Is it possible to give, in economics, an example of a relation ( set of ordered pairs) that is not a function? In mathematics, some relations ( sets of ordered pairs) are not functions. 

I know economists make use of functions. 

But do they also consider relations that are not functions. 

In which branch of economics could "non-functional" relations be useful?
I feel that your question might be a little broad, but there are certainly many areas of economics where non-functional relations are used. Two simple examples (there are many others):


One of the most fundamental models of behavior in economics relies on the idea that choices can be represented by "preferences" which mathematically are binary relations that do not have to be functions (they very often are not, e.g., $A \succ B \succ C$ is not a function, since $\succ = \{ (A,B), (A,C), (B,C)\}$.
Economics also relies on correspondences, or ``multivalued" functions (which, depending on the definition, can be just another way to view binary relations). Correspondences are used in a lot of subfields of economics but perhaps most notably in consumer theory, where the choice set of consumer with preference $\succeq$ and budget set $B$ can be a whole subset of the consumption space $C(\succ,B) \subseteq X$ (where $X$ denotes the consumption space).