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Allocative efficiency is achieved when the mix of goods and services produced in an economy best meets the needs and wants of consumers. In other words, it is where the value that consumers place on goods and services is equal to the cost of producing those goods and services. Allocative efficiency can be graphically represented using the demand and supply model, where allocative efficiency is achieved at the equilibrium point (where demand equals supply). At this point, the price consumers are willing to pay is equal to the price suppliers are willing to accept.

Productive efficiency refers to a situation where the production of goods and services is done at the lowest possible cost. In other words, it's about producing the maximum output from the available inputs (labor, capital, land, etc.) for minimizing the cost of production. Productive efficiency can be illustrated through the Production Possibility Frontier (PPF). When a nation is producing on the frontier, it means that it is using all its resources to the fullest, and it's not possible to produce more of one good without producing less of another good.

The key difference between the two concepts is that allocative efficiency is concerned with distributing resources to the production of goods and services where they provide the highest social utility, while productive efficiency is concerned with the minimum cost of production of these goods and services.

It is possible for a nation to be allocatively efficient but not productively efficient. This can happen when the goods and services produced successfully satisfy consumers' preferences, but resources are not used in the most efficient manner. For instance, a nation might be producing goods according to the preferences of its consumers, but the production methods could still be outdated or inefficient, using more resources than necessary. In such a case, the nation would be allocatively efficient (as it's satisfying consumer preferences) but not productively efficient (as it's not minimizing cost). It is important to note that attaining both allocative and productive efficiency is the ultimate goal of any economy. However, achieving one type of efficiency doesn't necessarily guarantee that the other type is also achieved. In conclusion, a nation can be producing in a way that is allocatively efficient (meeting the needs and wants of consumers) but can still be productively inefficient (inefficient use of resources). Therefore, it is crucial for nations to strive for both allocative and productive efficiency to achieve economic prosperity.