Question

2. Preferences. Let denote the consumers preference relation on C =Rn +: Answer the following: a. Sayis reexive, complete, but not transitive. Show that the consumers preferences could "cycle" (i.e., if for j = 1;2;3;:::;n; and consumption bundles xn we could have xj xj1 and x0 xn: b. Say is reexive, complete, and transitive. (i) Can indi⁄erence curves "cross"? (ii) what additional assumption rules this out. Show also that this assumption indeed does rule out crossing indi⁄erence curves. (iii) Show the consumer cannot "cycle" (i.a., part (a) cannot happen). (iv) Show that under "strictly monotonic" preferences, indi⁄erence curves cannot be "thick".

Answer 1

a).Consumption bundle are represented as x_j , where j=1,2,3,4 .....

Consumer preference is complete means two consumption bundle can be compared .

=> Relations such as  x_j ? x_j-1 ,  x_j ? x_{j-1 ,} x_j ? x_j-1 exists .

x_j ? x_j-1 (Given )

=> x₁? x₀ , x₂? x₁ .........x_n-1? x_n

=>x₀? x₁? x₂..................? x_n

But this cannot hold because transitivity does not hold .

Thus x₀? x_n can exist and hence Consumer Preference could cycle.

(b).(i). No, indifference curves cannot cross .

(ii). Assumption that "The further away the indifference curve lies, the higher the level of utility it denotes. Bundles of goods on a higher indifference curve are preferred by the rational consumer.

This assumption rules out crossing indifference curve.

(iii) To show that the consumer's preference cannot cycle .

Suppose we have consumption bundles x_j where j=1,2,3 .....,n then by the given relation x_j  ? x_j-1

we have x₀ ? x₁ and x₁? x₂............x_n-1 ? x_n

By transitivity ,

x₀ ? x₁ ? x₂ ? x₃............ ? x_n

So , x₀  ?   x_n cannot hold . Hence consumers preference cannot cycle.

(iv). Let x_j and x_j-1 be two consumption bundles such that x_j?x_j-1 (Strict monotonicity )

=> x_j?x_j-1 cannot hold .

=> x_j and x_j-1 cannot lie on the same indifference curve

and hence Indifference curve cannot be thick .