The gains from differentiated policies to control stock pollution when producers are heterogeneous.

Some major pollution problems including ground and surface water contamination, soil erosion, buildup of pesticides resistance, and even climate changes are frequently stock externality problems caused by heterogeneous sources (Hoel and Karp 2002; Xepapadeas 1992a). Many of these are nonpoint source pollution problems, so that it is not feasible to directly measure the pollution originating from each source. With information about production, it is feasible to design incentives to correct the externality problems (Shortle and Horan 2001; Xepapadeas 1992b). However, the optimal corrective measures (e.g., taxes of observable activities) may have to vary among sources to address their heterogeneity and the dynamics of the pollution stock buildup and its damage. Implementation of such complex policies has been minimal because of technical difficulty of implementation and the high transaction costs of policy design (Abdalla et al. 2007). However, improvements in monitoring and information technologies reduce the costs of implementing policies that track behavior of individual agents. Health and food quality concerns have led to development and adoption of new technologies to increase trace-ability of food-production activities. Wireless-based technologies (like Tiger JILL and Pocket JILL (1)) are being adopted in California to comply with chemical application reporting requirements.

There are some notable policy examples in which information technologies are used to differentiate among drivers in the case of traffic pollution and other externalities. In Singapore, for example, smart cards are used to assess location-specific driving fees, as part of an effort to reduce pollution and congestion externalities. Yet, policy makers prefer solutions that are simple to understand and easy to implement. Fully differentiated policies may require high transaction costs in terms of time spent negotiating, or political costs to reach a consensus. These costs are likely to increase the more diverse affected populations are or the greater the variation over time (Dixit 1996). The transaction costs may explain why policy makers may prefer second-best policies that do not vary over space and time very much, if at all. However, as new technologies lower the cost of implementation and monitoring, policy makers should consider the introduction of more efficient, differentiated policies, comparing the gains from these policies with the extra transaction costs they may entail. Policy design will benefit from better modeling that determines more accurately first-best policies as well as second-best policies and, consequently, provides quantitative estimates of the welfare losses caused by implementing policies that do not take into full consideration variations over space or time or both.

This article introduces a modeling framework that addresses the buildup of a stock pollution caused by heterogeneous agents. To make the analysis more concrete, we cast this framework in terms of an agricultural production problem concerning lands with varying quality where residues of applied input have accumulated, and the stock is a source of damage. We consider two strategies to control pollution: reducing the application of variable inputs and adopting precision (conservation) technology. While this model directly applies to pollution problems emanating from agricultural production, it can be easily modified to control stock pollution problems in industries such as mining and energy generation. In all of these industries, technological change has resulted in an emergence of precision technologies that increase the efficiency of variable inputs such as water and fuel and reduce polluting residues (Khanna and Zilberman 1997). New technologies also enable better policy formation by governments. Through improving data availability (e.g., use of wireless communication), reducing computational cost, and better monitoring (e.g., geographic information system [GIS], remote sensing), agencies can link unobserved pollution to observed action and institute policies that include best management practices as well as incentives for adoption of conservation strategies and reduction of input use.

Policy makers can induce first-best outcomes through incentives or permits that vary over time and space. This policy requires intensive and costly monitoring and constant modification. The optimal policies are then compared with second-best policies that are constant over land quality or time or both. (2) The article provides an analytical framework that allows determining the varying and nonvarying part of the policies optimally, and identifying conditions when the associated efficiency losses of the second-best policies are relatively small compared to first-best policies.

There is a strand of literature that focuses on measuring the cost-effectiveness of differentiated versus nondifferentiated policy instruments when land is heterogeneous. However, the articles reach different conclusions about the differences in the relative efficiency of alternative policies. Helfand and House (1995) analyze, in a static setting, the efficiency of different regulatory instruments to reduce nitrate leaching when pollution sources are heterogeneous. They found that uniform instruments do not lead to large welfare losses relative to a socially optimal solution. In contrast, Claassen and Horan (2001) use a market equilibrium simulation model to explore the differences in the relative efficiency of uniform and nonuniform input taxes when market prices are endogenous and found that differences in the relative efficiency of uniform and nonuniform taxes can be quite considerable.

Fleming and Adams (1997) use a dynamic programming approach to assess the importance of spatial variability in the design of efficient policies to control pollution. They evaluate the costs to producers of two different types of nitrogen taxes, a uniform tax and a tax that varies by location, to achieve a determined groundwater quality goal and found that the gains from a spatially differentiated tax are rather modest. However, although they use a dynamic setting, the policies they evaluate are not the optimal dynamic taxes. The initial value of the tax is adjusted in constant increments per year until the previously determined standard is achieved.

To our best knowledge, the previous studies that considered space overlooked the time dimension of pollution control. A dynamic framework is essential when evaluating the efficiency of alternative second-best instruments to control stock pollution. Uniform and differentiated policies have different capacities to affect the evolution of technology adoption and exit decisions over time to correct the buildup of pollution stock, and the heterogeneity of cultivated land will change over time. Therefore, the performance of uniform policies may be considerably affected by the dynamics and severity of the environmental problem.

Since the magnitude of the efficiency losses of the second-best policies cannot be determined analytically, we employ a numerical example that allows us to rank the different policy options for the studied case. The example is based on the waterlogging problem caused by irrigated cotton production in the San Joaquin Valley of California. Our empirical analysis shows that the efficiency losses of the different second-best policies depend particularly on the length of the planning horizon and the initial level of the pollution stock. In situations with significant initial environmental degradation, the imposition of a static but spatially differentiated tax leads to a smaller efficiency loss (in one example, 15%) in comparison to a spatially uniform policy that is adjusted over time (loss of 36%). However, if the initial pollution stock is sufficiently low, the ranking of these two policies is reversed, that is, the dynamic spatially uniform policy outranks the static spatially differentiated policy. Thus, if the environmental policy can be differentiated only in one dimension--either space or time--the optimal choice of the dimension depends on the state of the initial degradation.

We found other interesting results. It is clear a priori that the performance of any spatially uniform policy depends on the heterogeneity of the land quality. However, contrary to intuition, our results demonstrate that efficiency losses of a dynamic spatially uniform policy may decrease with an increase in the initial heterogeneity of the land quality. This result is explained by the fact that an optimal dynamic but spatially uniform tax increases over time, which drives the lowest quality land out of production and homogenizes the quality of the land that remains in production.

This article is organized as follows. We first introduce the economic model and define the optimal outcome from a social point of view. Next, we contrast this result with the optimal outcome from a private point of view, describe alternative policies to encourage farmers to behave optimally, and identify conditions when these policies may replace first-best policies at low cost. The following section presents the empirical part of the article and discusses the obtained results. The article closes with a summary and conclusions.

