Why Indifference Curves cannot Cross?

Yes they really cannot but before explain the reason why, I want to simply define indifference curves. Indifference curves are the curves indicating all combinations of consumption along which we are indifferent. For example, you want to buy a new house and you have two variables in terms of decision making: size and location. You have three choices: House A is really big but its location is not really appealing, House B has medium size and its location is acceptable and House C is small but its location is great. If you feel like choosing among A, B and C does not make any difference in terms of utility, then by definition you can show these 3 houses in same indifference curve.

To understand indifference curves like this, we have to be sure that our preferences are rational, by saying rational we assume that:

§ Completeness, we are always sure about our preferences, not vague about the desirability of any two alternatives.This simply means you cannot say I don’t know what I feel about choosing house B so we don’t have a place for uncertainty at this point.

§ Transitivity, it has same meaning that we used in mathematics, if you prefer product A to B and you prefer B to C then you prefer product A to C. It can exemplify like this product A gives you 10 utils (kind of 10 points of happiness), product B gives you 7 utils and C gives 5 utils.

§ Continuity, we do not have sudden jumps in our preferences, for example if we are giving 10 points to high quality product, we do not give 9 points to really low-quality product (assume ranking is 1-10)

§ Convexity, it is actually the fact that people are increasingly less willing to part with good y to get more x (while holding utility constant) seems to refer to the same phenomenon—that people do not want too much of any one good. Consider our example, you probably want house B but give up huge house A and house C having excellent location, so we love balance.

§ Non-satiation (Monotonicity), we always want more so we always prefer higher indifference curves.

Now we can answer why these curves cannot cross. Let's assume that these curves can cross and add one more curve to our curve

Assume we one product X in X axis and another product Y in Y axis. We have said that the points on the same indifference curves are equally attractive; this implies that

A=B=C and D=E=F

Moreover, we know that we always think "more is better" (non-satiation) and because of that we prefer higher indifference curves. The further from the origin the better. Considering this we can say that Point F is more attractive than point A because although both has same number of product Y, point F has more product X than point so more attractive. Then by following this approach we can write

F>A F>B F>C A>D B>D C>D

However, if we remember transitivity rule we have to note that if B>D and B=E ( since they are at the same point) then B>E, which contradicts with our assumption because we have said D, E and F are in the same indifference curve then equally attractive.

Actually, the main cause of this problem is the curve including points D, E and F because it is upward sloping. In this curve points D and F are equally attractive which makes no sense since point D (1,1) means you get 1 product X and 1 product Y , but F (3,9) means you get more and more is better. By revealing this we actually find another feature of indifference curves which is indifference curves downward sloping because of non-satiation assumption.

To sum up, because of the transitivity and non-satiation assumptions indifference curves cannot cross each other and because of non-satiation assmption indifference curves cannot upward sloping.