Two individuals live in the square [0,1] \times [0,1]. The location of individual i is (x_{i}, y_{i}), where 0 \leq x_{i} \leq 1 and 0 \leq y_{i} \leq 1; in this question (x_i, y_i) is a point in the square, not a consumption bundle. A social planner has to choose a location l = (l_x, l_y) \in [0,1] \times [0,1] for a swimming pool. If the pool is built at l, individual i obtains utility
u_i(l) = -|l_x - x_{i}| - |l_y - y_{i}|.
Let (x_1,y_1) = (0, 0.5) and (x_2,y_2) = (0.5, 1).
- Find the set of Pareto optimal locations in [0,1] \times [0,1]. Plot the utility possibility set and the utility possibility frontier in the (u_1, u_2) plane.
- Suppose instead that the planner chooses the pool’s location so as to maximize the smaller of the two individuals’ utilities. Which location or locations can the planner choose?