# Why is monotonic preference and monotonic utility function non-decreasing?

Obviously a monotonic function can be either nondecreasing or nonincreasing:

However, in Economics, a quick google search gives:

I am interested in the history or the motivation behind the econ definition that deviate from the definition in math. Why economists invent their own math terms?

• What's the invented term that deviates from the mathematical definition? You haven't presented any other definition of monotonic here beside than the usual mathematical meaning of "nonincreasing or nondecreasing". Oct 24 at 14:53
• Hi @Dodo. Regarding monotonicity of preferences, you must recognise that this is a choice-theoretical term. In other words, monotonic preferences draws on the fact that when people make choices about economic "goods." Since there economic goods (and not bads), we assume that welfare is at least increasing when any of these goods is increased. felixmunozgarcia.files.wordpress.com/2020/08/… Oct 24 at 15:15
• @NuclearHoagie In econ, a monotonic utility function can only be nondecreasing, not nonincreasing.
– dodo
Oct 24 at 15:29
• @dodo If the utility function is nondecreasing, it is absolutely correct to describe it as monotonic. Monotonic alone doesn't imply directionality, it must be the domain knowledge of the economic utility function that makes it nondecreasing. I don't think this is a re-definition of "monotonic", it's an additional specification of directionality by including "utility function" in a particular context. It would defy the common notion of "utility" to have more of something be less useful. Oct 24 at 15:44
• I usually read about increasing utility functions and monotonic preferences. Oct 24 at 16:23

Monotonicity is in economics defined the same way as in mathematics. According to Pemberton and Rau, Mathematics for Economists, pp 130;

function $$f$$ is said to be non-decreasing if $$f(x_1) \leq f(x_2)$$ whenever $$x_1. ... $$f$$ is said to be non-increasing if $$f(x_1)\geq f(x_2)$$ whenever $$x_1>x_2$$. A non-increasing or non-decreasing function is said to be weakly monotonic.

The definition is thus exactly the same (the wolfram alpha also talks about weak monotonicity). So in economics the definition of monotonic function exactly matches the definition in mathematics.

What might confuse you that in economics some functions already have restrictions put at them. For example, in most branches of economics, by default, it is assumed that $$U'>0$$. Hence any monotonic utility function is by definition increasing monotonic function because decreasing utility is not allowed by assumption. There are some subfields of economics where these assumptions might be relaxed, but by default any utility function should be assumed to be increasing function.

So monotonic utility function is not non-decreasing because in economics monotonicity is redefined. Monotonic utility is non-decreasing, because it is a utility function.

PS: Random working paper is also not good evidence for something applying to a whole field.

• MAke sense!!!!!
– dodo
Oct 24 at 21:06
• I think the P.S. part is uncalled for. Oct 24 at 23:20
• @MichaelGreinecker I toned it down
– 1muflon1
Oct 24 at 23:31
• Thank you. Much better. Oct 24 at 23:32