- Email: [email protected]
A Derivation of the Marginal Abatement Cost Curve
Journal of Environmental Economics and Management 37, 306-314 (1999) Article ID jeem.1999.1065,available online at http://www.idealibrary.corn 011 IDE
kL3
A Derivation of the Marginal Abatement Cost Curve Ross McKitrick Department of Economics, The Unirersityof Guelph, Guelph, Ontario, Canada NIG 2Wl E-mail: [email protected] Received March 11, 1997; revised July 28, 1998 The relationship between a firm’s technology and its marginal abatement cost (MAC) curve is explored. Even under the simplest specifications, the MAC curve will be kinked at some point except under a special assumption which, in reality, could easily be violated. The noridifferentiability implies that the choice of instrument under uncertainty may depend on the targeted level of emissions reduction. Also, stability conditions for dynamic tax mechanisms may be violated in the neighborhood of the kink point. A policy implication is that in some cases output restrictions are as efficient as emissions restrictions, in contrast to previous results. 0 1999 Academic Press Key Words: marginal abatement costs, cost functions, emissions functions.
1. INTRODUCTION
The marginal abatement cost curve (hereafter the MAC) links a firm’s emission levels and the cost of additional units of pollution reduction. While it is a key tool in environmental economics, its analytical properties are rarely explored. This article provides an expository derivation. An unexpected result is that even in the most basic case (a single firm with one pollutant and one abatement technology), a nonobvious necessary and sufficient condition must be assumed to ensure that the MAC is continuously differentiable. Since analyses of many aspects of pollution policy depend on an assumption of differentiability (e.g., Weitzman 181, Cropper and Oates [3], and countless others), identifying this condition facilitates theoretical consistency. Some analytical and policy implications are also discussed. For instance, a nondifferentiable point on the MAC can affect the stability of dynamic pollution policies under limited information and the relative advantage of price and quantity instruments under uncertainty. The model highlights a simple point. A firm producing a single output and emitting a single pollutant can control its emissions either by investing in pollution control equipment or reducing output. Since it cannot invest in negative levels of abatement equipment, a boundary condition applies on abatement effort, such that for some pollution reduction targets, and some specifications of technology, the firm will rely solely on reductions in output. Where the boundary condition no longer binds, the introduction of abatement equipment generates a nondifferentiable point on the MAC, changing the slope and causing a kink. This is ruled out only if the boundary point coincides with the unregulated emissions level. A kinked MAC curve also arises in models where numerous abatement activities can be combined. For instance, Fullerton et al. [4] discuss the case of electric utilities which use scrubbers, fuel switching, reallocation of production among 306 0095-0696/99 $30.00 Copyright 0 1999 hy Academic Press All rights of reproduction in any form resewed.
MARGINAL ABATEMENT COST CURVE
307
plants, coal washing, demand-side management, and so forth to reduce sulfur emissions. Some strategies have low fixed costs but high variable costs, while for others the reverse is true. If the marginal cost of one strategy depends on the other methods currently in place and how intensively they are used, the long run MAC may exhibit kinks and jumps, and path dependence may make it impossible to define the MAC analytically, even if each individual method is represented by a smooth MAC. But the analysis here shows that even when a firm has only one possible abatement technology, its MAC function may still be kinked. The firm has the option of reducing output, so it always has at least two means of reaching an emissions target. This ability to combine these methods gives rise to the potential nondifferentiability. 2 . THE MODEL
A firm or industry produces output y using inputs with a cost vector w and engages in nonnegative levels of pollution abatement1 activity a . Output sells for p per unit. Emissions e are generated based on the level of output and abatement. Total costs are denoted c. Thus profits are n-(p,y,w,a) =py
-
c(w,y,a)
and emissions are e
=
e ( y ,a ) .
