a dynamic general equilibrium analysis of environmental tax incidence
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A Dynamic General Equilibrium Analysis of Environmental Tax Incidence
Governments often choose to regulate environmental damage by taxing the
negative externality. The effect of this policy is that consumers, firms, capital owners,
and workers may share some of the tax burden (Garrett 2005). A partial equilibrium
model cannot adequately capture this incidence, but a general equilibrium model
succeeds at this task, especially when the taxes impact large segments of the economy
(Kotlikoff and Summers 1987). In addition, a general equilibrium model allows
government policy makers to understand more clearly the distributional effects of an
environmental tax (Garrett 2005).
Some previous literature uses a general equilibrium model to examine
environmental tax incidence. These models often assume that input factors, such as labor
and capital, are perfectly mobile across sectors (see Fullerton and Heutel 2005). Since
the results of these models differ when researchers relax this assumption, some
researchers, including Garrett (2005), build costs of adjustment into the model. This
change allows the model to capture the short-run effects of tax incidence with imperfect
factor mobility, but a drawback is that this static model does not reach a long-run
equilibrium. In addition, static models abstract from intertemporal issues of capital
formation, savings and investment decisions, and capitalization effects (Kotlikoff and
Summers 1987). The results of these static models substantially change in a dynamic
setting since a dynamic long-run model can capture some richer aspects of tax incidence
by accounting for intertemporal economic decisions, as well as the full effects of
adjustment costs.
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The purpose of this paper is to examine environmental tax incidence by
incorporating the framework of imperfect factor mobility of Garrett (2005) into a
dynamic general equilibrium model. By constructing a dynamic model, this paper
examines the short- and long-run effects and distributional burden of a one-time increase
in a pollution tax. Like Garrett (2005), this paper considers the following two distinct
possibilities: firms face capital adjustment costs when they alter their capital stock,
whereas dismissed workers bear the labor adjustment costs when changing jobs. First,
this paper models the capital adjustment costs using a framework similar to that of
Goulder and Summers (1989). However, rather than using the computable general
equilibrium approach of Goulder and Summers (1989), this paper derives an analytical
and graphical solution to the dynamic models. The second part of this paper then allows
firms to perfectly adjust all of their factors of production, but incorporates adjustment
costs for a worker into a dynamic structure. By developing the framework of Garrett
(2005) into a dynamic setting, this paper re-evaluates tax incidence.
The contributions of this paper are that it presents a dynamic general equilibrium
model for examining environmental tax incidence with imperfect factor mobility. This
paper borrows certain aspects from previous static models, including the Fullerton and
Heutel (2005) environmental extension of the Harberger (1962) model, and the Garrett
(2005) extension of relaxing the assumption of perfect factor mobility. Like Garrett
(2005), for simplification purposes this paper only considers one clean input into
production in the economy, which can represent either capital or labor. Nonetheless, this
paper differs from Garrett (2005), in that it presents a dynamic model to allow for the
attainment of transitional dynamics and a long-run equilibrium.
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The results for this paper show that the long-run solutions to a dynamic general
equilibrium model do not noticeably differ when accounting for the nature of the
adjustment costs. Garrett (2005) finds that capital adjustment costs for a firm lead to
changes in the prices and rates of return relative to the no adjustment cost case. On the
contrary, the result for labor adjustment costs for a worker is that prices and wages do not
change, but labor supply and pollution significantly decrease. Nonetheless, by
considering a dynamic model, this paper illustrates the attainment of a long-run
equilibrium and its transition path for the two types of adjustment costs. This paper finds
that the long-run result for both cases is the same as it is for a static model with no
adjustment costs.
The first section of this paper presents a dynamic general equilibrium model of
firms facing adjustment costs when they alter their capital stock. Section II considers a
less traditional case in the literature of displaced workers facing adjustment costs;
however, in this model, firms do not face adjustment costs when hiring or firing workers.
Finally, Section III concludes and ponders future research in this field.
