Question

Mac Rowe doesn’t sweat the petty stuff. In fact, he just cannot detect small differences. He consumes two goods, x and y. He prefers the bundle (x, y) to the bundle (x', y') if and only if (xy - x'y' > 1). Otherwise he is indifferent between the two bundles.
a.       Show that the relation of indifference is not transitive for Mac. (Hint: Give an example.)
b.      Show that the preferred relation is transitive for Mac.

Answer 1

a) To show that the bundles are not transitive, we need to take the bundles such that the consecutive bundles should be indifferent ,so the difference between the product be less than 1. For transitivity, the condition is a>b and B>c then a>c

Example: take A as (1,1) ,B As (1.5,1) and C as (1,2.25)

Now first we consider A and B ie x'y' =1*1=1

xy= 1.5*1= 1.5

Since xy -x'y' < 1, so he is indifferent between them.

Now the same process would be followed for B and C

xy= 2.25*1= 2.25

x'y'= 1.5*1= 1.5

Here also xy -x'y'< 1, so he is indifferent between B and C.

But when we consider A and C

x'y'= 1*1=1

xy= 2.25*1=2.25

So xy-x'y'>1 ,ie he prefers xy to x'y'. He prefers C to A and hence his preferences are not transitive in case of indifference.

b) Again we take 3 bundles A(1,1) ,B (2,3) and C(3,5)

xy= 2*3= 6

x'y'=1*1=1

So xy-x'y'>1, hence B is preferred to A.

For bundles B and C ,

xy= 3*5= 15

x'y'= 6

xy-x'y'>1, Herve C is preferred to B.

For bundles A and C

xy= 15

x'y'=1

xy-x'y'>1.

Hence C is preferred to A

Hence these are transitive.

(You can comment for doubts)