Problem 6. Consider a consumer with a continuous, increasing, quasi-concave utility function $u: \mathbb{R}^2_+ \to \mathbb{R}$. Fix $\bar{x} \in \mathbb{R}^2_+$, and denote its upper contour set as $U(\bar{x}) = \{y \in \mathbb{R}^2_+ : u(y) \ge u(\bar{x})\}$.
(a) Show $U(\bar{x})$ is a non-empty, closed and convex set in $\mathbb{R}^2_+$.
(b) Suppose $z \in \mathbb{R}^2_+$ satisfies $z \notin U(\bar{x})$. Then show that there exists non-zero $p \in \mathbb{R}^2_+$ such that $p \cdot z < p \cdot \bar{x}$. (Hint: use separating hyperplane theorem.)
