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Could someone please explain, what monotonicity is, and how it ensures that indifference curves do not cross?

Monotonicity in its simplest terms can be interpreted as being "more is better".

Be careful here though, because its not strictly true; if you assume that the demand for a good is non satiable (i.e. you can never have enough of the good), then this implies strict monotonicity, as utility is always strictly increasing (the marginal utility > 0 ).

However, monotonicity does not imply non-satiation, its only infers that in the range you are studying, that demand is insatiable and the more is better statement holds.

Now, in relation to you question on indifference curves, firstly, you have to know the assumptions behind them.

1. Preferences are complete - consumers can rank bundles in a preference order
2. Reflexivity - a bundle is at least as good as itself (this is just to stop choice paralysis)
3. Preferences are transitive -if you like A more than B, and B more than C, then you also like A more than C. (think stone paper scissors, although that isnt a technically transitive game )

Now, we know that for 2 goods, higher indifference curves are preferred to lower ones, because they entail a higher level of utility. So, from assumption 1, we rank any bundle on a higher utility curve as being better than a bundle on a lower one. So, in my crap attached diagram, we would strictly prefer A to C, as A is on the higher curve (IC2).

But, a property of the indifference curve is that it represents bundles that consumers are indifferent between, so we like A and B equally as much, but we must also like B and C equally.

And there lies the contradiction, because in fulfilling the ranking order of assumption 1, we violate assumption 3. We know that A should be preferred to C, but we are indifferent between A and B, and B and C. Thus if indifference curve cross, we cannot maintain all of these assumptions.

Hence you don’t need monotonicity to prove they cant cross, all that monotonicity does is give you the direction of improving utility (north east in this case), and the shape of the indifference curves would be determined by a second caveat of strict convexity.

However, you can also use the monotonicty approach as part of the argument about ranking, because as more is better, A is preferred to C, but through B they are equally preferred etc

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Originally submitted by raven100 on TSR Forums.