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Productive efficiency refers to a situation in which a firm or an economy is producing goods and services at the lowest possible cost. In other words, it is producing the maximum possible output using the available resources and technology, without any wastage.

Let's suppose there are two companies, Company A and Company B, which produce the same product. Company A uses advanced technology and skilled labor, and manages to produce 100 units at a cost of \(1,000. Company B, on the other hand, uses outdated technology and unskilled labor, producing 100 units at a cost of \)1,200. Therefore, Company A is more productively efficient than Company B, because it is producing the same output at a lower cost.

Allocative efficiency refers to a situation in which the distribution of resources and goods and services is optimized in the economy, meaning that they are allocated according to consumer preferences and are being put to their highest valued use. In other words, it is a state where the value of one more unit of a good or service is equal to the cost of producing that additional unit.

Let's consider a simple example. Suppose we have an economy with two goods – apples and oranges. Consumers have a preference for apples at a ratio of 2:1, meaning they are willing to give up two oranges for one apple. In an allocatively efficient economy, the cost of producing one apple would be exactly equal to the cost of producing two oranges. If the economy is producing at this ratio, it is achieving allocative efficiency, as resources are being allocated according to consumer preferences and the highest valued use. In summary, productive efficiency is about producing goods and services at the lowest possible cost, while allocative efficiency is about distributing resources in an optimal way according to consumer preferences and the highest valued use.