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MSWordDocWord.Document.69qndary 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, McDermott and Caulkins (1996) discuss the case of electric utilities which use scrubbers, fuel switching, reallocation of production among plants, coal washing, demand-side management, and so forth to reduce sulphur 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 non-negative levels of pollution abatement activity a. Output sells for p per unit. Emissions e are generated based on the level of output and abatement. Thus profits are
EMBED Equation.2 (1)
and emissions are
EMBED Equation.2 (2).
Assume emissions are convex-increasing in output, and convex-decreasing in abatement activities, so EMBED Equation.2 and EMBED Equation.2 . Assume also that EMBED Equation.2 , EMBED Equation.2 , EMBED Equation.2 . Finally, assume that, for all pairs (y,a) such that a=0,
EMBED Equation.2 (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 EMBED Equation.2 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 EMBED Equation.2 when a equals zero. This will happen if, at low levels of abatement effort, the absolute magnitude of EMBED Equation.2 becomes very small and/or EMBED Equation.2 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 EMBED Equation.2 is positive. Further intuition behind these conditions, and implications for the differentiability of the MAC, are developed graphically below.
Note that (3) is always satisfied if the marginal cost of the first unit of abatement activity is zero. So a stronger version of (3) would be
EMBED Equation.2 when a = 0 (3a).
In actual practice, a firms 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, 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 EMBED Equation.2 and EMBED Equation.2 , which imply an unregulated optimum (y*, a*) = (y*, 0), and emissions e*. A graph of the firms decision problem can be drawn using iso-profit lines in (y,a) space. Differentiating (1) and setting EMBED Equation.2 =0 yields an equation for the slope of the iso-profit line
EMBED Equation.2 (4).
When a>0, the iso-profit line has an inverted-u shape above the y axis. Formally:
EMBED Equation.2 ;
EMBED Equation.2 ;
and EMBED Equation.2 .
At a=0, assuming (3a) holds, the iso-profit lines are vertical:
EMBED Equation.2
and EMBED Equation.2 .
Thus, the iso-profit lines are semi-circles which converge concentrically to a point at (y*,0), which corresponds to profits EMBED Equation.2 . The lines are vertical as they meet the y axis. Examples are shown in Figure 1. The direction of increasing profits is towards the centre, at (y*, a*).
An emissions control standard is written
EMBED Equation.2 (5).
The iso-emission line in (y,a) space has the slope
EMBED Equation.2 (6).
Reasonable assumptions about technology yield diminishing returns to abatement effort, which implies the iso-emissions constraint is convex upwards. It is graphed as the line labeled e1 in Figure 1, and it shows combinations of a and y which yield emissions e1. For a given level of output, emissions fall as abatement rises, thus the direction shown indicates movement into regions of lower emissions.
The firms optimization problem is to maximize EMBED Equation.2 subject to EMBED Equation.2 . This yields first-order conditions
EMBED Equation.2 (7).
A comparison of (7) with (4) and (6) shows that this defines the locus of tangencies where the slope of the iso-profit line equals the slope of the iso-emissions line. It is labeled Zy* in Figure 1. The optimal choice of output and abatement, given the constraint e1, is the point (y1, a1), with associated profits EMBED Equation.2 . The true cost to the firm of meeting this target is the change in the level of profits, EMBED Equation.2 .
The importance of (3) can now be explained in graphical terms. Under (3a), assuming EMBED Equation.2 ensures that the iso-profit lines cross the y-axis vertically, therefore an upward-sloping emissions constraint will always be tangent to an iso-profit line at interior points in the non-negative (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 iso-profit line to cross the y-axis at an angle, as long as the slope of the iso-emissions line at the same point is less than or equal to the slope of the iso-profits line (compare (3) and (7)). Under this condition, the tangency point must be at a non-negative 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 EMBED Equation.2 , showing the level of abatement required to achieve emissions e given output level y. Note also that EMBED Equation.2 and EMBED Equation.2 . Substituting into the profit function for a and taking partial derivatives yields
EMBED Equation.2 (8)
Along the tangency locus (7), it must be the case that EMBED Equation.2 . 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 being optimally adjusted,
EMBED Equation.2 (9).
(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 Figure 2 as MAC1. 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 EMBED Equation.2 is positive at a = 0, it will be the case that EMBED Equation.2 at the unregulated emissions level, hence the firms 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 EMBED Equation.2 is positive, a can be held constant and only y adjusted to achieve an emissions target. But y* is defined such that small variations do not change profits. However, once out of the neighborhood of y*, price no longer equals marginal cost, so there is no way to rRoot Entrye~?`?2WordDocumentI:i+00#C:\y DocumentsMYDOCU~1!1_!t~e*ENVObjectPool37.doc00#C:\$1??ENVTAX~1SummaryInformationCMAC.docNSCMAC.DOCC($!`mMy Docume~e
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If (3) does not hold then neither will (7), so as the iso-profit lines converge to the centre in Figure 3, tangencies such as the one on e1, may reach a corner solution, say at y2. 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 y2, before moving to positive levels of a. Along the black line the slope of the iso-profit line is lower than the slope of the iso-emissions line, so
EMBED Equation.2 ,
which implies
EMBED Equation.2 ,
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
EMBED Equation.2 EMBED Equation.2 .