The Economic Model

Consider an agricultural region where a single crop is produced by profit-maximizing production units (fields), using lands of varying qualities and a variable input (water). Heterogeneity is denoted by [epsilon], [epsilon] [member of] [[epsilon].sub.0], [epsilon].sub.1]], reflecting land quality. In our model, it represents a measure of land quality, but it may be vintage in the case of machinery, or coal quality in the case of electricity generation. It is assumed that higher [epsilon] corresponds to higher land quality. The distribution of land quality is known. Its density function is denoted by l([epsilon]) > 0, [for all] [epsilon] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where X denotes the number of acres of land in the region. For simplicity, we concentrate on the case where two different technologies i, i = 1, 2, are available. The subscript i = 1 represents the precision technology, and i = 2 denotes the traditional technology. The share of land cultivated with technology i at any moment of calendar time t with quality [epsilon] is denoted by [x.sub.i](t, [epsilon]).

We assume constant returns to scale with respect to land. Thus, the production function per acre under technology i is f(hi ([epsilon])([u.sub.i] (t, [epsilon])). The variable [u.sub.i](t, [epsilon]) is the input per acre applied with technology i, i = 1, 2, and [h.sub.i]([epsilon]) denotes technical efficiency of input use, that is, the fraction of applied input that is utilized by the crop. It depends on the technology and the land quality. The function f(*) has regular properties of a neoclassical production function, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the subscript of a function with respect to a variable denotes its partial derivative. We assume that precision technology uses the input more effectively at each land quality, [h.sub.1]([bar.[epsilon]]) > [h.sub.2]([bar.[epsilon]]), [for all][epsilon] [[[epsilon].sub.0], [[epsilon].sub.1]]. (3) Additionally, we assume that land quality increases effectiveness of input use irrespective of the technology used, that is, [dh.sub.i]([epsilon])/d[member of] > 0, i = 1, 2. For example, in the case of irrigation, a larger fraction of the applied water will end up as drain water when the soil is sandy than when the soil is heavy. A transition from gravitational irrigation to drip will increase wateruse efficiency.

The product and input prices are denoted by p and c, respectively, and are assumed to be constant. Each technology differs in its operational cost per acre, denoted by [I.sub.i]. Operational cost includes the cost of inputs such as labor (e.g., costs of extra monitoring in some cases of precision farming), the rental or annualized cost of the equipment (e.g., costs of technology maintenance), and the cost of licensing or other fees associated with the used technology. We assume that operational cost per acre is higher for the precision technology than for the traditional technology, that is, [I.sub.1] > [I.sub.2].

The applied input that is not utilized by the crop can be a source of environmental degradation. Pollution per acre is given by [[gamma].sub.i]([epsilon]) [[gamma].sub.i] (t, [epsilon]), where [[gamma].sub.i] ([epsilon]) is the pollution coefficient per unit of applied input with technology i. It depends on the land quality and on technology choice i, i = 1, 2. We assume that the precision technology has a lower pollution coefficient than the traditional one, [[gamma].sub.2]([bar.[[epsilon]]) > [[gamma].sub.1]([bar.[[epsilon]]), [for all] [bar.[epsilon]] [member of] [[epsilon].sub.0], [[epsilon].sub.1]]. Knowledge of agricultural production suggests that effective uses of the input increase with land quality, and therefore the no-utilized part of the input responsible for pollution has to decrease with land quality, that is, d [y.sub.i]([epsilon])/d[epsilon] < 0. Pollution accumulates over time causing economic losses to the society. Let s(t) denote the stock of the pollutant at time t. The monetary damage per period resulting from the pollution stock is denoted by m(s(t)), with m(0) = 0, [m.sub.s], > 0, [m.sub.ss] [greater than or equal to] 0.

It would have been also possible to consider the case where the pollution increases with the quality of the land. This alternative corresponds frequently to cases of a contaminating byproduct of agricultural activities, when pollution originates from production itself. In this case pollution equals [alpha] f(*), where a is a pollution output ratio. One example is animal waste, often a contaminating byproduct of animal production. Space limitations lead us to focus our analysis on the case of a contaminating input, but major lessons apply to other cases.

The Socially Optimal Outcome

It is assumed that a social planner exists and maximizes the present discounted value of the net margin of agricultural production over time while taking into account the social-economic losses due to the accumulation of the pollutant. Given the regional focus of the analysis, we assume that prices are neither influenced by regional production decisions nor by the production of the externality. (4)

The social planner's decision problem is given by

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where a dot over a variable denotes the operator d/dt. The parameter so denotes the pollution stock at the initial point of calendar time, [delta] is the social discount rate, and [zeta], 0 < [zeta] < 1, is the decay rate of the pollutant stock.

The presence of the double integral in the objective function and of an integrodifferential equation in the constraints of problem (1) makes an analytical solution very difficult. To enhance the analytical tractability, Xabadia, Goetz, and Zilberman (2006) proposed a method that allows solving certain mathematical optimization problems in two stages. This method is applicable to problem (1) because the control variable is distributed over time and some qualitative aspect, but the state variable is distributed only over time. In their article, Xabadia, Goetz, and Zilberman analyze the first-best, long-run adoption pattern for different types of technologies from a theoretical perspective. The proposed method, however, is not only useful for a theoretical analysis, but also for empirical work as it allows solving problem (1) more easily by numerical techniques. In this article, we show how the two-stage method can be employed for a theoretical analysis, and how it can be reformulated so that it can be used for policy design and policy evaluation.

In the first stage, the social planner determines the optimal level of input use and the optimal technology choice at each land quality e for a prespecified level of emissions, z. In the second stage, the optimal intertemporal solution of the previously obtained optimal spatial solution is derived. The solution of the spatial social planner's decision problem in the first stage is given by the value function V(z) defined as:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To simplify the notation, the arguments t and e of the functions will be suppressed.

A solution of the problem (2) has to satisfy the following necessary conditions:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4) pf([h.sub.i][u.sub.i]) - c[u.sub.i] - [I.sub.i] - [lambda][[gamma].sub.i][u.sub.i] + [[upsilon].sub.i+2] - [[upsilon].sub.5] = 0, [for all][epsilon]

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the Lagrange multiplier [lambda] is interpreted as the shadow cost of the prespecified level of the aggregated emissions, z, and [[upsilon].sub.1], ... [[upsilon].sub.5] are the Kuhn-Tucker multipliers associated with the constraints on the control variables. The necessary condition (3) indicates for an interior solution that at every land quality and for each technology, the input should be applied up to the point where the value of the marginal product per acre equals the sum of the marginal cost of input use and of the marginal cost of generated emissions. Equation (4) governs the optimal choice of technology at every land quality e. Since the production and pollution functions are linear in land, the technology that leads to a higher quasi-rent per acre, defined as pf([h.sub.i] [u.sub.i]) - c[u.sub.i] - [I.sub.i] - [lambda][[gamma].sub.i] [u.sub.i], will be completely preferred to the technology with the lower quasi-rent. Hence, the technology that yields the highest quasi-rent should be adopted on the entire land available with quality [epsilon]. In the case where there is a marginal land quality, [[epsilon].sup.*.sub.m], that is, the quasi-rent is equal to zero, it will be optimal to abandon the lands with a quality below [[epsilon].sup.*.sub.m].

