency encourages free trade agreements between countries due to the cost of production on a global scale.  In a perfect model, each country would be allowed to produce a product, wanted by other countries, which they are best suited to produce. 

AE fails in two scenarios when including trade.  A country may produce the most efficient product, but the product is not wanted by other countries.  Who cares if the USA can produce fake vomit if no one wants fake vomit?

The other scenario is when two or more countries equally produce the same product.  This is where trade can positively influence AE.  Suppose the US, Spain and Germany are all part of the same trade alliance and all produce bread equally well.  For AE to be properly achieved only one of the countries can produce it.  The other countries must produce some other product which might not be as efficient for them, but it tends to be more efficient for the alliance.  Thus trade has moved the alliance toward AE. 

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