This tool shows UMP and EMP when preferences are convex, i.e., when the upper-contour set \( \{\, (x,y)\in\mathbb{R}^2_+ : u(x,y) \ge \bar u \,\} \) is convex. Each of the four utilities considered in the tool has a convex upper-contour set, so the set of optimal bundles is always convex. These are the definitions of UMP and EMP.
\[ \text{(UMP)}\qquad \max_{(x,y) \in \mathbb{R}^2_+}\; u(x,y) \quad\text{s.t.}\quad p_x x + p_y y \le M \] \[ \text{(EMP)}\qquad \min_{(x,y) \in \mathbb{R}^2_+}\; \bigl(p_x x + p_y y\bigr) \quad\text{s.t.}\quad u(x,y) \ge \bar u \]Choose a utility function and a problem, then move the prices and income and watch the optimal bundle respond.
What to look for
- Find the interior tangency on the smooth arm. Does the MRS equal the price ratio there?
- Move the price ratio until the optimum sits at a kink in EMP. Observe the range of price ratios for which it remains optimal.
- Slide a price slowly. Does demand jump, or move continuously?
- Switch between UMP and EMP, and observe the nature of the optimal bundle.