To analyze how the optimal technology choice and input use are affected over time, the value function V(z), obtained in the first stage, is maximized over time. Hence, the social planner's decision problem is given by

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where an asterisk indicates that the variable (function) takes on the optimal value of the first-best solution. The aggregated emissions, z, becomes the decision variable of the second stage. Thus, it now depends on t. The current value Hamiltonian is given by H [equivalent to] V(z) - m(s) - [phi](z - [zeta]s), where [phi] denotes the costate variable and has been multiplied by minus one in order to be positive. The first-order conditions read as

(7) [V.sub.z] - [phi] = 0, [??] [lambda](t) = [lambda](t)

(8) [lambda] = ([delta] + [zeta])[phi] - [m.sub.s]

(9) [??] = z - [zet]s, s(0) = [s.sub.0].

Equation (7) indicates that the marginal value of the aggregate emissions of the agricultural region should equal the shadow cost of the pollution stock, [phi](t), which, in turn, is equal to the shadow cost of the spatial allocation problem, [lambda](t). Equation (8) suggests that the cost of a one-period delay in generating a marginal unit of pollutant stock will be the additional discounting and decay of the shadow cost minus the temporal marginal social cost of the pollutant stock [m.sub.s].

Characterization of the First-Best Policy

Let us now assume that there are many competitive farmers in the area, and the pollution stock is a result of the emissions of every individual farmer. Since each individual farmer does not consider the externality, they will select their variable input and technology per unit of land to maximize their net income. Without an explicit policy intervention, their behavior will not lead to the social optimum. The following proposition defines a policy that assures at every moment of time the optimal applied amount of input and technology choice at every land quality [epsilon].

PROPOSITION 1. For a given amount of aggregated emissions z, and provided that the amount of input used and technology choices can be observed at each land quality [epsilon], an optimal policy can be obtained by a dynamic, technologically and spatially differentiated input tax [[tau].sup.*.sub.i] (t, [epsilon]), i = 1, 2, given by [[tau].sup.*.sub.i](t,[epsilon]) = [phi](t)[[gamma].sub.i]([epsilon]), i = 1, 2.

Proof: When a temporal, technological, and spatially differentiated input tax [tau](t,[epsilon]) is implemented, the farmers' private decision problem is to choose at every moment of time the input use and technology that maximizes their net income. The aggregate net income (ANI) of all farmers, ANI ([[tau].sub.i](t, [epsilon])), is defined as

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order conditions read as

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(12) pf([h.sub.i] [u.sub.i]) - c[u.sub.i] - [I.sub.i] - [[tau].sub.i][u.sub.i] + [[upsilon].sub.i+2] - [[upsilon].sub.5] = 0, i = 1,2.

Analyzing the first-order conditions of the private problem (11)-(12) and of the different stages of the social problem (3)-(5) and (7)-(9) shows that the optimal private choice of the technology employed [x.sub.i](t, [epsilon]) and of the level of input use [u.sub.i](t, [epsilon]) coincide with the socially optimal value of [x.sup.*.sub.i](t, [epsilon]) and [x.sup.*.sub.i](t, [epsilon]), provided that the input tax [[tau].sup.*.sub.i](t, [epsilon]) is set equal to [lambda](t)[[gamma].sub.i]([epsilon]) = [phi](t)[[gamma].sub.i]([epsilon]), i = 1, 2. Q.E.D.

Considering the heterogeneity of land implies that one should take into account both the spatial allocation and the temporal allocation in designing the correct policy in order to achieve the social optimum. Ignoring any of these two aspects will lead to inefficient outcomes.

One alternative to differentiated variable input taxation that will lead to the optimal policy is the introduction of tradable pollution permits. In this case, pollution at each location will be computed based on variable input use, technology, and quality; and the amount of permits to be distributed at each period at no cost is equal to the optimal level of emissions, [z.sup.*](t). The trading among profit-maximizing polluters will lead to the optimal outcome introduced earlier if abatement will be priced according to marginal cost pricing. The tradable permit scheme may be politically feasible in situations where taxation is not a viable option. (5)

Second-Best Nondifferentiated Policies

The optimal policy requires differentiation among land qualities, technologies, and changes over time, which may entail high transaction, monitoring, and control costs, and may encounter political constraints. Instead, the government may consider second-best policies that require less information or adjustments over time. In many occasions, the policy maker may not observe choices at the field level, but may be able to tax variable input (by monitoring and taxing the sellers of variable inputs). In this case, the regulator cannot implement an optimal technological and site-specific policy and instead has to implement an optimal, technological, and spatially uniform policy. In the remaining part of the article, we will refer to it simply as the optimal spatially uniform policy. If it is feasible to vary the tax over time, the policy maker enacts a dynamic but spatially uniform input tax, denoted by [[tau].sup.DU] ou (t). First, consider the case where there is only one technology available. The following proposition specifies the characteristics of the optimal, spatially uniform input tax.

PROPOSITION 2. For a given prespecified level of aggregated emissions, z, there is a unique spatially uniform tax rate [[tau].sup.DU] (t; z) that achieves z with the lowest loss in efficiency compared to all other spatially uniform tax rates. The optimal spatially uniform input tax is smaller than the optimal differentiated input tax imposed at the marginal land quality, [[tau].sup.*](t, [[epsilon].sup.*.sub.m]), that is, [[tau].sup.DU] (t) < [[tau].sup.*](t, [[epsilon].sup.*.sub.m]).

The proof is presented in the Appendix (Xabadia, Goetz, and Zilberman 2008). A spatially uniform input tax causes a distortion at the extensive margin. The minimum land quality for agricultural production to be profitable decreases from [[epsilon].sup.*.sub.m](first-best policy) to [[epsilon].sup.DU.sub.m] (spatial uniform policy). Consequently, additional land for which quality lies between [[epsilon].sup.DU.sub.m] and [[epsilon].sup.*.sub.m] comes into production, that is, the imposition of the optimal spatially uniform input tax leads to an increase in the cultivated land, and therefore it mitigates the exit effect of a newly imposed tax. Similarly, it stimulates the entry of farms that would not have entered the sector in the presence of a fully differentiated policy.

For policy analysis, it is convenient to reformulate the value function, V(z), of problem (2) as a value function of the aggregate net income ANI([tau]) and the collected taxes T ([tau]), that is, as V([tau]) = ANI([tau]) + T([tau]) since it allows the regulator to design and evaluate policies that are based on different tax regimes. In contrast, it would not have been possible to define and evaluate different policies based on the same aggregated emissions. The equivalence of V(z) and V([tau]) is formally established and demonstrated in a corollary in the Appendix (Xabadia, Goetz, and Zilberman 2008).