Assume that emissions are convex-increasing in output and convex-decreasing in abatement activities, so ey > 0, el,y > 0, e, < 0, and eua > 0. Assume also that c y > 0, c J J > 0 , and c , 2 0. Finally, assume that, for all pairs ( y , a ) such that a = 0, c u ( w , y , a ) I- ( p - c y ) e u / e y . (3) This expression is the necessary and sufficient condition referred to in the Introduction, and much of the analysis to follow focuses on its meaning. The firm will only operate at levels of output where price exceeds or equals marginal cost; hence the marginal revenue term ( p - c J ) is nonnegative.‘ The ratio of the derivatives of e is negative, so the inequality states that the marginal cost to the firm of the first unit of abatement effort must be bounded above by some nonnegative amount. It is violated if c,(w, y , a ) > - ( p - cl,)eu/el, when a equals zero. This will happen if, at low levels of abatement effort, the absolute magnitude of e, becomes very “small” and/or c , becomes “large”; in other words, if for the initial levels of abatement, marginal units of a are very costly and/or relatively ineffective. It will also happen if marginal revenue is zero and c, is positive. Further intuition behind these conditions and implications for the differentiability of the MAC are developed graphically below. The term “abatement” is used by some authors to mean activities applied to reducing pollution, arid by others to mean a unit reduction in emissions. The two uses are not equivalent. The terminology here is as follows. “Abatement,” “abatement effort,” arid “abatement activity” refer to a costly undertaking which reduces emissions subject to diminishing returns. It is denoted throughout as a. The other usage is denoted here as “emission reductions.” It is implicitly defined as (e* - e) where the * denotes the unregulated private optimum level of emissions e. Equality of price arid marginal cost does riot generally hold when emissions are constrained, as shown later in this article.
308
ROSS McKITRICK
Note that (3) is always satisfied if the marginal cost of the first unit of abatement activity is zero. A stronger version of (3) would be c,
=
0 when a
0.
=
( 3a)
In actual practice, a firm’s first unit of pollution abatement effort can involve large variable costs, especially if the technology is “lumpy,” so both (3a) and (3) may be violated. The MAC will first be derived assuming (3a) holds, and then the consequences when neither it nor (3) hold for some range of abatement activity will be explored. In the absence of controls on emissions, the firm chooses ( y , a ) such that p = cy and c, = 0, which imply an unregulated optimum ( y * , a*) = ( y * , O), and emissions e*. A graph of the firm’s decision problem can be drawn using isoprofit lines in ( y , a ) space. Differentiating (1) and setting dn- = 0 yields an equation for the slope of the isoprofit line:
When a > 0, the isoprofit line has an inverted-u shape above the y axis. Formally:
< y*
3
da/dy
> 0,
y =y*
3
da/dy
=
y =y*
3
da/dy
< 0.
y
0,
and
At a
=
-
0, assuming (3a) holds, the isoprofit lines are vertical: y
< >y*
and y =y*
-
da/dy
da/dy
=
=
o/o.
Thus, the isoprofit lines are semicircles which converge concentrically to a point at ( y * , 01, which corresponds to profits n-*. The lines are vertical as they meet the y
axis. Examples are shown in Fig. 1. The direction of increasing profits is toward the center, at ( y * , a*). An emissions control standard is written
The isoemission line in ( y , a ) space has the slope da
-ey
dy
el
-~ -
> 0.
Reasonable assumptions about technology yield diminishing returns to abatement effort, which implies the isoemissions constraint is convex upwards. It is graphed as the line labeled el in Fig. 1, and it shows combinations of a and y which yield emissions e,. For a given level of output, emissions fall as abatement rises; thus the direction shown indicates movement into regions of lower emissions.
309
MARGINAL ABATEMENT COST CURVE
a
Z
Tr
I
1
YI
\
\%
I
ILI
I
I
Y*
Y
FIG. 1. Optimal output and abatement combinations.