I. Model with Capital Adjustment Costs for Firms
Previous examples in the literature, both theoretical and empirical, illustrate why
firms may face adjustment costs when altering their clean factors of production, such as
labor and capital (see Summers 1981, Goulder and Summers 1989, Hamermesh 1989,
and Hamermesh and Pfann 1996). The present model uses many aspects of the Goulder
and Summers (1989) model, except this model offers less generality since it does not
include both capital and labor as clean factors of production. Like Garrett (2005), this
model uses only one clean factor of production in order to attain an analytical solution.
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Goulder and Summers (1989) include more generality in their model, but they solve their
model computationally. Like Fullerton and Heutel (2005) and Garrett (2005), this model
examines a general equilibrium economy when the government exogenously increases its
tax on pollution. Incorporating the features of Goulder and Summers (1989) that output
is separable between the production and adjustment costs, and that production is less than
output when firms adjust their capital stock, the production functions are the following:
XXX IKX = (1.1)
YYY IZKYY = ),( (1.2)
where X is the clean good, Y is the dirty good, XK and YK are capital inputs into
production, Z is the pollution input into production, X and Y are capital adjustment
cost functions for the firm, andXI and YI are the net investments in capital. Net
investment, which includes the depreciation rate of capital over time, adheres to the
following transition equations:
XX IK =& (1.3)
YY IK =& (1.4)
where XK& and YK
& are the rates of change in capital in each sector of the economy. The
resource constraint for capital is the following:
KKK YX =+ (1.5)
where K is the total fixed capital stock in the economy (Harberger 1962). Although the
aggregate capital stock is fixed, capital in each sector can vary. Since this model assumes
a fixed capital stock, households do not face a savings decision, and consequently, the
household intertemporal elasticity of substitution does not make a difference in their
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maximization problem. Thus, the following equation characterizes the maximization
problem of a representative consumer:
)//()/(
)//()/(
XYXY
Uppppd
YXYXd= (1.6)
where U is the intratemporal elasticity of substitution between the two goods, and this
elasticity depends on the two output prices,Xp and Yp (Harberger 1962). Since markets
are perfectly competitive, households and firms take these two output prices as given
when making their optimizing decisions. Also, firms face no market price for pollution
except for a tax, so the price of pollution,Zp , is equal to the governmental tax on
pollution,Z (Fullerton and Heutel 2005). In order to rule out the case when the input
usage of pollution is infinite (i.e. 0=Z ), I assume that the government levies a positive
tax on pollution in the initial equilibrium (Fullerton and Heutel 2005). Thus, the profit-
maximizing problem for the representative firm for the clean good is to choose XI in
maximizing the following objective function:
XXXXXX KrIKp )( (1.7)
subject to Equation (1.3),
whereXr is the net return to capital owners when they invest in the clean good. This
paper uses the most basic version of quadratic adjustment costs for both the clean and
dirty good (Cahuc and Zylberberg 2004). As a result, the adjustment costs take the
following form:
2)(XXXX IbI = (1.8)
2)( YYYY IbI = (1.9)
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whereXb and Yb are positive constants. Therefore, the Hamiltonian for the clean good
becomes the following:
][))(( 2 XKXXXXXXX IKrIbKpH X+= (1.10)
The first-order condition for the control variable of net investment is the
following:
XKXXXpIb =2 (1.11)
The interpretation of (1.11) is that the representative firm optimizes so that the
marginal adjustment cost times the output price, or the marginal cost of investment,
equals the shadow price on capital, or the marginal benefit of investment. Next, the co-
state equation for the state variable, capital, is the following:
rpXKX=+ & (1.12)
where r is the discount rate. Equation (1.12) says that the price of capital plus the rate of
change of the shadow price of capital, equals the discount rate1. Thus, in this arbitrage
condition, the marginal cost of investing in capital, or discount rate, equals the sum of
two marginal benefits: the instantaneous return of capital and the change in the shadow
value over time. I plot Equations (1.12) and (1.3) on a phase diagram to characterize the
steady state path of capital in the clean sector, with the following result:
1One aspect to note is that this discount rate is constant, and this feature of the model is somewhat
inconsistent with the fixed capital stock assumption.
