Open Access to Groundwater

OPTIMAL AQUIFER MANAGEMENT

4.3 Open Access to Groundwater

The details of characterizing the optimal path are in Appendix 4.A.

annual crop farmers no longer use groundwater. Type-P farmers purchase surface water rights from the Type-A farmers since the cost of extraction is larger than the return of the annual crops. In wet year, Type-P farmers can purchase enough water to irrigate all their land without groundwater extraction, while in a dry year, they need to pump groundwater as the total surface water supply cannot meet the demand for all permanent cropland.

I consider the optimization problem of a representative permanent crop farmer when the water depth isH > h^a. The price of surface water equals the return to annualsr^ain the wet year when water is abundant and it equals the pumping costz(H)in a dry year when water is scarce. The farmer’s profit function at periodt is:

y_t^p(kt,Ht)=r^pkt−r^a(kt−s^H)+r^pkt−z(Ht)(kt−s^L) −itC^k (4.30) Because planting new permanent cropsit affects future profits, the farmer chooses {it}^∞_t=

0

to maximize the sum of future profits: Í∞

t=1y_t^p(kt,Ht). Like the social planner, the Type-P farmer will use all land to grow permanent crops(k = l) as long asH < h^s and only grow k^s acreage of permanents while using the surplus surface water in the wet year to grow annuals. h^s and k^s are the steady state of water depth and permanent crop acreage in the competitive case:

h^s = 2r^p−r^a−C^k(¹_ρ+δ−1)

α (4.31)

k^s = l

L^p(R+S^L) (4.32)

Note that, in the equilibrium above the steady state stock of permanent cropsk^sis identical for all permanent crop farmers. In fact, it does no have to be so; however, a necessary condition for the steady state is that the stock of permanent crops be Í

k^p = R+S^L. If H < h^s,italways takes a corner solution, so the permanent crop farmers fully use their land kt = l.

Social planner vs. Open access

With an efficient surface water market, the inefficiency of the open access regime mainly comes from the commons problem. Individual farmers ignore the impact of their extraction

on the water level in the aquifer and pump too much. Optimal management improves social welfare in two ways if we compare the key water levelH^sandH^awithh^s andh^a.

First,h^a−H^a ≈ ^ρ(L_α(1−ρ)^p−SL) >0. The social planner stops applying groundwater to the annual crops earlier than the individual farmers. That not only delays depletion of the aquifer, but also reallocates some groundwater from annual crops now to higher-return permanent crops in the future.

Second,h^s−H^s = _1−ρ^ρR > 0. The social planner stops depleting the aquifer at a water depth shallower than the individual farmers. The savings from lower pumping costs are larger than the value of depleting extra stock of groundwater. However, individual pumpers deviate from the socially optimal extraction whenH = H^s because of the commons problem.

In theory, both channels of efficiency gain could be significant. Efficiency gain from the first channel is positively correlated to the scale of permanent cropsL^pand negatively correlated with surface water supply in the dry yearS^L. WhenL^pincreases or S^L decreases, the gain from reallocating water from annual to permanent crops is larger since the permanent crops yield higher return. This channel is likely to be underestimated if people ignore the sharp difference between annual and permanent crops and estimate a smooth water demand function.

The second channel could have a huge influence if natural recharge R is large. Since the steady state permanent crop acreage equals toR+S^L, a larger natural recharge means a larger equilibrium permanent crop acreage and a larger amount of groundwater to be extracted at the steady state. That raises the social planner’s incentive to maintain a higher water table and thus a lower pumping cost. On the opposite, whenRis small enough, the equilibrium extraction is so low that even the social planner feels unnecessary to save water for a lower pumping cost. WhenR= 0, the socially optimal steady state water depth is the same as the competitive case. There is no gain from optimal management in this case.

The first channel of efficiency gain becomes significant when water depth approachesH^a and the second channel of efficiency gain is important when water depth is close enough

to the steady state. For a water depth that is very low, the efficiency gain of groundwater management occurs in a far future so the discounted welfare improvement from optimal management is small.

However, for a water depth close to H^a, according to this model, the efficiency gain from optimal management can be significant as it reallocates groundwater from low-value annuals to high-value permanents. Studies that fail to consider the difference of the two types of crops will take a smooth water demand function and hence miss the efficiency gain from this channel. For example, in Knapp, Weinberg, et al. (2003), they estimated a water demand function using Kern County Water Agency 1998 report, when both annual and permanent crops were farmed in the county and the annual crops were always the marginal crops. The estimated water demand function then only captures the demand elasticity of annual crops and ignores the inflexible water demand from high-value permanent crops. That biases down the estimate of efficiency gain.

The second channel of efficiency gain usually does not add too much to the welfare im- provement of optimal management. Even though the equilibrium recharge is significant and the difference of equilibrium water depth between optimal management and common access regime is large as in Knapp and L. J. Olson (1995), the time required to reach the steady state is so long that the discounted value of the efficiency gain is tiny. This is often the case for large aquifer with plenty of groundwater stock to deplete before water becomes scarce. For small aquifers, water is scarce; therefore, it is important to manage the steady state water depth as the efficiency gain is realized immediately.