The firm’s optimization problem is to maximize py I el. This yields first-order conditions p
-
cy
-ell
-
-
c ( w , y , a ) subject to e ( y , a )
ey
(7)
efl
A comparison of (7) with (4) and (6) shows that this defines the locus of tangencies where the slope of the isoprofit line equals the slope of the isoemissions line. It is labeled Zy* in Fig. 1. The optimal choice of output and abatement, given the constraint e l , is the point (yl, all, with associated profits r l .The true cost to the firm of meeting this target is the change in the level of profits, r * - r l . The importance of (3) can now be explained in graphical terms. Under (3a), assuming c,(w, y , 0) = 0 ensures that the isoprofit lines cross the y axis vertically; therefore an upward-sloping emissions constraint will always be tangent to an isoprofit line at interior points in the nonnegative ( y , a ) space. Consequently the firm will use positive amounts of abatement equipment, rather than output reductions alone, to respond to all levels of required emissions reductions (or pollution charges). Assumption (3) weakens the condition slightly compared to (3a), allowing the isoprofit line to cross the y axis at an angle, as long as the slope of the isoemissions line at the same point is less than or equal to the slope of the isoprofits line (compare (3) and (7)). Under this condition, the tangency point must be at a nonnegative abatement level. To derive the MAC function, first note that it shows the marginal benefit to the firm of generating one more unit of pollution. Assume e ( y , a ) can be inverted to yield a function a = a(y, el, showing the level of abatement required to achieve emissions e given output level y . Note also that a y > 0 and a, < 0. Substituting into the profit function for a and taking partial derivatives yields dU
de
310
ROSS McKITRICK
FIG. 2. Marginal abatement cost functions
Along the tangency locus (7), it must be the case that ( p - c y ) = -c,ey/eu. This plus (6) implies that the term in the brackets is zero (this is an application of the envelope theorem). Thus, when output and abatement are both optimally adjusted, dn-
- -- -
de
da
c, -. de
(9)
Equation (9) is the MAC curve corresponding to the locus of optimal output-abatement pairs in (7). It is easily verified that the MAC curve is downward sloping. An example is drawn in Fig. 2 as MAC,. This is the classical representation of the marginal abatement cost function: continuously differentiable and meeting the horizontal axis. Assumption (3) is necessary and sufficient to yield this construction in the perfectly competitive case. Now consider the implications of relaxing (3). If the firm is not perfectly competitive and c, is positive at a = 0, it will be the case that p - cy > 0 at the unregulated emissions level; hence the firm’s MAC does not meet the horizontal axis, and even initial emissions reductions must be costly to the firm. Otherwise, it must be the case that, starting from an unregulated emissions level, a competitive firm can always reduce emissions slightly at no cost. Even if c, is positive, a can be held constant and only y adjusted to achieve an emissions target. But y* is defined so that small variations do not change profit^.^ However, once out of the neighborhood of y * , price no longer equals marginal cost, so there is no way to reduce emissions without reducing profits. If (3) does not hold then neither will (7), so as the isoprofit lines converge to the center in Fig. 3, tangencies such as the one on el, may reach a corner solution, say at y,. Between it and y * , (7) no longer holds. In effect, this is the region in which the firm would like to use “negative” abatement effort combined with output reductions to meet the target. The firm instead reduces emissions using output reductions alone, following the heavy black line segment between y* and y , , I am grateful to an Associate Editor for this point.
311
MARGINAL ABATEMENT COST CURVE a
Yl
Y*
YZ
Y
FIG. 3. Corner solutions
before moving to positive levels of a . Along the black line the slope of the isoprofit line is lower than the slope of the isoemissions line, so
which implies
so (3) is violated. As mentioned above, this will happen whenever, at low levels of abatement effort, marginal units of a are very costly and/or relatively ineffective, so the firm controls emissions solely through reductions in output. From (8) we have -ey/ea
> ( p - c,)/c,
-
d.ir/de < - c , d a / d e .
That is, the slope of the MAC curve wherever (3) is violated is lower than that associated with interior tangencies; geometrically that means the MAC is steeper downward between e , and e*, where e2 is the emissions level associated with output at y,. The MAC function if (3) is violated over some interval (O,e,) is written
This is drawn as MAC, in Fig. 2. Clearly, (10) is nondifferentiable at the kink point e2.
