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Allocative efficiency occurs when the production of goods and services is distributed in such a way that maximizes consumer satisfaction or social welfare. In other words, the quantity and combination of goods produced are such that no other mix of goods could better satisfy consumers without harming anyone.

Productive efficiency occurs when the goods and services are produced at the lowest possible cost. This means that the production process is optimized in such a way that no input resources are wasted, and there is no possibility of producing more goods using the same quantity of resources, nor of producing the same quantity of goods using fewer resources.

It is important to understand that allocative efficiency and productive efficiency are not the same things. Allocative efficiency focuses on the satisfaction of consumers or society, whereas productive efficiency focuses on the optimization of the production process to minimize the cost of producing goods and services.

It is theoretically possible for a nation to achieve allocative efficiency while being productively inefficient. This could happen when the resources are allocated in such a way that the satisfaction of consumers is maximized, but the production process is not optimized to produce goods at the lowest possible cost. An example of this situation could be a nation producing a high-quality product with excellent durability, but also at a higher production cost which might not be passed on to the consumers.

Although it is possible for a nation to achieve allocative efficiency with productive inefficiency, such a situation is likely to be unsustainable in the long run. Productively inefficient firms could face significant challenges in terms of competition and cost management, which could eventually lead to economic challenges for the nation. In addition, achieving allocative efficiency would require a high level of central planning and coordination, which may be difficult to achieve in practice. Therefore, a more realistic approach would be to strike a balance between allocative and productive efficiency to achieve the best possible outcome for both consumers and producers. In conclusion, it is possible for a nation to produce in a way that is allocatively efficient but productively inefficient, at least in the short run. However, it is unlikely to be sustainable in the long run, and a balanced approach including both types of efficiency would be more desirable for economic growth and stability.