Question

Anna consumes two goods, x and y. She prefers the bundle (x, y) to the bundle (x ' , y' ) if and only if (xy − x 'y'  > 1). Otherwise she is indifferent between the two bundles.

a. Show that the relation of indifference (∼) is not transitive for Anna.

b. Show that the preferred relation () is transitive for Anna.

Answer 1

a)

Let there be three bundles:

$\small X_1:= (x_1, y_1) \hspace{3em}X_2:= (x_2, y_2) \hspace{3em}X_3:= (x_3, y_3)$

such that

$\small X_1 \sim X_2 \hspace{3em } \text{and} \hspace{3em } X_2 \sim X_3$

$\small X_1 \sim X_2 \Rightarrow x_1y_1 - x_2y_2 \leq 1 \hfill (1)$

and $\small X_2 \sim X_3 \Rightarrow x_2y_2 - x_3y_3 \leq 1 \hfill (2)$

(1) + (2) gives from RHS:

$\small x_1y_1 - x_3y_3 \leq 2 \hfill (3)$

However for $\small X_1 \sim X_3 \hspace{1em }, x_1y_1 - x_3y_3 \leq 1$

which is inconsistent with equation (3)

Hence relation of indifference (∼) is not transitive for Anna.