The application of the corollary allows us to characterize the efficiency losses of the optimal spatially uniform policy for the given level of the aggregated emissions, z. We define aggregate tax payment for the optimal spatially uniform policy, T([[tau].sup.DU]), in every moment of time as the amount of collected taxes given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the superscript DU denotes the evaluation of the variables at the values that correspond to the optimal spatially uniform policy. Taking into account that taxes revert to society, the value function is given by the aggregate net income of the farmers, ANI, plus the collected taxes, T([[tau].sup.DU]), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6). Hence, the efficiency losses of the optimal spatially uniform policy are given by

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Defining g as the land quality where [[tau].sup*]([bar.[epsilon]]) = [[tau].sub.DU], allows us to rewrite equation (13) as the sum of three components

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the evaluation of the integral, one has to keep in mind that the generated pollution has to be identical under both policies, and therefore, welfare losses depend only on the net benefits of production. The first integral corresponds to the effect of the change in the extensive margin, while the second and third integrals correspond to the effect of the change in the intensive margin. The first integral is negative since it indicates the ANI and the related foregone taxes of the land that would not be cultivated in the presence of the optimally differentiated tax. Graphically this integral corresponds to the area A of figure 1. This implies that the changes in the extensive margin partially compensate the welfare losses of not choosing the amount of input optimally. Moreover, farmers who cultivate land with a quality ranging from [[epsilon].sup.*.sub.m] to [bar.[epsilon]] apply input above its optimal value since the uniform tax applied is below the optimal differentiated tax. Therefore, the second integral is also negative, and the corresponding change in welfare is presented by area B of figure 1. In other words, the introduction of the optimal spatial uniform tax increases the net benefits of production on land that has quality inferior to [bar.[epsilon]]. However, the farmers with a land quality superior to [bar.[epsilon]] will use input below its optimal value, and thus the third integral is positive (see area C of figure 1). Hence, the overall welfare loss of the introduction of the optimal spatially uniform policy is given by the area C minus areas A and B. This result is the minimum welfare loss that the social planner can achieve if he/she implements the optimal spatially uniform policy.

[FIGURE 1 OMITTED]

A high value of d [gamma]/d [epsilon] in absolute terms implies that emissions increase considerably with a decrease in the land quality. Therefore, in order to achieve the prespecified level of emissions, [[epsilon].sup.DU.sub.m] will not differ substantially from [[epsilon].sup.*.sub.m] and the area A will be small. In a similar way, area C will be higher, the higher the changes are in the optimally adjusted production over space. Finally, efficiency losses also depend on the size of the land that is affected, that is, l([epsilon]). Figure 1 depicts the case where the land quality is uniformly distributed, that is, every land quality has the same number of hectares. However, if the land quality does not follow a uniform distribution, these areas will be weighted by l([epsilon]) and the welfare loss will change accordingly. We can summarize the previous discussion in the following way: A spatially uniform policy will result in higher efficiency losses the higher are:

(a) the changes of the optimally adjusted production over space,

(b) the changes of the emissions of the optimally adjusted production over space,

(c) and the number of hectares where the changes in i) and ii) take place.

If the regulator does not know the precise distribution of the land quality, he/she cannot implement the optimal spatially uniform policy and, therefore, the chosen spatially uniform policy leads to higher welfare losses. In this respect, the optimal spatially uniform tax presents a second-best solution, and all other spatially uniform taxes that are not derived by an optimization process over space and time can only be third best.

Comparing first- and second-best policies, one sees that the informational requirement for the design of first- or second-best polices is identical. However, the implementation of the optimal spatially uniform policy is easier than the implementation of the first-best policy. It requires less information since the regulator only needs to know the distribution of the land quality within the region but not the land quality of each farm.

Returning to the case where there are two technologies available, the introduction of a spatially uniform tax may change the land quality from where farmers switch from the traditional technology to the precision technology. Hence, it may lead to an additional distortion of the distribution of the two employed technologies over space. To achieve the optimal level of aggregated emissions, [z.sup.*], the optimal spatially uniform tax has to be set below the optimal differentiated input tax associated with the traditional technology, [[tau].sub.2], and above that of the precision technology, [[tau].sub.1].

Consequently, the optimal spatially differentiated policy favors the employment of the traditional and more polluting technology. As the graphical analysis is not substantially different from the case where there is only one technology, we do not present it here in order to save space.

When the government is not able to change the policy continuously, it may introduce a static but spatially differentiated policy. It will consist of a static, technology, and spatially differentiated tax on the variable input (applied water or fertilizer), denoted by [[tau].sup.SS.sub.i] ([epsilon]). The following proposition identifies the situations where efficiency losses of a static policy in comparison with a dynamic policy can be important.

PROPOSITION 3. A constant policy over time results in higher efficiency losses if

(a) the difference between the initial value of the stock and the steady-state stock is large, or

(b) the decay rate of the pollutant [zeta] is small, and changes in the emissions significantly affect the monetary damages ([m.sub.s] is large) and the value function ([V.sub.z] is large).

The proof is presented in the Appendix (Xabadia, Goetz, and Zilberman 2008). It shows that the optimal static tax is in between the initial and the final value of the spatially and temporally differentiated tax. Therefore, the efficiency losses of a static policy are smaller, the closer the initial value of the pollutant stock is to the steady-state value of the stock. Hence, if the environmental damages are severe at the point of time when the first policy actions are taken, it is recommendable to design a dynamic and not a static instrument. Moreover, the divergence of the two policies will be higher if the decay rate is low and the marginal damages of pollution are large, since the policy change will conduce to an important alteration of pollution damages. In the same way, a large marginal value of aggregated emissions implies that the adjustment of the emissions due to the imposition of a static tax will result in considerable changes in the benefits.

The previous literature (Claassen and Horan 2001; Fleming and Adams 1997; Helfand and House 1995) disagreed about the magnitude of efficiency losses caused by a switch from first- to second-best policies. In retrospective, the different conclusions of the literature may be explained by differences in the underlying exogenous conditions as described in Propositions 2 and 3.

Nevertheless, a general assessment of the order of magnitude of differences in social welfare of the different taxes is not possible (Claassen and Horan 2001). Therefore, a numerical analysis needs to be conducted to rank the different policies and to determine the magnitude of the inefficiencies of each policy resulting from a deviation from the socially optimal solution when the spatial or temporal or both dimensions of the problem are neglected. With this purpose, the next section analyzes the waterlogging problem in California's San Joaquin Valley.

