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?

  • $\begingroup$ In "I know mathematicians make use of functions. But do they also consider relations that are not functions.", the "they" seems to be refer to "mathematicians". Do you mean to ask whether mathematicians make use of relations that are not functions? Or more in line with you last line whether "economists" do? $\endgroup$ – Martin Van der Linden Apr 19 '19 at 12:43
  • $\begingroup$ @MartinVanderLinden. It was a mistake. Thanks. $\endgroup$ – Saint James Apr 19 '19 at 12:46
  • $\begingroup$ You could edit your question if it was a mistake. $\endgroup$ – Martin Van der Linden Apr 19 '19 at 13:24
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    $\begingroup$ I'm not sure how much asking this in the context of economics adds. Unless you look at really abstract example, most examples from math will have applications to economics. $\endgroup$ – Acccumulation Apr 19 '19 at 17:37

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):

  1. 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 $\succ = \{ (A,B), (A,C), (B,C)\}$.
  2. 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).
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  • $\begingroup$ Example 1 can be represented as a function though, relating the set of alternatives $S$ to the powerset of $S$. It is not very unusual to see it represented this way. It's perfectly fine (mathematically) to have a function relate elements to sets. It's just not a function from the set to itself in that case. Example 2 has the same issue. $\endgroup$ – John Doucette Apr 19 '19 at 16:50
  • $\begingroup$ Sure. Every relation $R \subseteq (A\times B)$ can be somewhat "equivalently" represented as a function $f \colon A \rightarrow 2^B$ such that for all $a \in A$ and all $b \in B$, we have $b \in f(a)$ if and only if $(a,b) \in R$. It remains that $\succ$ is not itself a function. I think it's fair to say that binary relations like $\succ$ are "useful" in economics. Since the OP's question was about "relations that are not functions and are useful in economics" that hopefully provides a good example. $\endgroup$ – Martin Van der Linden Apr 19 '19 at 17:30

Relations which are not functions are extremely, extremely, extremely common in all fields of study, and economics is no exception. Some examples are:

  • Most people buy multiple different products, so the relation "buys" is not a function.
  • Most companies employ multiple different people, so the relation "employs" is not a function.
  • Some people work for multiple different companies, so the relation "works for" is not a function, either.
  • Some pieces of property are jointly owned by multiple people, so the relation "is owned by" is not a function.
  • A single product can be sold at multiple different prices, so the relation "costs" is not a function.
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