312
ROSS McKITRICK
3. ANALYTIC AND POLICY IMPLICATIONS
If a large amount of abatement effort is required to achieve an initial unit of emissions reduction, or if the threshold effort is sufficiently costly, the range from to e2 to e* may encompass the emission reduction targets implemented by a regulator. Admittedly, reasonable assumptions about the firm would place the kink point close to the unregulated level, while most regulatory targets are likely to be well below that. But if efficiency dictates an emissions level between e2 and e*, a regulation which specifies output reductions is not (in the single firm case) suboptimal. This contrasts with the argument in Helfand [5, p. 629-6301 in which an output restriction on a single firm is always less efficient than a direct emissions standard. There are implications for instrument choice if a MAC function is kinked and is steeper close to the unregulated emissions level. The Weitzman [81 analysis shows that, under uncertainty, whether a price-based (e.g., tax or subsidy) or a quantitybased instrument (e.g., standards or permits) is preferred depends on the relative slopes of the marginal damages and marginal abatement cost functions in the neighborhood of the target emissions level. If the slope of the MAC function changes across the range of emissions, the regulator’s preference for any one instrument would be sensitive to the amount of emissions control required. Hence, if the MAC function is steeper at low levels of abatement, it would (ceteris paribus) shift the regulator’s preference toward using a price instrument (see [ l , Chap. 51) for modest emission reduction targets. The stability of some dynamic procedures for determining the optimal pollution tax when control costs are privately known depends on the slope of the MAC function. Conrad [21 analyzes iterative tax adjustment mechanisms under myopic behavior and foresight. In both cases the condition for stability of the optimum becomes less assured the steeper the MAC. If the MAC is kinked, stability may depend on the level of emissions control achieved and whether the optimum is close to the kink point. Karp and Livernois [6] analyze polluter responses to a linear tax-adjustment rule with a fixed emissions target. In the steady state of the open-loop mechanism with asymmetric firms, polluters do not reach efficient emission levels because of strategic responses to the tax-adjustment mechanism. If a kink in the MAC were in the vicinity of the equilibrium tax level, it would have the interesting effect of moving high-cost firms further away from their optimal emissions level, but would move low-cost firms towards their^.^ In the Markov perfect equilibrium, there are multiple steady states, and an increase in the slope of the MAC can shrink the region in which stable paths originate, depending on the other parameters of the model. 4. CONCLUSIONS
The primary purpose of this paper is an exposition of the conditions under which a continuously differentiable MAC will exist. Even a simple case, with one firm and one abatement technique, requires some care to ensure analytical consistency. Some authors include assumption (3a) in the specification of their models, alSee their Fig. l [6, p. 431,
MARGINAL ABATEMENT COST CURVE
313
though the implication for the smoothness of the MAC is not stated as an objective. Generally, neither (3) nor (3a) is assumed, and consequently the slope of the MAC function may not be well defined. This can have implications for instrument choice under uncertainty and the stability of dynamic policy regimes. The main policy implication of this analysis is that, in the region where the nonnegativity constraint on abatement activity binds, policies which require output reductions may in fact be no less efficient than direct emission standards. A somewhat unrelated contribution of this model is that it provides a simple graphical explanation for recent evidence that costs of pollution regulation can be systematically ~ v e r s t a t e d .In ~ Fig. 1, if a naive regulator were to ask a firm how much it would cost to reduce emissions from e* to el, the firm could “truthfully” calculate the dollar value of the amount of a located where a vertical line going up from y* meets the line el. This, of course, would be an overestimate. Even reporting the dollar value of a, would be incorrect. The true cost estimate is the change in profits, n-* - v1,and depending on the slope of the profit function, this may differ substantially from the nominal value of a,. APPEND IX : NOM ENC LATURE
d.1 P
Y C(.>
W
a e(.)
*
ZY*
firm profits output price output firm costs prices of inputs abatement effort emissions function the firm’s unregulated optimum the locus of tangencies defining optimal output-abatement pairs
ACKNOWLEDGMENTS I thank, without implicating, John Livernois, two anonymous referees, and an Associate Editor for detailed and helpful comments.
REFERENCES 1. W. Baumol and W. Oates, “The Theory of Environmental Policy,” 2nd ed., Cambridge Univ. Press, New York, 1992. 2. K. Conrad, Incentive Mechanisms for Environmental Protection under Asymmetric Information: A Case Study, Appl. Econom. 23 (1991). 871-880. 3. M. Cropper and W. Oates, Environmental Economics: A Survey, J . Econom. Lit. 30 (19921, 675-740.
’See, for example, Morgenstern et al. [7] who estimate that true economic costs to firms average only 13 cents for every dollar of reported abatement expenditures.
314
ROSS McKITRICK
4. D. Fullerton, S. McDermott, and J. Caulkins, Sulfur Dioxide Compliance of a Regulated Utility, J . Enriron. Econom. Management 34 (1997), 32-53. 5. G. Helfand, Standards vs. Standards: The Effects of Different Pollution Restrictions, Am. Econom. Rer.. 81 (1991). 622-634. 6. L. Karp and J. Livernois, Using Automatic Tax Changes to Control Pollution Emissions, J . Enriron. Econom. Management 27 (1994). 38-48. 7. R. Morgenstern, W. Pizer, and J. Shih, “Are We Overstating the Real Economic Costs of Envirom mental Protection?” Resources for the Future Discussion Paper 97-36, June 1997. 8. M. Weitzman, Prices vs. Quantities, Rec. Econom. Stud. 41 (1974), 477-91.
