This post follows up on the previous two (see here and here), by explaining why indifference curves don't cross each other. As noted in those earlier posts, indifference curves are the way that we represent the decision-maker's preferences in the constrained optimisation model. An indifference curve connects all of the bundles of goods that provide the decision-maker with the same amount of utility (satisfaction, or happiness).
We'll stick with using the consumer choice model as our illustrative model, and show what happens when indifference curves cross, as shown in the diagram below. The indifference curve I1 represents some higher level of utility than the indifference curve I1. So, the bundle of goods C is better than Bundle A, because it lies on a higher indifference curve - Bundle C provides the consumer with more utility than Bundle A. Bundle C is also clearly better than Bundle A because Bundle C has the same amount of Good X, but more of Good Y. And, as we assumed earlier, more is always better than less.
The problem comes in when we compare Bundle A and Bundle C with Bundle B. Bundle A and Bundle B are just as good as each other. They are both on the indifference curve I0, so they must provide the consumer with the same amount of utility. Bundle B and Bundle C are just as good as each other. They are both on the indifference curve I1, so they must provide the consumer with the same amount of utility.
So, to summarise, Bundle A is just as good as Bundle B (same utility), and Bundle B is just as good as Bundle C (same utility). That means that Bundle A must be just as Good as Bundle C. And yet, we started this example by saying that Bundle C is clearly better than Bundle A. Clearly this makes no sense. Bundle A and Bundle C cannot simultaneously provide the same utility when Bundle C is better than Bundle A.
That is because when indifference curves cross, it violates the mathematical property of transitivity. I prefer to say that it simply makes no sense. So, while indifference curves need not necessarily be parallel, they certainly can't cross each other.
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