Question

Determine if completeness and transitivity are satisfied for the following preferences defined on x = (x₁, x₂) and y = (y₁, y₂).
x ≽ y iff x₁ > y₁ or x₁ = y₁ and x₂ > y₂.
(Hints: 1- You have to use z = (z₁, z₂) to prove or disprove transitivity. 2- You can disprove by a counter example)

Answer 1

TRANSITIVITY

Transitivity means, if a consumer prefers 'x' basket of good over 'y' and 'y' over 'z', then he will prefer 'x' over 'z'.

Given; x=(x1,x2) and y=(y1,y2) and taking z=(z1,z2)

Note- x>y means x is strictly preferred over y.

If x1>y1 and y1>z1, then x1>z1.

Similarly, if x2>y2 and y2>z2, then x2>z2.

Hence, if x>y and y>z, then x>z

Transitivity proved.

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COMPLETENESS

The axiom of completness states that if a consumer is given two baskets to make a choice, x and y, then one of following statements will be true-

• x>y
• y>x
• x ~ y

Note- x>y means x is strictly preferred over y.

x~y means consumer is indifferent between x and y.

In this case

x1>y1 and x2>y2

or

x1=y1 and x2=y2

or

y1>x1 and y2>x2

Then

• x>y or
• y>x or
• x ~ y

is true respectively.

Hence completeness proved.

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