Question

Prove the following statements: (a) indifference curves can never intersect. (b) an indifference curve is never positively-sloped.

Answer 1

(a)  indifference curves can never intersect

This is one of the properties of indifference curves. The IC can never intersect each other because at the point of tangency the higher IC gives them as much as of two commodities that the lower IC will give.

In this diagram, you can see that these two IC intersect each other. We know the point of intersection is on IC1 as well as on IC2 also. So we can say that all the bundles of good X and Y give equal utility of we get from both IC's. But if you look at any point above the intersection point, IC1 is higher than IC2 which means the consumer is better on IC1, and similarily any point below intersection point IC2 is higher than IC1 which means the consumer is better off on IC2. This contradicts our results in which we say that both consumers stay at the same utility level on both IC. Therefore two ICs cannot intersect.

(b) an indifference curve is never positively-sloped

An IC can never be upward sloping. This is because if consumer wants to consume more of good X than he have to give up some units of good Y. He cannot consume more of both goods. That's why IC is downward sloping.