kL3
A Derivation of the Marginal Abatement Cost Curve Ross McKitrick Department of Economics, The Unirersityof Guelph, Guelph, Ontario, Canada NIG 2Wl E-mail: [email protected] Received March 11, 1997; revised July 28, 1998 The relationship between a firm’s technology and its marginal abatement cost (MAC) curve is explored. Even under the simplest specifications, the MAC curve will be kinked at some point except under a special assumption which, in reality, could easily be violated. The noridifferentiability implies that the choice of instrument under uncertainty may depend on the targeted level of emissions reduction. Also, stability conditions for dynamic tax mechanisms may be violated in the neighborhood of the kink point. A policy implication is that in some cases output restrictions are as efficient as emissions restrictions, in contrast to previous results. 0 1999 Academic Press Key Words: marginal abatement costs, cost functions, emissions functions.
1. INTRODUCTION
The marginal abatement cost curve (hereafter the MAC) links a firm’s emission levels and the cost of additional units of pollution reduction. While it is a key tool in environmental economics, its analytical properties are rarely explored. This article provides an expository derivation. An unexpected result is that even in the most basic case (a single firm with one pollutant and one abatement technology), a nonobvious necessary and sufficient condition must be assumed to ensure that the MAC is continuously differentiable. Since analyses of many aspects of pollution policy depend on an assumption of differentiability (e.g., Weitzman 181, Cropper and Oates [3], and countless others), identifying this condition facilitates theoretical consistency. Some analytical and policy implications are also discussed. For instance, a nondifferentiable point on the MAC can affect the stability of dynamic pollution policies under limited information and the relative advantage of price and quantity instruments under uncertainty. The model highlights a simple point. A firm producing a single output and emitting a single pollutant can control its emissions either by investing in pollution control equipment or reducing output. Since it cannot invest in negative levels of abatement equipment, a boundary condition applies on abatement effort, such that for some pollution reduction targets, and some specifications of technology, the firm will rely solely on reductions in output. Where the boundary condition no longer binds, the introduction of abatement equipment generates a nondifferentiable point on the MAC, changing the slope and causing a kink. This is ruled out only if the boundary point coincides with the unregulated emissions level. A kinked MAC curve also arises in models where numerous abatement activities can be combined. For instance, Fullerton et al. [4] discuss the case of electric utilities which use scrubbers, fuel switching, reallocation of production among 306 0095-0696/99 $30.00 Copyright 0 1999 hy Academic Press All rights of reproduction in any form resewed.
MARGINAL ABATEMENT COST CURVE
307
plants, coal washing, demand-side management, and so forth to reduce sulfur emissions. Some strategies have low fixed costs but high variable costs, while for others the reverse is true. If the marginal cost of one strategy depends on the other methods currently in place and how intensively they are used, the long run MAC may exhibit kinks and jumps, and path dependence may make it impossible to define the MAC analytically, even if each individual method is represented by a smooth MAC. But the analysis here shows that even when a firm has only one possible abatement technology, its MAC function may still be kinked. The firm has the option of reducing output, so it always has at least two means of reaching an emissions target. This ability to combine these methods gives rise to the potential nondifferentiability. 2 . THE MODEL
A firm or industry produces output y using inputs with a cost vector w and engages in nonnegative levels of pollution abatement1 activity a . Output sells for p per unit. Emissions e are generated based on the level of output and abatement. Total costs are denoted c. Thus profits are n-(p,y,w,a) =py
-
c(w,y,a)
and emissions are e
=
e ( y ,a ) .
