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.]
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(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.
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