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


by Xabadia, Angels^Goetz, Renan U.^Zilberman, David

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 instrume