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Figure (1) says that if the clean sector has an initially high level of capital stock
and a low value or shadow price of capital, then capital owners stop investing in the clean
sector. This disinvestment means that capital in the clean sector becomes scarcer and the
shadow value of capital increases, allowing a movement along the transition path to the
steady state. Alternatively, if the clean sector has an initially low level of capital stock
and a high value of capital, then capital owners invest more in the clean sector. This
increase in investment means that capital becomes less scarce, which decreases the value
of capital, and as a result, the clean sector moves along the transition path to the steady
state. Figure (1) also shows that if the clean sector begins at different initial values than
the ones previously mentioned, then it does not reach a steady state. This paper next
derives the equations for the dirty sector in order to analyze the comparative statics of an
increase inZ .
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The representative firm for the dirty good faces a slightly different but analogous
profit maximization problem. This firms objective is to choose bothY
I and Z in
maximizing the following objective function:
ZKrIZKYp ZYYYYYY )),(( (1.13)
subject to Equation (1.4).
whereYr is the net return to capital owners when they invest in capital usage in the dirty
good. Adjustment costs for this problem follow the type specified in Equation (1.9).
Therefore, the Hamiltonian for the dirty good becomes the following:
][))(),(( 2 YKZYYYYYYY IZKrIbZKYpH Y += (1.14)
The first-order conditions for the two control variables of net investment and
pollution are as follows:
YKYYYpIb =2 (1.15)
ZYZ
Yp =
)((1.16)
The interpretation of (1.15) is the same as it is for the clean good: the marginal
cost of purchasing capital equals the marginal benefit of investment. The interpretation
of (1.16) is that the output price times the marginal product of pollution equals the price
of pollution (i.e. the marginal benefit of pollution equals the marginal cost). The co-state
equation for the state variable of capital is the following:
rK
Yp
YK
Y
Y =+
&
)((1.17)
Equation (1.17) states that the sum of the two returns on capital, the instantaneous
marginal benefit and the rate of change of the shadow price, equals the marginal cost of
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investing, or the discount rate. This equation is slightly more complicated than the co-
state equation for the clean good, Equation (1.12), because the production function for the
dirty good has two inputs. Next, I plot Equations (1.17) and (1.4) on a phase diagram.
The following graph characterizes the transition dynamics and steady state for the dirty
good:
The interpretation of the transition path to the steady state for Figure (2) is exactly
the same as it is for Figure (1). Since Figures (1) and (2) characterize the general
equilibrium of this economy, this paper now evaluates the equilibrium when the
government exogenously increases an environmental tax. From Equation (1.16), when
Z increases, the output price increases because one of the input prices is now more
expensive. Also, the marginal product of pollution increases because the dirty firm uses
less pollution. When the rate of change of the shadow price of capital is zero, the
discount rate equals the output price times the marginal product of capital (see Equation
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(1.17)). Since pollution decreases, when holding the amount of capital stock constant and
applying the fact that the production functions are constant returns to scale, then the
marginal product of capital decreases. WhetherYK
& is positive or negative depends on
the magnitudes of the elasticity of substitution for firms, Y , between the two production
inputs, and the elasticity of substitution for consumers,U , between the two output
goods. This occurrence is similar to what Garrett (2005) derives in the static general
equilibrium model with no adjustment costs.
This paper considers the case whenUY > and acknowledges that the result for
the other case is merely the opposite. The fact thatUY > means that the percentage
change ofYp is larger in magnitude than the percentage change of the marginal product
of capital, since consumers need a larger percentage change to switch. The result is that
the line of 0=YK
& shifts to the right (see Figure (2)). Initially firms cannot adjust their
capital stock; but since the price is larger than the marginal product of capital, capital
becomes more valuable, and the shadow value of capital in the dirty sector increases.
Consequently, capital owners increase investment, which causes the value of capital to
decrease and the dirty sector to transition to a new steady state (see Figure (2)). Since the
price of the clean good,Xp , falls in the short run and the discount rate is constant, the
line of 0=XK
& shifts to the left (see Figure (1)). As shown in Figure (1), the transition to
a new steady state in the clean sector is the opposite of what occurs in the dirty sector.