Numerical Analysis

The drainage problem has been a major constraint for the sustainability of irrigated agriculture in parts of the central valley of California. Soils in this area are characterized by an impermeable layer that impedes the percolation of the irrigated water (drain water) below a certain depth. As a result, irrigation water accumulates at the subsurface leading to a considerable reduction of crop yields of all farmers within the watershed. (7) Research and policies aiming to solve this problem have been carried out since the 1930s, and the efforts have intensified since the Kestersen debacle of 1985. An overview of the problem and studies presenting solutions appear in Dinar and Zilberman (1991). Some of the solutions in the book and later studies are static but address heterogeneity across locations (Khanna, Isik, and Zilberman, 2002), while others provide policy solutions that vary over time but ignore heterogeneity (Shah, Zilberman, and Lichtenberg 1995; Weinberg, Kling, and Wilen 1993). Comparison of alternative solutions suggests that introduction of more efficient solutions require extra costs and the gain from improved outcome has to be evaluated to assess whether the implementation effort is worthwhile. Our numerical analysis does not aim to provide an extremely detailed study but rather to use its parameters to compare the benefits of alternative policies that vary in their degree, their adjustment to changes in time, and across locations.

The numerical analysis is based on the case of cotton, produced on 400,000 irrigated acres in the west side of the San Joaquin Valley in California. Farmers can choose between i irrigation technologies: i = 1 denotes drip irrigation and i = 2, furrow irrigation. Following Khanna, Isik, and Zilberman (2002), we specify the production function as f(*) = Max[0, -1589 + 2311(hi([epsilon])[u.sub.i] ([epsilon]))--462[([h.sub.i]([epsilon])[u.sub.i]([epsilon])).sup.2]], where [u.sub.i](t, [epsilon]) denotes the amount of applied water per acre associated with technology i, and [h.sub.i]([epsilon]) is the irrigation effectiveness, that is, the fraction of the applied water that is effectively utilized by the crop. Within the context of this study, land quality stands for the capacity of the land to retain the applied water such that it is available for crop uptake and does not reach the impermeable layer where it leads to a depletion of the water-storage capacity. Thus, flat and heavy soils are assigned high values of [epsilon], while steeper lands and sandy soils are assigned lower values of [epsilon]. The land-quality index is calibrated such that it coincides with the irrigation effectiveness of the traditional technology, that is, [h.sub.2]([epsilon]) = [epsilon]. Given this calibration, the quality of the land of the considered irrigated area ranges from 0.2 (steep and sandy soils) to 0.8 (flat and heavy soils). It is assumed that land quality is uniformly distributed, with an average land quality of 0.5. Khanna, Isik, and Zilberman (2002) also provide information that allows calibrating the efficiency of drip irrigation with constant elasticity, which is given by [h.sub.1] ([epsilon]) = [[epsilon].sup.0.1]. The part of the applied water that is not utilized by the crop can percolate below the crop-root zone. Following Khanna, Isik, and Zilberman (2002), the pollution coefficient (drainage) for each technology is specified as [[gamma].sub.1]([epsilon]) = [(1 - [h.sub.1]([epsilon])).sup.1.074], and [[gamma].sub.2]([epsilon]) = [(1 - [h.sub.2]([epsilon])).sup.1902]. The cotton price is assumed to be $0.65 (U.S.) per pound, and water price is $55 (U.S.) per acre-foot (AF). Operational costs of furrow are taken to be $500 (U.S.) per acre, while operational costs of drip are $633 (U.S.) per acre. Similar values of the parameters and coefficients have been used by Schwabe, Kan, and Knapp (2006).

The social discount rate is set equal to 0.04. Let g denote the water-storage capacity of the land. It is assumed that the production function is independent of the stock of drain water s(t), while the top level of the drain water is below the crop-root zone, that is, the water-storage capacity is not yet depleted. Above this level, where s(t) > [bar.s], the soil is not productive anymore, and f(*) = 0 for s(t) > [bar.s].

Comparison of Differentiated Versus Nondifferentiated Policies

Assessing the magnitude of inefficiencies of the second-best policies and their rankings requires incorporating the biophysical and economic parameters of each particular problem. This section analyzes the performance of alternative policies in the context of the drainage problem. In this way, we can determine the magnitude of the inefficiencies of deviations from the socially optimal solution. With this purpose, the optimal private outcome and the social outcome are computed for a water-storage capacity of 5 feet. Moreover, to investigate the magnitude of these inefficiencies, we determined the optimal static spatially differentiated tax [[tau].sup.SS.sub.i] ([epsilon]), the optimal dynamic spatially uniform tax, [[tau].sup.DU] (t); and the optimal static spatially uniform tax, [[tau].sup.SU] , and the social welfare obtained, denoted by W([[tau].sup.DU] (t)), W([[tau].sup.SS.sub.i] ([epsilon])), and ([[tau].sup.SU), respectively. Table 1 summarizes the optimal taxes of the different policies considered, and it illustrates the implications of these policies on technology adoption, land allocation, and welfare.

As shown in table 1, in the absence of any drainage control policy, the private outcome leads to the cultivation of the entire land of the region. Drip irrigation will be adopted at the lowest-quality lands, up to a land quality of 0.47, which represents 46% of the available land of the region, while furrow irrigation will be used at 54% of the land. Annual water use in the region is 1,308 thousand AF, cotton production is 517 million pounds, and the generated drain water is 178 thousand AE This production plan leads to a complete depletion of the water-storage capacity in about eleven years, leading to a discounted sum of private welfare of $369 millions (U.S.).

Fully Differentiated Policy

To induce farmers to behave optimally from a social point of view, the social planner could impose a spatially and temporally differentiated water tax, given by [phi](t)[(1 - [[epsilon].sup.0,]).sup.1.074] and [phi](t)[(1 - [epsilon]).sup.1.902] for drip and furrow irrigation, respectively. The optimal value of the shadow cost at each point in time, [phi](t), is determined by the parameter values of the model.

In our example, the optimal spatially and temporally differentiated taxes in the first year on water applied with drip irrigation are given by $17.9/AF, $7.6/AF, and $2.3/AF on the low(e = 0.2), average- ([epsilon] = 0.5), and high-quality ([epsilon] = 0.8) lands, respectively (see table 1), and the optimal taxes in the first year on water applied with furrow irrigation are $90.8/AF, $37.1/AF, and $6.5/AF, respectively. The introduction of the fully differentiated tax scheme increases the share of drip-irrigated land from approximately 45.9% to 72.2% compared to the case where no policy is implemented. Applied water decreases initially by 17.1% and drainage by 52.7%. As shown in Proposition 1, the optimal spatially and temporally differentiated water tax increases over time at the discount rate, (8) leading to a substitution of furrow irrigation initially by drip and afterward by idle land, until sixty-five years, when the drain-water-storage capacity is completely depleted. The implementation of the socially optimal policy leads to a welfare gain of 64.2%, where the welfare gains are computed as WG ([[tau].sup.*.sub.i](t, [epsilon])) = with W ([[tau].sup.*.sub.i] (t, [epsilon])) - W (0)/W (0) W(0) being the welfare of the baseline scenario, that is, the welfare obtained when no policy is implemented.

