2 Preferences Let E denote the consumers preference relation

2. Preferences. Let E denote the consumer\'s preference relation on C =RI. Answer the following: a. Say > is reflexive, complete, but not transitive. Show that the consumer\'s preferences could \"cycle\" (i.e., if for j = 1, 2, 3, ..., n, and consumption bundles In we could have Tj > Tj-1 and co > In. b. Say is reflexive, complete, and transitive. (i) Can indifference curves \"cross\"? (ii) what additional assumption rules this out. Show also that this assump- tion indeed does rule out crossing indifference curves. (iii) Show the consumer cannot \"cycle\" (i.a., part (a) cannot happen). (iv) Show that under \"strictly monotonic\" preferences, indifference curves cannot be \"thick\".

Solution

(i) No, two indifference curves cannot cross each other. If we assume that two ICs (IC1) & (IC2) cross each other at point R. The Point of intersection(R) must indicate different levels of satisfaction since it lies on IC1 as well as on IC2 curves.But it is impossible.Hence two indifference curves cannot intersect or cross each other.

(ii)More is better is the additional assumption that rules this out.Suppose an individual is offered two almost identical bundles A, B then definitely individual will choose bundle B. But point B happens to be the intersection point then his preference will be depicted by two ICs which is impossible

(iii)Non satiation precludes circular Indifference curves.

(iv) Suppose that IC1 is thick enough to contain both a and b because both are on IC1 but prefers b to a by more is better assumption because b lies above and to the right of a. Because of this contradiction, ICs cannot be thick