Figures (1) and (2) illustrate that the long-run result of the case when UY > is
that the capital in the dirty sector increases, while capital in the clean sector decreases.
Also, the increase in the pollution tax leads to less pollution and a more ambient
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environment. The shadow values of capital return to their initial level in the long run, but
these two values differ in the short run to entice capital owners to invest inY
K . The
higher pollution price and the same rental rate of capital leads to less long-run output in
the dirty industry. Also, since the clean representative firm uses less capital, its long-run
output decreases as well. Finally, the price of the dirty good is higher than the initial
equilibrium, while the price of the clean good returns to its initial level, meaning that
consumers pay a higher price for the dirty good and bear some of the tax burden in the
long run.
II. Model with Labor Adjustment Costs for Workers2
This model differs from the one in Section I in a few ways. First, the clean factor
of production is now labor instead of capital. More importantly, firms no longer directly
face adjustment costs when altering the clean factor of production; instead, workers face
adjustment costs when they move from one sector of the economy to the other (Garrett
2005). Jacobson, LaLonde, and Sullivan (1993) show empirically that displaced workers
often deal with a number of difficulties in their reemployment period, including lower
wages, less labor time, and possible transition costs. Their study also finds that some
displaced workers never again attain the same level of earnings. Garrett (2005) provides
a number of intuitive reasons why workers may face adjustment costs. These reasons
include search costs, moving to a new area, investing in new job training or education,
and dealing with the psychological issues of losing a job, which can even result in suicide
(Garrett 2005).
2The numbering of the equations in this section is not always sequential. In order to highlight the
connections between equations in this section and those in Section I, the equation numbers here parallel
those in Section I (e.g., Equation (2.16) in this section is analogous to Equation (1.16) in Section I).
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The constant returns to scale production functions for labor simplify to the case of
perfect factor mobility for firms:
XLX= (2.1)
),( ZLYY Y= (2.2)
where the symbols represent the same aspects of the Section I model, except thatXL and
YL are labor inputs into production. The resource constraint for labor is the following:
LLL YX =+ (2.5a)
whereXXXX NLL += (2.5b)
andYYYY NLL += (2.5c).
Also,X
andY
are labor adjustment costs for workers,XN and YN are the net flows
of labor between sectors, and L is the fixed stock of labor in the economy. Thus,
workers experience the adjustment costs through a loss of labor available to supply to the
market. Since workers cannot adjust to their new jobs costlessly, transition costs absorb
some of their labor time. Equations (2.5b, c) show that a firm employs a dismissed
worker from the other sector immediately and costlessly, but once the worker is in the
new sector, he or she is not immediately productive and does not receive a wage. The
representative firm hires displaced workers knowing that before the workers can be
productive in the new sector, they confront moving costs and/or a job retraining period.
Nonetheless, since firms hire workers immediately, the specification of Equations (2.5b,
c) does not encapsulate a search cost.
The net flows of labor abide by the following transition equations:
XX NL =& (2.3)
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YY NL =& (2.4)
where XL&
and YL&
are the net changes in labor supply over time in each sector of the
economy. The adjustment costs for workers conform to the following basic quadratic
structure (Cahuc and Zylberberg 2004):
2)(XXXX NdN = (2.8)
2)( YYYY NdN = (2.9)
where Xd and Yd are positive constants. Hence, the structure of the adjustment costs is
the same as it is in Section I.
II.A. Representative Firms Problem
As in Section I, the prices of the clean good, dirty good, and pollution areXp ,
Yp , and Z , respectively. The unconstrained profit maximization problem for the
representative firm in the clean industry is to chooseX
L in maximizing the following
objective function:
XXXX LwLp (2.7)
whereXw is the net wage to workers in the clean industry. The first-order condition
becomes the following:
XX wp = (2.11)
so that the output price or marginal benefit equals the wage or marginal cost of labor.
