Why indifference curve cannot intersect?

Answer 1

Because if they intersected, they would not be "indiffernece" curves. Imagine two intersecting ICs, call them x and y. That means that all points on y have the same utility. Also, all the points on x have the same utility, but that number is different from x. (From this we know that X1 = X2 = X3 = X4 etc etc and the same thing with Y1 = Y2 etc etc where the subscripts are points on the curves). We can refer to each curve by its utility value, call them x and y. So being that they are different curves, we expect them to have different utilities. So, U(x) != U(y). In words, the utility of x is never equal to the utility of y. Now imagine a point at which they cross. We would obviously have a point, let's call it A, where this does not hold true. More importantly though, we need to remember that ALL POINTS ON AN INDIFFERENCE CURVE HAVE THE SAME UTILITY. So back to this interesection at point A, we get X1 = X2 = XA. Also, Y1 = Y2=YA where A is the intersection. From this, we transitively know that X1 = X2 = Y1 = Y2. This creates on obvious problem. It would mean that every point on the two separate indifference curves would have the same utility, which is the complete opposite of the first rule of indifference curves, that all points on them have the same utility.