Indifference curves, and why two of them cannot cross

What an indifference curve is. Take a consumer choosing between two goods, say plates of idli (on the horizontal axis) and cups of filter coffee (on the vertical axis). A consumer usually cannot measure satisfaction in numbers, but she can rank bundles of the two goods. An indifference curve is the curve joining all the bundles that give her the same level of satisfaction, so that she is indifferent among them.

• It slopes downward. To stay equally well off with one more plate of idli, she must give up some coffee; the amount of coffee she will give up for one more idli is her marginal rate of substitution (MRS).
• It is usually convex to the origin. As she gets more idli and less coffee, she is willing to give up less and less coffee for each extra idli: the MRS diminishes.
• An indifference map. Her preferences over all bundles are shown by a family of such curves. A curve lying further from the origin stands for a higher level of satisfaction.

Can two indifference curves cross? No. Two indifference curves of the same consumer can never intersect, because a crossing would force two contradictory rankings of the same bundles.

Proof by contradiction. Suppose, contrary to the claim, that two curves IC₁ and IC₂ cross at a bundle A. Our own figures:

1. A and B lie on the same curve IC₁, so the consumer is indifferent between A and B.
2. A and C lie on the same curve IC₂, so she is indifferent between A and C.
3. If B is as good as A and A is as good as C, then B must be as good as C: she should be indifferent between B and C.
4. But B and C have the same quantity of idli (6 plates), while B has 2 more cups of coffee than C. A consumer who prefers more to less must rank B above C.

Steps 3 and 4 contradict each other: B cannot be both equal to C and better than C. So the starting supposition was wrong, and two indifference curves cannot cross.

The assumptions behind the result.

• Monotonic preferences: of two bundles, the consumer prefers the one with more of at least one good and no less of the other. This is what makes B better than C.
• A consistent ranking: bundles that are each as good as a third bundle are as good as each other. This is what makes B and C equal in step 3.

If either assumption failed, the proof would not go through; for a consumer with well-behaved preferences of this kind, each bundle lies on exactly one indifference curve.

Conclusion. An indifference curve joins the bundles that give a consumer equal satisfaction. Two such curves cannot cross: at a crossing point one bundle would be equally good as two other bundles, one of which contains more of a good than the other, and a consumer who prefers more to less cannot rank them both equally and unequally at once.