3. Dune (water rights) Arrakis is a planet without much water. There are two uses for water: consumption by local inhabitants, which creates local utility u(w) = 8w 1 2 w2 for w units of water, and spice mining, which creates profits (w) = 4w 1 2 w2. (a) Suppose that we care equally about utility and profits and that there is a total of w = 4 units of water on Arrakis. What is the first-best allocation of water? (b) What is the first-best allocation of water if w = 10? (c) Suppose that the spice miners have enforced "property rights" that let them take as much water as they like before local inhabitants are allowed to take water. Spice miners maximize their profits and do not care about the local inhabitants' utility. If there are w = 8 units of water, how much water will the inhabitants of Arrakis consume? Is this efficient? (d) Recalculate your answer to (c) for w = 12. Do the property rights lead to the efficient outcome? Use your results to explain how water scarcity determines the welfare costs of the spice miners' control of water rights on Arrakis