Next, the unconstrained profit maximization problem for the representative firm for the
dirty good is to chooseYL and Z in maximizing the following objective function:
ZLwZLYpZYYYY ),( (2.13)
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whereYw is the net wage in the dirty industry. Maximizing this equation yields the
following two first-order conditions:
Y
Y
Y wL
Yp =
)((2.15)
ZYZ
Yp =
)((2.16)
The interpretation of Equation (2.15) is that the output price times the marginal
product of labor, or marginal benefit of labor, equals the wage or marginal cost of labor
in the dirty industry. The first-order condition for pollution is the same as it is in Section
I (see Equation (1.16)). Rewriting these two first-order conditions yields the following
result:
Z
YY w
Z
Y
L
Y
=
)(
)(
(2.18).
This equation shows that the representative firm in the dirty industry optimizes so that the
marginal rate of transformation equals the price ratio.
II.B. Consumers Optimization Problem
Representative consumers maximize the following utility function over an infinite
time horizon:
dsYXCeU sst
t
),()(
= (2.6)
where is the rate of time preference, and ),( YXCs is the overall consumption of the
two goods. As long as the function ),( YXCs is HD(1), the utility functional form
specifies the special case when the intertemporal elasticity of substitution,U , is
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infinity. Also, like in Section I, the intratemporal elasticity of substitution is U . Thus,
consumers maximize Equation (2.6) subject to the transition equations, Equations (2.3)
and (2.4), and the following static budget constraint:
YXYYXXYX LwLwYpXp +++=+ (2.19)
whereX and Y are the shares of firm profits in each industry, which the
representative consumer receives exogenously. The solutions to the Lagrangian problem
of choosing how much of each good to buy are the following:
Xs p
X
C=
)((2.20)
Ys pY
C=
)((2.21)
where is the Lagrange multiplier on Equation (2.19). Since binds, rearranging
Equations (2.20) and (2.21) yields the following familiar result:
Y
X
s
s
p
p
Y
CX
C
=
)(
)(
(2.22).
This equation shows that the representative consumer optimizes so that the marginal rate
of substitution between the two goods equals the price ratio. The consumer uses the
necessary condition of Equation (2.22) to obtain an indirect utility function. The
following indirect utility function is a function of prices and income:
),,( YXYYXXYX LwLwppV +++ (2.23).
Thus, the Hamiltonian, or intertemporal optimization problem, for the representative
consumer becomes the following:
][][)( XLXL NNVH YX ++= (2.24)
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whereYX NN = from Equation (2.5a). Thus, the first-order condition for choosing the
net flow of labor is the following:
I
VwNd
I
VwNd YYYLXXXL YX
=
)(2
)(2 (2.25)
whereI
V
)(is the marginal value of income. This necessary condition shows that the
consumer intertemporal optimization problem takes into account the shadow price of
labor supply in each industry, the marginal value of income, and the product of the
marginal adjustment cost and the wage in each industry. Thus, consumers maximize
their indirect utility so that the marginal benefit of working in the clean sector is equal to
the marginal benefit of working in the dirty sector, net of the marginal adjustment costs in
each industry.
II.C. The Equilibrium
The co-state equations for the two state variables, XL and YL , are the following:
X
X
L
LXw
I
V
=+
&
)(
(2.26)
Y
Y
L
LYw
I
V
=+
&
)(
(2.27).
Thus, for each industry, the present discounted value of the sum of the instantaneous
return on working and the rate of change of the value of labor equals the shadow value of
labor. Next, I plot the equations characterizing the equilibrium on two graphs (see
Equations (2.3) and (2.26) in Figure (3) and Equations (2.4) and (2.27) in Figure (4) on
the following page).
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The interpretations of the transition paths to the steady state in Figures (3) and (4)
are very similar to that of Figures (1) and (2). The only difference from the graphs in
Section I is that the 0=& lines are downward sloping because a higher labor stock
implies a lower wage and lower shadow value of labor (see Equation (2.15)). Since
Equations (2.16) and (1.16) are the same, previous analysis shows that an increase in the
pollution tax leads to an increase in both the price of the dirty good and the marginal
product of pollution. The decrease in pollution causes the marginal product of labor to
decrease since the production function is constant returns to scale. Therefore, from
Equation (2.15), the sign of the wage in the dirty industry is ambiguous, and similar to
analysis in Section I, this sign depends on the magnitudes ofY and U .
