Question

Could a nation be producing in a way that is allocatively efficient, but productively inefficient?

Short answer

No, a nation can't produce in a way that is allocatively efficient but productively inefficient.

Step-by-step solution

Productive Efficiency Concept 

Productive Efficiency means to produce product combinations, which are through best efficient utilisation of given resources & technology. No more units of a good can be produced without sacrificing other good, with given resources & technology.

These are the points on Production Possibility curve, which shows product combinations that can be produced with efficient utilisation of given resources & technology.

Points below PPC comprise less of either or both goods, show productive inefficiency.

Allocative Efficiency Concept

Allocative Efficiency means that commodities' combinations being produced are in accordance with their social demand.

It shows the resources are efficiently allocated or distributed in production of those goods, which are in coordination with patterns of society' demand.

It denotes match between resources production & consumption preferences.

Detailed Explanation 

Allocatively Efficient point is one of the productively efficient point on PPC. So, a point that is productively efficient may or may not be allocatively efficient.

But, since allocative efficient point is one among productively efficient point. So, there can't be production in a way that is allocatively efficient but not productively efficient.