I think it is not possible. For example efficiency is connect to productivity and on this way to scale economy. On this way if a firm (a farm in this case) is no efficient it is impossible that it could be competitive because to be productive is a condition to compete. There is a remote possibility in some market where there is an oligopoly condition where there is no a true competition, firms can have a lower productivity and can be “competitive” at the same time.

actually two gentlemen above me are discussing different side of efficiency. The earlier talks about comparative efficiency, where one (or several) firm(s) producing more outputs given the same number of inputs, or take less inputs given the same number of outputs. Then it can be said that those firms are the most efficient among others. The efficiency level of other firms in the group then are measured comparatively by taking the efficient firms as benchmark.

The latter talks about the efficiency level when there are standards outputs for all firms. Let's say that all electric companies posses the same generator with maximum capacity 10MW per hour. However, due to some constraints company A could achieve 9MW per hour, while company B could achieve 7MW per hour. Compared to the maximum capacity of the generator, none of them are efficient. While if we neglect the maximum capacity, it's clear that company A is more efficient than company B.

Competitiveness is directly and proportionately related to productivity and the latter shares the same relationship with efficiency. Michael Porter develops some of these relations in his different books and chapters of books.

If all farms competing are subject to similar economic and institutional conditions, I would suggest that if a farm is productive or efficient, it must be competitive. Given LAND heterogeneity and being agricultural output highly dependent on land or soil quality, a farmer may efficiently use all factors of production, but not be as productive as another farm with a better soil.

So answering your question, "it depends". A farm may be efficient and not "as productive" as another, and hence not competitive. The relation between competitiveness and productivity is clear cut, but not between efficiency and competitiveness.

If you include institutional aspects, such as differential taxation or different transaction costs, being productive and competitive is not so clear. One farm may be extremely productive within its gates and lose competitiveness due to transportation costs, transaction costs or taxation.

Hope this helps to answer your question.

Very deep question & very interesting views by experts. Based on considerable research competitiveness, I feel that the relative concept of competitiveness has relevance across levels (see our papers at link below) and goes much beyond efficiency.  Efficiency can be a necessary condition in some contexts, but not sufficient for competitiveness. Best of learning @ exciting concept of competitiveness.

http://scholar.google.co.in/scholar?as_q=&as_epq=&as_oq=competitiveness&as_eq=&as_occt=any&as_sauthors=Momaya+K&as_publication=&as_ylo=&as_yhi=&btnG=&hl=en&as_sdt=0%2C5

What if i want to define a relationship between efficiency and competitiveness for banks?

I guess, the comparative efficiency concept will be applicable here?