Throughout the world, volumetric pricing of irrigation water - where water is priced based on the quantity (volume) of the water used - is the exception rather than the rule (Boss and Walters 1990); the reason being that irrigation water is often unmetered. Riparian rights, under which farmers divert water freely from a nearby stream or pump water from an underlying aquifer, are pervasive. Unmetered water is usually priced on a per area basis (where each season or each year a fixed, prespecified amount is paid for any cultivated hectare), or it is priced by taxing observable outputs or inputs (Bowen and Young 1986, 211; Tsur and Dinar 1995), or pricing attempts are ignored altogether.
If the entire cost of water supply is borne by the water users, then it makes no difference if water inputs are unobservable: the farmers themselves take into account the true cost of water in their input/output decisions. However, a problem arises when the costs of water include non-private components due to scarcity and extraction cost externalities. Such externalities can be the result of declining water tables that lead to higher pumping costs in the future (temporal externality); or can be associated with costs borne by the manager of a large-scale irrigation project such as conveyance costs from the water source to farmer's fields, maintenance and operation costs, and (imputed) investments costs (delivery cost externalities). Another common situation occurs when many users pump groundwater from a shared aquifer via private wells (spatial cost externality).
In the case where true water costs contain non-private components, farmers' water allocation decisions will typically be greater than the corresponding socially optimal allocations, and some type of regulatory intervention may be desirable. When individual water use is observable, in principle it is straightforward to include the non-private costs in the price of water. When individual water use is unobserved (moral hazard), pricing water indirectly through observed instruments like outputs and/or other inputs may affect water allocation decisions (see Bowen and Young 1986).(1) However, unless such a pricing mechanism makes use of all available information concerning production technologies, such instruments will likely be inaccurate proxies for water use and the indirect water fees may be off target and could even increase inefficiencies. This is likely to occur when individual farmers are heterogeneous in their production technologies and a single per unit output tax is imposed. The water pricing problem is exacerbated if farmers are heterogeneous (or homogeneous) in their production technologies and the technologies are private information, that is, known to the farmer but not to the water authority (adverse selection). For examples of analytically similar problems see Besanko and Sappington (1987) or Smith (1995). Further complications arise when regulation entails transaction costs associated with administration, monitoring, and enforcing activities.
We offer in this paper a water pricing scheme (under moral hazard and adverse selection) that depends only on observable outputs. In the absence of implementation (transaction) costs, the pricing procedure achieves first-best allocations: that is, the optimal allocation when individual water applications are observed. With transaction costs, first-best allocations are not possible and second-best allocations might be attainable. Results of the numerical example suggest that transaction costs might possibly explain the types of water pricing institutions currently observed.
The idea underlying the pricing procedure is rather straightforward. If the regulator had complete information on the relevant parameters of farmers' production technologies, that information could be used to infer each farmer's water use as a function of that farmer's (observed) output and the situation would then be as if water use were observed. Lacking complete information about farmers' production technologies, the regulator's actions (the pricing mechanism) are conditioned on the farmers' unobservable technologies. This is accomplished by appropriately imposing (nonlinear) taxes on outputs. Our procedure follows closely the direct revelation mechanism approach of Guesnerie and Laffont (1984), and Laffont and Tirole (1987).
The next section formalizes the problem. Section III describes the pricing mechanism. Section IV offers a numerical example and Section V concludes. Some technical derivations are relegated to the Appendix.
II. THE PROBLEM
Consider n farmers - indexed i = 1, 2...., n - producing a homogeneous crop y using water as an input. Farmers' water response functions (after maximizing out all variable inputs other than water) are
[y.sup.i] = g([w.sup.i], [[Beta].sup.i]), i = 1, 2,..., n,
where g(..) is the water response function, [w.sup.i] is water input and [[Beta].sup.i] [element of] [0,1]] parameterizes the farmer's water response function. The water response function is assumed thrice continuously differentiable in both arguments, increasing and strictly concave in w, strictly increasing in [[Beta].sup.i], with [g.sub.12] = [[Delta].sup.2]g/[Delta][w.sup.i][Delta][[Beta].sup.i] [greater than or equal to] 0 (subscripts denote partial derivatives). The parameter [[Beta].sup.i] is referred to as the farmer's type and is viewed as a scalar index of variables like farming ability or soil quality. With g(..) increasing in [[Beta].sup.i], for a given water application level, a higher level of [[Beta].sup.i] (a higher type) is associated with higher yields. See Smith (1995) for an example of how a parameter like [[Beta].sup.i] has been quantified.
For a given [[Beta].sup.i], g can be inverted to give [w.sup.i] = W([y.sup.i], [[Beta].sup.i]), that is, W(..) satisfies W(g(w, …