Partially Differentiated Policies

Table 1 shows that the optimal static spatially differentiated tax scheme would require imposing a tax of $31/AF, $13.1/AF, and $4/AF, respectively, on water applied with drip irrigation at the low-, average-, and high-quality lands, and of $156/AF, $64.2/AF, and $11.2/AF on water applied with furrow irrigation. Since the tax cannot be adjusted over time in response to the increased depletion of the water-storage capacity, this policy measure is initially harsher than the optimal spatially and temporally differentiated tax. Consequently, applied water decreases initially by 31.9% compared to the decrease of 17.1% with the optimal instrument. Compared to the private outcome, the implementation of a static spatially differentiated tax increases the adoption of drip irrigation only by 16.5%, from 45.9% to 62.4%; while 17.6% of the land must be retired from production. Retirement of land causes a decrease in cotton production of 17.7% in the first year. The welfare gain of a static tax, WG([[tau].sup.SS.sub.i]([epsilon])), is 55.4%, that is, the imposition of a static spatially differentiated tax increases the welfare relative to the private outcome by 55.4%. In order to analyze the efficiency of second-best policies, we compute their efficiency loss with respect to the first-best policy, given by EL([[tau].sup.**].sub.i]) = [WG([[tau].sup.*.sub.t]((t, [epsilon])) - WG([[tau].sup.**.sub.i])) / WG([[tau].sup.*.sub.i](t, [epsilon])), where [[tau].sup.*.sub.i] corresponds to the tax of the spatially and temporally differentiated policy and [[tau].sup.**.sub.i] corresponds to the tax of the analyzed second-best policies. The relative efficiency loss of the static spatially differentiated tax is 13.8%, implying that the imposition of a static tax achieves about 86.2% of the welfare gains that would be obtained with the optimal spatially and temporally differentiated tax scheme and, thus, 13.8% of the potential welfare gains are lost.

Alternative forms of second-best policies are optimal dynamic spatially uniform taxes. These taxes evolve over time but do not differentiate according to land quality. Our analysis finds that these dynamic spatially uniform taxes are not capable of stimulating the same level of adoption of drip irrigation as spatially differentiated policies. The optimal dynamic spatially uniform tax has to be initially set at $21.3/AF, which is in between the optimal spatially and temporally differentiated tax on applied water with drip irrigation and on applied water with furrow irrigation at the low and average land quality, ([epsilon] = 0.2)and ([epsilon] = 0.5). Hence, this policy favors the use of furrow irrigation above the socially optimal level. The initial share of land with drip irrigation is only 50%, compared to the 72.2% of the socially optimal policy. Although the spatially uniform policy is able to cut the initial water use by about the same percentage as the socially optimal policy (16.7%), the drain-water flow is only cut back by 37.1%, compared to the cut back of 52.7% achieved by the socially optimal policy. The implementation of a dynamic spatially uniform tax leads to a welfare gain of 41.3%, relative to no intervention, and results in an efficiency loss of 35.7%, compared to the optimal spatially and temporally differentiated policy.

Completely Nondifferentiated Policy

Finally, when the regulator is not in the position to implement neither a spatially differentiated nor a temporally differentiated policy, a static and spatially uniform tax on applied water of $23.4/AF must be imposed on all land qualities. As shown in table 1, the share of land irrigated with drip or furrow technology decreases from 45.9% to 37.8%, and from 54.1% to 40.4%, respectively. Consequently, the share of idle land increases to 21.8%, which, in turn, leads to a decrease in cotton production by 22.1%. Since the drain-water flow decreases only by 46.8%, the water-storage capacity is depleted ten years later than in the private outcome, that is, the duration of agricultural production is extended by only ten years. Compared to no intervention, the welfare gain of a static spatially uniform tax is 32%, which is equivalent to an efficiency loss of 50.1%, that is, the implementation of this policy would attain less than half of the welfare gains of the optimal policy.

Sensitivity Analysis

One should expect that the magnitude of the inefficiency of spatially uniform policies depends on the heterogeneity of the land quality within the region. Thus, in order to measure to what extent the heterogeneity of land affects the performance of the analyzed policies, the decision problem was solved for different underlying land-quality distributions obtained by varying the variance of the distribution. The land-quality distribution is characterized by the beta distribution, since it allows a wide variety of shapes. Besides the uniform distribution of the land quality, where the two parameters of the beta distribution, denoted by [alpha] and [beta] are equal to 1, three n-shaped distributions ([alpha] = [beta] > 1) and two u-shaped distributions ([alpha] = [beta] < 1) were used. However, for each prespecified distribution, the optimal private outcome produces a different amount of drain-water flow; therefore, a comparison of spatially uniform policies is not straightforward. In order to allow for comparisons, the private decision problem was solved by setting the initial water-storage capacity for each of the land distributions, such that all private outcomes have identical welfare levels.

[FIGURE 2 OMITTED]

The efficiency losses of the second-best policies are depicted in figure 2. It shows that land heterogeneity has a significant impact on the efficiency losses of the different policies. With low land heterogeneity (variance = 0.01), the efficiency loss of a static but spatially differentiated tax is 11.69% and of a dynamic but spatially uniform tax is 18.30%. However, as land heterogeneity increases, the efficiency loss of the spatially uniform policies augments. In the case of the dynamic spatially uniform tax, the efficiency losses reach a peak for a variance of the land quality of 0.03. At this level of variance, the land quality is uniformly distributed ([alpha] = [beta] = 1). A further increase in land heterogeneity produces the somehow counterintuitive result that the efficiency losses tend to decrease as the initial heterogeneity of land increases further. This development can be explained as follows: If the land-quality distribution is u-shaped ([alpha] = [beta] < 1), the cultivated land of the region is initially concentrated at the lower and upper ends of the distribution. However, as time passes by, the dynamic spatially uniform tax increases and the low-quality land is taken out of production. Thus, the quality distribution of the remaining cultivated land becomes more homogeneous. Consequently, the inaccuracy of the dynamic spatially uniform tax is reduced over time and the overall welfare losses are lower. In contrast, the static spatially differentiated tax reaches its peak for much higher values of the variance of land quality so that the ranking of these two policies is reversed from a variance of land quality of 0.05 onward.

As shown in the Appendix (Xabadia, Goetz, and Zilberman 2008), the ranking of the policies is not only determined by the land heterogeneity, but also by the initial water-storage capacity. The higher the initial water-storage capacity, the less favorable is the static spatially differentiated tax in comparison with the dynamic spatially uniform tax. Thus, the intersection between the dynamic spatially uniform tax and the static spatially differentiated tax depicted in figure 2 shifts to the left as the water-storage capacity increases. In other words, high values of the land heterogeneity and of the initial water-storage capacity suggest employing a dynamic spatially uniform policy instead of a static spatially differentiated tax. Finally, figure 2 also shows that the efficiency loss of the static and spatially uniform tax is always higher than any other policy considered.

