1. (17 %)

Consider a utility function $U: \mathbb{R}+^2 \to \mathbb{R}$, $U(x1, x2) = |x1 - x_2|$.

This utility function represents a preference relation $\succcurlyeq$. Equivalently, for any bundles $(x1, x2), (y1, y2)$,

$$(x1, x2) \succcurlyeq (y1, y2) \text{ if and only if } U(x1, x2) \geq U(y1, y2).$$

Consider the following properties.

Definition. A preference relation $\succcurlyeq$ is convex if for any $\alpha \in (0,1)$

$$\begin{array}{l} (x1, x2) \succcurlyeq (z1, z2) \text{ and } (y1, y2) \succcurlyeq (z1, z2) \\ \Rightarrow (\alpha x1 + (1 - \alpha)y1, \alpha x2 + (1 - \alpha)y2) \succcurlyeq (z1, z2). \end{array}$$

Definition. A preference relation $\succcurlyeq$ is homothetic if for any $t > 0$

$$\begin{array}{l} (x1, x2) \succcurlyeq (y1, y2) \\ \Rightarrow (t x1, t x2) \succcurlyeq (t y1, t y2). \end{array}$$

(a) True/False/Uncertain: Is \(\succsim\) transitive?