groundwater - A Three-Period Groundwater Model Susan E...

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A Three-Period Groundwater Model Susan E. Stratton These notes develop a three-period groundwater model. I strongly recommend trying to work this out yourself before looking at these notes. In other words, use these notes as a guide if you get stuck or to check that you’ve done it right. 1 The problem set-up Groundwater exists in aquifer underground. If we pump groundwater to the surface, we can use it for some purpose, giving us a benefit of B ( y ) where y measures groundwater extractions. The amount of water in the aquifer is given by x . Because these variables change over time, we’ll index them with t and refer to y t as groundwater extraction in time t and x t as the stock of water in the aquifer at time t . The total cost of pumping y t units of groundwater to the surface depends on both the amount of groundwater we pump and the level of the groundwater. As the amount of water in the aquifer increases, the level rises. This means water doesn’t have to pumped as far to reach the surface and hence costs less to pump. Therefore, we write the cost of pumping as C ( x t , y t ) with C x < 0 (high stock = high level = low pumping cost) and C y > 0 (each unit has to be pumped). Our goal is to maximize the net present value of benefits minus costs, but we have to remember that as we extract water, the amount in storage goes down. The relationship between extraction and water in storage is x t +1 = x t - y t + g ( y t ) (1) where g ( y t ) is recharge. Recharge has two components – natural recharge and return flows. The idea is that when we pump water to the surface and “use” it, only some of the water is actually consumed. The water that isn’t consumed will eventually percolate back down to the aquifer. This means that g y > 0 – recharge is higher the more we pump. 2 Our maximization problem We want to maximize t (1 + r ) - t [ B ( y t ) - C ( x t , y t )] subject to the equation of motion x t +1 = x t - y t + g ( y t ) . Let’s write out a three-period Lagrangian.
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