the care with which water is managed, and general scarcity of water including the level of extraction. In particular, when less water is extracted from the basin, a smaller fraction of this extracted water is likely to flow back, as most water is then either absorbed by plants or lost to evapo-transpiration.

# **5.3 Stochastic recharge of the water basin, and water demand**

There is no uncertainty in the model as applied so far. In reality, for India and elsewhere, the recharge rate R can be highly variable. It is pointed out (e g by Banerji et al (2006)) that some water basins in India are replenished only every 10 years or so. Additionally, water demand may vary with the amount of natural rainfall, thus leading to variable values of F’ in our model and putting different pressures on extraction over time.

Uncertainty will tend to put an extra premium on groundwater value as one recognizes that groundwater will be pumped in some future scarcity situations, at high pumping cost (low water table). Indeed, groundwater may have a large fraction of its value in the form of a buffer against variations in rainfall and other factors affecting water demand. Some indications exist that such motivations can constitute a large fraction of its value. (Tsur and Graham-Tomassi (1991), for example, calculate in some of their examples that the buffer stock value of groundwater can constitute more than 80 percent of its total value.)<sup>11</sup> In the context of the model above, such concerns can partly be accommodated directly, and may in part require model extensions. In my model, an effect of stochastic supply and demand could be to increase externality costs of water withdrawals, perhaps dramatically, as one gets close to emptying the aquifer (with costs of further withdrawals approaching infinity as the basin becomes empty). In theory these costs can be avoided by simply not withdrawing water in such cases. But this raises the need for additional demand-side considerations. When one cannot withdraw water, its marginal value in agriculture can be very high (a likely outcome in a drought). This opportunity cost of not withdrawing “normal” quantities is then part of the cost of current withdrawals.<sup>12</sup> This can be a substantial factor in particular when aquifer dryout and water value are highly correlated, which is of course highly likely.

A separate aspect of this issue is that some aquifers are more complex as there may be restrictions on the water flow across the aquifer. Athanassoglou et al (2010) combine this assumption with an assumption that extracting agents are heterogeneous. On this basis they claim to show that a socially optimal policy (implemented as a Markov perfect equilibrium) takes a quite similar form to that of an aquifer with perfect flow. Since this work is preliminary, more research is here clearly needed.

# **5.4 Groundwater value has additional components**

So far we have concentrated on the productive value of water as an input in agriculture. Water may have other values or uses as described e g in National Research Council (1997) and as presented above. These arguments tend to enhance the net groundwater value whenever they are significant. They of course include the general extractive supply value of water to households and industry. But they also include other in situ value items such as supporting

> 11 Tsur and Zemel (1995) model implications of uncertainty for groundwater valuation, focusing on the possibility of aquifer “rundown” due to over-exploitation. 12 This factor is largely what drives the main result in Tsur and Graham-Tomassi (1991).