Hence, our analysis shows that: (a) the ranking of the instruments can be reversed for different levels of heterogeneity, and (b) for the same level of heterogeneity, the efficiency loss of the different policies depends on the severity of the initial environmental problem.

Summary and Conclusions

The continuing improvement in computation and communications technologies has been expanding the capacity to establish more precise policy controls (incentives, regulations) and institutions (tradable permits) for better management of stock pollution problems. Policy makers need to calculate the gains from improved precision to determine whether the introduction of tools capable to adjust to variation over space and time are worth the extra economic and political efforts they may entail. This article develops a method that utilizes two-stage optimal control techniques to determine optimal parameters of policies to control stock pollution caused by heterogeneous producers with different information. The analytical framework permits either designing the optimally differentiated policy or optimally nondifferentiated policies. Within an empirical setup, this approach allows for a quantification of the gains from more precise policies.

We applied this approach to the well-studied problem of waterlogging in irrigated cotton production in the central valley of California to assess the order of magnitude of gains from more precise policies that take advantage of improved computation and communication technologies. The numerical analysis shows gains from policy interventions can be significant. For instance, there is a 65% increase in social welfare due to optimal intervention, relative to nonintervention, when the water-storage capacity is 5 feet and there is no disposal of drain water. We assume that regulators can choose between three different policy formulations. The first is a variable static policy that varies based on quality and technology differences, the second is a dynamic but spatially uniform policy, and the third and final case is a static and spatially uniform policy.

The static but spatially differentiated tax is initially able to achieve almost the same level of technology adoption than the socially optimal policy. However, the infeasibility of adjusting the policy over time makes it necessary to increase the harshness of the policy measure in the initial periods leading to a welfare loss of 14% compared to the socially optimal policy. The imposition of a dynamic but spatially uniform policy favors the cultivation of the low-quality land and the employment of more polluting technology. Moreover, it shifts the intensity of production from high- to low-quality lands in comparison with an optimal differentiated tax. Thus, this policy does not stimulate the same level of water-saving technology adoption as spatially differentiated policies. The efficiency loss of the dynamic but spatially uniform policy is 36%. Finally, the welfare loss from the static and uniform policy relative to the optimal policy is greater than 50%. While differentiated policies are not widely used to address stock pollution problems at the present, they are being considered, and this article provides a method for evaluation and suggests that they provide significant gain.

The results also show that the efficiency losses of the different policy measures and their rankings depend on the initial pollution stock (unutilized drainage-storage capacity in our case) and on the degree of land heterogeneity. The loss from static policies tends to increase with most of the considered values for land heterogeneity. In the case of the dynamic but spatially uniform policy, the efficiency loss also increases initially with land heterogeneity, but beyond a certain and not extreme level, the efficiency loss declines with heterogeneity such that the dynamic spatially uniform policy is superior to all other second-best policies considered. The superiority of the dynamic spatially uniform policy is reinforced by an increase in the initial value of the water-storage capacity.

The model presented in this article considers the case of financial incentives in the form of taxes to correct an externality problem. However, the framework allows extending our analysis to evaluate the efficiency of tradable rights. With heterogeneity and differentiated technologies, the main challenge in introducing trading systems is the development of an accounting system that weighs input use in different locations according to quality and technology. The presented analysis could also be expanded by incorporating uncertainty using the Dixit-Pindyck (1994) real-option model. Similarly, it may be interesting to consider time lags between the polluting activities and the presence of the pollutant in the environmental media, or the evolution of technological progress in production or pollution abatement. Future research should compare differentiated and nondifferentiated policies when these additional effects are present.

[Received November 2006; accepted March 2008.]

References

Abdalla, C., T. Borisova, D. Parker, and K. Saacke Blunck. 2007. "Water Quality Credit Trading and Agriculture: Recognizing the Challenges and Policy Issues Ahead." Choices 22(2):117-24.

Claassen, R., and R.D. Horan. 2001. "Uniform and Non-Uniform Second-Best Input Taxes." Environmental and Resource Economics 19(1):1-22.

Dinar, A., and D. Zilberman, eds. 1991. The Economics and Management of Water and Drainage in Agriculture. Norwell, MA: Kluwer Academic Publishers.

Dixit, A.K. 1996. The Making of Economic Policy: A Transaction Cost Politics Perspective. Cambridge, MA: MIT Press.

Dixit, A., and R. Pindyck. 1994. Investment Under Uncertainty. Princeton, NJ: Princeton University Press.

Fleming, R.A., and R.M. Adams. 1997. "The Importance of Site-Specific Information in the Design of Policies to Control Pollution." Journal of Environmental Economics and Management 33(3):347-58.

Helfand, G.E., and B.W. House. 1995. "Regulating Nonpoint Source Pollution Under Heterogeneous Conditions." American Journal of Agricultural Economics 77(4):1024-32.

Hoel, M., and L. Karp. 2002. "Taxes Versus Quotas for a Stock Pollutant." Resource and Energy Economics 24(4):367-84.

Khanna, M., and D. Zilberman. 1997. "Incentives, Precision Technology and Environmental Protection." Ecological Economics 23(1):25-43.

Khanna, M., M. Isik, and D. Zilberman. 2002. "Cost-Effectiveness of Alternative Green Payment Policies for Conservation Technology Adoption with Heterogeneous Land Quality." Agricultural Economics 27(2):157-74.

Schwabe, K.A., I. Kan, and K.C. Knapp. 2006. "Drainwater Management for Salinity Mitigation in Irrigated Agriculture." American Journal of Agricultural Economics 88(1):133-49.

Shah, F., D. Zilberman, and E. Lichtenberg. 1995. "Optimal Combination of Pollution Prevention and Abatement Policies: The Case of Agricultural Drainage." Environmental and Resource Economics 5(1):29-49.

Shortle, J.S., and R.D. Horan. 2001. "The Economics of Nonpoint Pollution Control." Journal of Economic Surveys 15(3):255-90.

Weinberg, M., C.L. Kling, and E. Wilen. 1993. "Water Markets and Water Quality." American Journal of Agricultural Economics 75(2):278-91.

Xabadia, A., R.U. Goetz, and D. Zilberman. 2006. "Control of Accumulating Stock Pollution by Heterogeneous Producers." Journal of Economic Dynamics & Control 30(7):1105-30. doi:10.1016/j.jedc.2005.04.002.

--. 2008. "AJAE Appendix: The Gains from Differentiated Policies to Control Stock Pollution when Producers are Heterogeneous." Unpublished manuscript. Available at: http://agecon.lib.umn.edu/.

Xepapadeas, A.P. 1992a. "Environmental Policy Design and Dynamic Nonpoint-Source Pollution." Journal of Environmental Economics and Management 23(1):22-39.