Assume that emissions are convex-increasing in output and convex-decreasing in abatement activities, so ey > 0, el,y > 0, e, < 0, and eua > 0. Assume also that c y > 0, c J J > 0 , and c , 2 0. Finally, assume that, for all pairs ( y , a ) such that a = 0, c u ( w , y , a ) I- ( p - c y ) e u / e y . (3) This expression is the necessary and sufficient condition referred to in the Introduction, and much of the analysis to follow focuses on its meaning. The firm will only operate at levels of output where price exceeds or equals marginal cost; hence the marginal revenue term ( p - c J ) is nonnegative.‘ The ratio of the derivatives of e is negative, so the inequality states that the marginal cost to the firm of the first unit of abatement effort must be bounded above by some nonnegative amount. It is violated if c,(w, y , a ) > - ( p - cl,)eu/el, when a equals zero. This will happen if, at low levels of abatement effort, the absolute magnitude of e, becomes very “small” and/or c , becomes “large”; in other words, if for the initial levels of abatement, marginal units of a are very costly and/or relatively ineffective. It will also happen if marginal revenue is zero and c, is positive. Further intuition behind these conditions and implications for the differentiability of the MAC are developed graphically below. The term “abatement” is used by some authors to mean activities applied to reducing pollution, arid by others to mean a unit reduction in emissions. The two uses are not equivalent. The terminology here is as follows. “Abatement,” “abatement effort,” arid “abatement activity” refer to a costly undertaking which reduces emissions subject to diminishing returns. It is denoted throughout as a. The other usage is denoted here as “emission reductions.” It is implicitly defined as (e* - e) where the * denotes the unregulated private optimum level of emissions e. Equality of price arid marginal cost does riot generally hold when emissions are constrained, as shown later in this article.
308
ROSS McKITRICK
Note that (3) is always satisfied if the marginal cost of the first unit of abatement activity is zero. A stronger version of (3) would be c,
=
0 when a
0.
=
( 3a)
In actual practice, a firm’s first unit of pollution abatement effort can involve large variable costs, especially if the technology is “lumpy,” so both (3a) and (3) may be violated. The MAC will first be derived assuming (3a) holds, and then the consequences when neither it nor (3) hold for some range of abatement activity will be explored. In the absence of controls on emissions, the firm chooses ( y , a ) such that p = cy and c, = 0, which imply an unregulated optimum ( y * , a*) = ( y * , O), and emissions e*. A graph of the firm’s decision problem can be drawn using isoprofit lines in ( y , a ) space. Differentiating (1) and setting dn- = 0 yields an equation for the slope of the isoprofit line:
When a > 0, the isoprofit line has an inverted-u shape above the y axis. Formally:
< y*
3
da/dy
> 0,
y =y*
3
da/dy
=
y =y*
3
da/dy
< 0.
y
0,
and
At a
=
-
0, assuming (3a) holds, the isoprofit lines are vertical: y
< >y*
and y =y*
-
da/dy
da/dy
=
=
o/o.
Thus, the isoprofit lines are semicircles which converge concentrically to a point at ( y * , 01, which corresponds to profits n-*. The lines are vertical as they meet the y
axis. Examples are shown in Fig. 1. The direction of increasing profits is toward the center, at ( y * , a*). An emissions control standard is written
The isoemission line in ( y , a ) space has the slope da
-ey
dy
el
-~ -
> 0.
Reasonable assumptions about technology yield diminishing returns to abatement effort, which implies the isoemissions constraint is convex upwards. It is graphed as the line labeled el in Fig. 1, and it shows combinations of a and y which yield emissions e,. For a given level of output, emissions fall as abatement rises; thus the direction shown indicates movement into regions of lower emissions.
309
MARGINAL ABATEMENT COST CURVE
a
Z
Tr
I
1
YI
\
\%
I
ILI
I
I
Y*
Y
FIG. 1. Optimal output and abatement combinations.
