Author: amit

This tool shows UMP and EMP when preferences are satiated: there is a bliss point \( (\delta_x, \delta_y) \) where utility peaks, and utility falls as the bundle moves away from it. Preferences are therefore non-monotone — more is not always better (local nonsatiation fails). When the bliss point is affordable, the consumer always buys it, whatever their income, and leaves unspent any income beyond its cost, so the budget need not bind and Walras’ law fails. 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 how distance is measured and a problem, then move the bliss point, prices, and income, and watch the optimal bundle respond.

What to look for

• Lower the prices or raise income until the bliss point is affordable. Does the consumer still always spend all income?
• Make the bliss point unaffordable. Where does the optimum sit relative to the budget line and the bliss point?
• Switch the distance measure between Euclidean and taxicab, and observe how the optimum changes.

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.

This tool shows UMP and EMP when preferences are non-convex, i.e., when the upper-contour set \( \{\, (x,y)\in\mathbb{R}^2_+ : u(x,y) \ge \bar u \,\} \) is non-convex. Each of the four utilities considered in the tool has a non-convex upper-contour set. These are the definitions of UMP and EMP.

\[ \text{(UMP)}\qquad \max_{(x,y) \in \mathbb{R}^2_+}\; u(x,y) \quad\text{s.t.}\quad p_x x + p_y y \le M \] \[ \text{(EMP)}\qquad \min_{(x,y) \in \mathbb{R}^2_+}\; \bigl(p_x x + p_yy\bigr) \quad\text{s.t.}\quad u(x,y) \ge \bar u \]

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 tangency point, if it exists. Is it the best bundle?
• Slide a price through the knife-edge. Does demand move smoothly, or jump?
• Switch between UMP and EMP, and observe the nature of the optimal bundle.