In order to compare the results from the Section I and Section II models, again
this paper considers the case when UY > . Since the magnitude of the percentage
change of Yp is larger than the percentage change of the marginal product of labor, the
wage in the dirty sector increases. From Equation (2.27), the increase in the wage causes
the line of 0=YL
& to shift to the right (see Figure (4)). The shadow value of labor in the
dirty industry increases initially, and since workers receive a higher wage in that industry,
over time they shift their labor supply from the clean to the dirty industry. This increase
in labor supply causes the wage to decrease and the dirty sector returns to a steady state
(see Figure (4)). The increase in the pollution tax causes the wage in the clean sector to
decrease when UY > , and from Equation (2.26), the line of 0=XL
& shifts to the left
(see Figure (3)). Again, like in Section I, the transition to a new steady state in the clean
sector moves in the opposite direction as it does in the dirty sector.
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Figures (3) and (4) illustrate that in the long run, the labor supply in the clean
sector decreases, whereas the labor supply in the dirty sector increases when UY > .
While the shadow values of labor and wages change in the short run to attract workers to
move to the dirty sector, they eventually return to the initial equilibrium level. Also, like
in Section I, the increase in the environmental tax accomplishes the policy goal of
decreasing pollution. By similar reasoning as in Section I, the long-run output in both
industries decreases when UY > . The fact that wages return to their initial level
means that the price of the clean good also returns to its initial level, but the price of the
dirty good is higher in the long run than initially because the input price of pollution is
higher. Thus, like in Section I, consumers pay a higher price for the dirty good and bear
some of the burden of the tax in the long run. Finally, an interesting but expected result
is that both capital owners in Section I and workers in this section only experience tax
incidence in the short run as they face the adjustment costs.
III. Conclusion
This paper examines tax incidence when the government increases an
environmental tax; it accomplishes this goal using two dynamic models with imperfect
factor mobility. This paper finds that the long-run equilibrium result is independent of
the nature of the adjustment costs, and that this equilibrium is the same as it is in a static
model with no adjustment costs. Nonetheless, due to the adjustment costs, the models in
this paper derive a different result in the short run than the static model with no
adjustment costs. In both the capital adjustment cost case for a firm, and the labor
adjustment cost case for a worker, the shadow values change in the short run. This short-
run change induces capital owners to change their investment and workers to move to a
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different sector. Over time the shadow values in each sector and the price of the clean
sector return to their initial level, but the increase in the price of the dirty good in the long
run means that consumers share some of the tax burden. Importantly, the pollution tax
accomplishes the policy goal of reducing the negative externality. Finally, these results
are independent of whether firms face adjustment costs via their production functions or
workers face adjustment costs via the labor resource constraint.
Researchers can find a number of ways to extend this paper in future research.
The first way is to allow for more generality in the model by including both capital and
labor as clean inputs into production (see Goulder and Summers 1989 and Fullerton and
Heutel 2005). Another way to extend the model is to use different functional forms for
the adjustment costs other than quadratic adjustment costs. This research can examine if
the results of this paper depend on the structure of the adjustment costs. Finally,
researchers can relax the assumption of a fixed stock of labor or capital to allow for
savings decisions in the model.
The results of this paper launch certain policy implications. Clearly, taxing a
negative externality like pollution leads to a lower amount of the externality. The
problem with this taxation is that certain economic agents bear the burden of the tax.
Using a dynamic model, this paper shows that regardless of the nature of the adjustment
costs, the long-run incidence only affects consumers who want to buy the dirty good.
Therefore, in the long run, the externality is internalized, and governments can view
direct taxation as an effective and economically costless way of reducing negative
externalities. (Look at last paragraphs of first two sections, especially at the end of
Section II.)
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