The firm’s optimization problem is to maximize py I el. This yields first-order conditions p
-
cy
-ell
-
-
c ( w , y , a ) subject to e ( y , a )
ey
(7)
efl
A comparison of (7) with (4) and (6) shows that this defines the locus of tangencies where the slope of the isoprofit line equals the slope of the isoemissions line. It is labeled Zy* in Fig. 1. The optimal choice of output and abatement, given the constraint e l , is the point (yl, all, with associated profits r l .The true cost to the firm of meeting this target is the change in the level of profits, r * - r l . The importance of (3) can now be explained in graphical terms. Under (3a), assuming c,(w, y , 0) = 0 ensures that the isoprofit lines cross the y axis vertically; therefore an upward-sloping emissions constraint will always be tangent to an isoprofit line at interior points in the nonnegative ( y , a ) space. Consequently the firm will use positive amounts of abatement equipment, rather than output reductions alone, to respond to all levels of required emissions reductions (or pollution charges). Assumption (3) weakens the condition slightly compared to (3a), allowing the isoprofit line to cross the y axis at an angle, as long as the slope of the isoemissions line at the same point is less than or equal to the slope of the isoprofits line (compare (3) and (7)). Under this condition, the tangency point must be at a nonnegative abatement level. To derive the MAC function, first note that it shows the marginal benefit to the firm of generating one more unit of pollution. Assume e ( y , a ) can be inverted to yield a function a = a(y, el, showing the level of abatement required to achieve emissions e given output level y . Note also that a y > 0 and a, < 0. Substituting into the profit function for a and taking partial derivatives yields dU
de
310
ROSS McKITRICK
FIG. 2. Marginal abatement cost functions
Along the tangency locus (7), it must be the case that ( p - c y ) = -c,ey/eu. This plus (6) implies that the term in the brackets is zero (this is an application of the envelope theorem). Thus, when output and abatement are both optimally adjusted, dn-
- -- -
de
da
c, -. de
(9)
Equation (9) is the MAC curve corresponding to the locus of optimal output-abatement pairs in (7). It is easily verified that the MAC curve is downward sloping. An example is drawn in Fig. 2 as MAC,. This is the classical representation of the marginal abatement cost function: continuously differentiable and meeting the horizontal axis. Assumption (3) is necessary and sufficient to yield this construction in the perfectly competitive case. Now consider the implications of relaxing (3). If the firm is not perfectly competitive and c, is positive at a = 0, it will be the case that p - cy > 0 at the unregulated emissions level; hence the firm’s MAC does not meet the horizontal axis, and even initial emissions reductions must be costly to the firm. Otherwise, it must be the case that, starting from an unregulated emissions level, a competitive firm can always reduce emissions slightly at no cost. Even if c, is positive, a can be held constant and only y adjusted to achieve an emissions target. But y* is defined so that small variations do not change profit^.^ However, once out of the neighborhood of y * , price no longer equals marginal cost, so there is no way to reduce emissions without reducing profits. If (3) does not hold then neither will (7), so as the isoprofit lines converge to the center in Fig. 3, tangencies such as the one on el, may reach a corner solution, say at y,. Between it and y * , (7) no longer holds. In effect, this is the region in which the firm would like to use “negative” abatement effort combined with output reductions to meet the target. The firm instead reduces emissions using output reductions alone, following the heavy black line segment between y* and y , , I am grateful to an Associate Editor for this point.
311
MARGINAL ABATEMENT COST CURVE a
Yl
Y*
YZ
Y
FIG. 3. Corner solutions
before moving to positive levels of a . Along the black line the slope of the isoprofit line is lower than the slope of the isoemissions line, so
which implies
so (3) is violated. As mentioned above, this will happen whenever, at low levels of abatement effort, marginal units of a are very costly and/or relatively ineffective, so the firm controls emissions solely through reductions in output. From (8) we have -ey/ea
> ( p - c,)/c,
-
d.ir/de < - c , d a / d e .
That is, the slope of the MAC curve wherever (3) is violated is lower than that associated with interior tangencies; geometrically that means the MAC is steeper downward between e , and e*, where e2 is the emissions level associated with output at y,. The MAC function if (3) is violated over some interval (O,e,) is written
This is drawn as MAC, in Fig. 2. Clearly, (10) is nondifferentiable at the kink point e2.