--. 1992b. "Optimal Taxes for Pollution Regulation: Dynamic, Spatial and Stochastic Characteristics." Natural Resource Modeling 6(2):139-70.

(1) See Orange Enterprises, Inc., www.orangesoftware.com for further details.

(2) Without loss of generality, we only consider the case where the policies are continuously differentiated or not. The case of weak and discretely differentiated policies (only a very few land qualities) is not taken into account as it would not affect the analytical framework of this method. Simply note that the range of the integrals over time or space has to be divided into the sum of integrals where the range of each integral corresponds to the partial range where the instrument is constant.

(3) The inequality does not include [[epsilon].sub.1] since it may be the case that [h.sub.1]([[epsilon].sub.1]) = [h.sub.2]([[epsilon].sub.1]).

(4) Any changes in consumer surplus as a result of the imposition of a tax are considered to be zero since our study deals with a small agricultural region. In other words, it is assumed that prices remain constant and the utility of the consumers is quasi-linear with respect to environmental quality.

(5) This economic model does not allowing banking of the permits as it would require adding another decision variable (banking), and a stock variable (banked permits). We think that these considerations are an interesting extension of this work, and we would like to thank one of the reviewers for drawing our attention to this point.

(6) For a precise definition of ANI, please see equation (10), where ANI([[tau].sup.DU]) is obtained from ANI([tau]([epsilon])) by replacing [[tau].sub.i]([epsilon]) with the optimal spatially uniform tax.

(7) In other words, the water-storage capacity of the land can be considered as a shared good of all farmers, and thus the optimal private depletion of the water-storage capacity does not coincide with the socially optimal depletion strategy.

(8) The maximum tax rates on applied water that the farmer can withstand if drip irrigation is used are given by $19.5/AF, $26.7/AF, and $30.6/AF at the low-, average-, and high-quality lands. If furrow irrigation is used, these values change to $0/AF, $17.6/AF, and $61.1/AF, respectively. When water taxes exceed these maximum tax rates, the ANI turns negative, and farmers have to cease production. Thus, whenever the increase in the shadow cost leads to optimal tax values, which are higher than these maximum tax rates, they may be established in practice at the maximum tax rates, producing the same outcome.

Angels Xabadia and Renan U. Goetz are assistant and associate professors, respectively, in the Department of Economics, University of Girona, Spain. Goetz is also affiliated with the Centre de Referencia en Economia Analitica, Spain. David Zilberman is professor and Robinson Chair in the Department of Agricultural and Resource Economics and Member of the Giannini Foundation of Agricultural Economics, University of California, Berkeley.

The authors are grateful to Associate Editor Stephen Swallow and three anonymous referees for useful remarks that helped to define the economic contribution of the article more precisely. They also acknowledge the support of Ministerio de Ciencia y Tecnologia Grant AGL 2007-65548, INIA Grants (SUM 2006-00019-C02-01, RTA04-141-C2-2), and Generalitat de Catalunya Grants (XREPE and 2005SGR213). Table 1. Effects of Alternative Policies on the Level of Adoption and Welfare (where s = 5) Time of Spatially and Static Exhaustion Temporally Spatially (Years) Differentiated Differentiated

Tax Tax

65 35 Value of variables in the first year: Tax (furrow irrigation)

[epsilon] = 0.2 90.8 156.0

[epsilon] = 0.5 37.1 64.2

[epsilon] = 0.8 6.5 11.2 Tax (drip irrigation)

[epsilon] = 0.2 17.9 31.0

[epsilon] = 0.2 7.6 13.1

[epsilon] = 0.2 2.3 4.0 Marginal land quality 0.213 0.306 Switching land quality 0.646 0.680 Share of drip (%) 72.2 62.4 Share of furrow (%) 25.7 20.0 Share of idle land (%) 2.1 17.6 Applied water ([10.sub.3] AF) 1,085 891

(-17.1) (a) (-31.9) Drain-water flow ([10.sub.3] AF) 84 57

(-52.7) (-67.8) Yield ([10.sub.6] lb.) 506 425

(-2.2) (-17.7) Discounted sum of economic variables Aggregate net income ([10.sub.3] $) (ANI) 326,922 309,700

(-11.4) (-16.1) Collected taxes ([10.sub.3] $) (T) 279,145 263,683 Welfare ([10.sub.3] $) (W) 606,067 573,383 Welfare Gain (WG)(%) (64.2) (55.4) Efficiency loss (EL)(%) -- 13.8 Time of Dynamic Static Exhaustion Spatially Spatially (Years) Uniform Uniform

Tax Tax

55 21 Value of variables in the first year: Tax (furrow irrigation)

[epsilon] = 0.2 21.3 23.4

[epsilon] = 0.5 21.3 23.4

[epsilon] = 0.8 21.3 23.4 Tax (drip irrigation)

[epsilon] = 0.2 21.3 23.4

[epsilon] = 0.2 21.3 23.4

[epsilon] = 0.2 21.3 23.4 Marginal land quality 0.252 0.331 Switching land quality 0.552 0.558 Share of drip (%) 50.0 37.8 Share of furrow (%) 41.4 40.4 Share of idle land (%) 8.6 21.8 Applied water ([10.sub.3] AF) 1,091 942

(-16.7) (-28.0) Drain-water flow ([10.sub.3] AF) 112 94

(-37.1) (-46.8) Yield ([10.sub.6] lb.) 471 403

(-9.0) (-22.1) Discounted sum of economic variables Aggregate net income ([10.sub.3] $) (ANI) 162,055 166,039

(-56.1) (-55.0) Collected taxes ([10.sub.3] $) (T) 359,345 321,234 Welfare ([10.sub.3] $) (W) 521,400 487,273 Welfare Gain (WG)(%) (41.3) (32.0) Efficiency loss (EL)(%) 35.7 50.1 Time of Exhaustion Private (Years) Outcome

(No Policy)

11 Value of variables in the first year: Tax (furrow irrigation)

[epsilon] = 0.2

[epsilon] = 0.5 --

[epsilon] = 0.8 -- Tax (drip irrigation) --

[epsilon] = 0.2 --

[epsilon] = 0.2 --

[epsilon] = 0.2 -- Marginal land quality 0.2 Switching land quality 0.475 Share of drip (%) 45.9 Share of furrow (%) 54.1 Share of idle land (%) -- Applied water ([10.sub.3] AF) 1,308 Drain-water flow ([10.sub.3] AF) 178 Yield ([10.sub.6] lb.) 517 Discounted sum of economic variables Aggregate net income ([10.sub.3] $) (ANI) 369,084 Collected taxes ([10.sub.3] $) (T) -- Welfare ([10.sub.3] $) (W) 369,084 Welfare Gain (WG)(%) -- Efficiency loss (EL)(%) 100 (a) All values in parentheses correspond to the percentage of change with respect to the private outcome.