a prediction about which direction your answers would change and justify why your prediction makes coonomic 3. Suppose there are two large -scale agricultural water users : Barbara and Dianne . Lety denote water use. Barbara has a utility function of water use, (y) = 5y - 0.25y- and Dianne has a utility function of water use Up (y) = 20 - 1. 15 (a) What are Barbara and Dianne's respective demands for water ? (Find each person's demand curve for y.) (b) Suppose the supply of water is uncertain . In each period , there is a 40% chance of there being 2 acre-feet of water available, a 40% chance of there being 5 acre-feet of water available, and a 20% chance of there being 10 acre-feet of water available. i. What is the expected value of water availability Ely]? ii. Suppose Barbara and Dianne are cach contemplating creating a groundwater storage project. The project would be purely for storage: there would be no additional water yield. Derive the value of stabilization this storage would provide Barbara. Derive the value of stabilization this storage would provide Dianne. Label your answers with the correct units . (c) Now suppose the climate permanently changes and droughts become more severe . In particular, the probability of water availability is such that in each period, there is a 40% chance of there being 0 acre-feet of water available, a 40% chance of there being 5 acre-feet of water available, and a 20% chance of there being 10 acre-feet of water available. i. What is the new expected value of water availabilityEly]? ii. Re-derive the value of stabilization of groundwater storage for both Barbara and Dianne as in part (b) ii. above (d) When the weather becomes drier (moving from part (b) to part (c)), both Barbara and Dianne's values of stabilization should increase. For whom is the increase larger in proportional terms? 4. Suppose the government seeks to maximize social welfare by extracting groundwater from an aquifer holding 6 million acre-feet in storage. The inverse demand for the water isp(q.) = 182 - 3q, where of is million acre -feet extracted at time . The cost of extracting groundwater is c(qt, S.) = (15q - 25,)q where S, is the volume of groundwater in storage at timet. The government uses an interest rate of 100 % and does not care about any period after the second DEC . (a) What are the state and control variables in this problem? (b) Write down the social welfare maximization problem. Derive the optimality conditions (first -order conditions (c) Solve for the optimal levels of water extraction in each period:qf and q;. 5. Consider an infinite -horizon groundwater management problem where It is the stock of groundwater at time , 7 is the quantity of groundwater extracted at timet, g(3,) gives the amount of total recharge to the aquifer, B (y:) is the benefit of extractingye, and C(It, y:)