312
ROSS McKITRICK
3. ANALYTIC AND POLICY IMPLICATIONS
If a large amount of abatement effort is required to achieve an initial unit of emissions reduction, or if the threshold effort is sufficiently costly, the range from to e2 to e* may encompass the emission reduction targets implemented by a regulator. Admittedly, reasonable assumptions about the firm would place the kink point close to the unregulated level, while most regulatory targets are likely to be well below that. But if efficiency dictates an emissions level between e2 and e*, a regulation which specifies output reductions is not (in the single firm case) suboptimal. This contrasts with the argument in Helfand [5, p. 629-6301 in which an output restriction on a single firm is always less efficient than a direct emissions standard. There are implications for instrument choice if a MAC function is kinked and is steeper close to the unregulated emissions level. The Weitzman [81 analysis shows that, under uncertainty, whether a price-based (e.g., tax or subsidy) or a quantitybased instrument (e.g., standards or permits) is preferred depends on the relative slopes of the marginal damages and marginal abatement cost functions in the neighborhood of the target emissions level. If the slope of the MAC function changes across the range of emissions, the regulator’s preference for any one instrument would be sensitive to the amount of emissions control required. Hence, if the MAC function is steeper at low levels of abatement, it would (ceteris paribus) shift the regulator’s preference toward using a price instrument (see [ l , Chap. 51) for modest emission reduction targets. The stability of some dynamic procedures for determining the optimal pollution tax when control costs are privately known depends on the slope of the MAC function. Conrad [21 analyzes iterative tax adjustment mechanisms under myopic behavior and foresight. In both cases the condition for stability of the optimum becomes less assured the steeper the MAC. If the MAC is kinked, stability may depend on the level of emissions control achieved and whether the optimum is close to the kink point. Karp and Livernois [6] analyze polluter responses to a linear tax-adjustment rule with a fixed emissions target. In the steady state of the open-loop mechanism with asymmetric firms, polluters do not reach efficient emission levels because of strategic responses to the tax-adjustment mechanism. If a kink in the MAC were in the vicinity of the equilibrium tax level, it would have the interesting effect of moving high-cost firms further away from their optimal emissions level, but would move low-cost firms towards their^.^ In the Markov perfect equilibrium, there are multiple steady states, and an increase in the slope of the MAC can shrink the region in which stable paths originate, depending on the other parameters of the model. 4. CONCLUSIONS
The primary purpose of this paper is an exposition of the conditions under which a continuously differentiable MAC will exist. Even a simple case, with one firm and one abatement technique, requires some care to ensure analytical consistency. Some authors include assumption (3a) in the specification of their models, alSee their Fig. l [6, p. 431,
MARGINAL ABATEMENT COST CURVE
313
though the implication for the smoothness of the MAC is not stated as an objective. Generally, neither (3) nor (3a) is assumed, and consequently the slope of the MAC function may not be well defined. This can have implications for instrument choice under uncertainty and the stability of dynamic policy regimes. The main policy implication of this analysis is that, in the region where the nonnegativity constraint on abatement activity binds, policies which require output reductions may in fact be no less efficient than direct emission standards. A somewhat unrelated contribution of this model is that it provides a simple graphical explanation for recent evidence that costs of pollution regulation can be systematically ~ v e r s t a t e d .In ~ Fig. 1, if a naive regulator were to ask a firm how much it would cost to reduce emissions from e* to el, the firm could “truthfully” calculate the dollar value of the amount of a located where a vertical line going up from y* meets the line el. This, of course, would be an overestimate. Even reporting the dollar value of a, would be incorrect. The true cost estimate is the change in profits, n-* - v1,and depending on the slope of the profit function, this may differ substantially from the nominal value of a,. APPEND IX : NOM ENC LATURE
d.1 P
Y C(.>
W
a e(.)
*
ZY*
firm profits output price output firm costs prices of inputs abatement effort emissions function the firm’s unregulated optimum the locus of tangencies defining optimal output-abatement pairs
ACKNOWLEDGMENTS I thank, without implicating, John Livernois, two anonymous referees, and an Associate Editor for detailed and helpful comments.
REFERENCES 1. W. Baumol and W. Oates, “The Theory of Environmental Policy,” 2nd ed., Cambridge Univ. Press, New York, 1992. 2. K. Conrad, Incentive Mechanisms for Environmental Protection under Asymmetric Information: A Case Study, Appl. Econom. 23 (1991). 871-880. 3. M. Cropper and W. Oates, Environmental Economics: A Survey, J . Econom. Lit. 30 (19921, 675-740.
’See, for example, Morgenstern et al. [7] who estimate that true economic costs to firms average only 13 cents for every dollar of reported abatement expenditures.
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4. D. Fullerton, S. McDermott, and J. Caulkins, Sulfur Dioxide Compliance of a Regulated Utility, J . Enriron. Econom. Management 34 (1997), 32-53. 5. G. Helfand, Standards vs. Standards: The Effects of Different Pollution Restrictions, Am. Econom. Rer.. 81 (1991). 622-634. 6. L. Karp and J. Livernois, Using Automatic Tax Changes to Control Pollution Emissions, J . Enriron. Econom. Management 27 (1994). 38-48. 7. R. Morgenstern, W. Pizer, and J. Shih, “Are We Overstating the Real Economic Costs of Envirom mental Protection?” Resources for the Future Discussion Paper 97-36, June 1997. 8. M. Weitzman, Prices vs. Quantities, Rec. Econom. Stud. 41 (1974), 477-91.