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