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 downwards between e2 and e*, where e2 is the emissions level associated with output at y2. The MAC function if (3) is violated over some interval (0, e2) is written
EMBED Equation.2 (10).
This is drawn as MAC2 in Figure 2. Clearly, (10) is non-differentiable at the kink point e2.
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 EMBED Equation.2 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 EMBED Equation.2 and e*, a regulation which specifies output reductions is not (in the single firm case) sub-optimal. This contrasts with the argument in Helfand (1991, p. 629-630) 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 an MAC function is kinked and is steeper close to the unregulated emissions level. The Weitzman (1974) analysis shows that, under uncertainty, whether a price-based (e.g. tax or subsidy) or a quantity-based instrument (e.g. standards or permits) is preferred depends on the relative slopes of the marginal damages and marginal abatement cost functions in the neighbourhood of the target emissions level. If the slope of the MAC function changes across the range of emissions, the regulators 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 regulators preference towards using a price instrument (see Baumol and Oates 1988, chapter 5) 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 (1991) analyses iterative tax adjustment mechanisms under myopic behaviour and foresight. In both cases the condition for stability of the optimum becomes less assured, the steeper is 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 (1994) 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 theirs. 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 in 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, although the implication for the smoothness of the MAC is not stated as an objective. Generally, neither (3) nor (3a) are 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 non-negativity 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 overstated. In Figure 1, if a nave regulator were to ask a firm how much it would cost to reduce emissions from e* to EMBED Equation.2 , 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 EMBED Equation.2 . This of course would be an overestimate. Even reporting the dollar value of EMBED Equation.2 would be incorrect. The true cost estimate is the change in profits, EMBED Equation.2 , and depending on the slope of the profit function, this may differ substantially from the nominal value of EMBED Equation.2 .
References.
1. W. Baumol and W. Oates, The Theory of Environmental Policy, 2nd ed. Cambridge University Press, New York (1992).
2. K. Conrad, Incentive Mechanisms for Environmental Protection under Asymmetric Information: A Case Study, Applied Econ. 23, 871-880 (1991).
3. M. Cropper and W. Oates, Environmental Economics: A Survey, J. Econom. Lit. 30, 675-740 (1992).
4. G. Helfand, Standards vs. Standards: The Effects of Different Pollution Restrictions, Am. Econ. Rev. 81, 622-634 (1991).
5. L. Karp and J. Livernois, Using Automatic Tax Changes to Control Pollution Emissions, J. Environ. Econom. Management 27, 38-48 (1994).
6. R. Morgenstern, W. Pizer, and J. Shih. Are We Overstating the Real Economic Costs of Environmental Protection? Resources for the Future Discussion Paper 97-36, June 1997.
7. M. Weitzman, Prices vs. Quantities, Rev. Econom. Stud. 41, 47791 (1974).
Figure 1
OPTIMAL OUTPUT AND ABATEMENT COMBINATIONS
Figure 2
MARGINAL ABATEMENT COST FUNCTIONS
Figure 3
Corner Solutions
List of Terms and Symbols
EMBED Equation.2 firm profits
p output price
y output
EMBED Equation.2 firm costs
w prices of inputs
a abatement effort
e(.) emissions function
* the firms unregulated optimum
Zy* the locus of tangencies defining optimal output-abatement pairs
The term abatement is used by some authors to mean activities applied to reducing pollution, and by others to mean a unit reduction in emissions. The two uses are not equivalent. The terminology here is as follows. Abatement, abatement effort and 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 and marginal cost does not generally hold when emissions are constrained: see below.
I am grateful to an Associate Editor for this point.
See their Figure 1, page 43.
See, for example, Morgenstern, Pizer and Shih (1997), who estimate that true economic costs to firms average only 13 cents for every dollar of reported abatement expenditures.
PAGE
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=:1)WYCTITLE OF THIS PAPERRoss McKitrickRoss McKitrick> 1 ܥhc eRJ"LD41xZ"|0ƘXAC0<lBBxlllhB|?Ҋh:,hBBlTl
A DERIVATION OF THE MARGINAL ABATEMENT COST CURVE
Ross McKitrick
Department of Economics,
The University of Guelph,
Guelph ON
Canada N1G 2W1.
e-mail: rmckit@css.uoguelph.ca
Telephone: (519) 824-4120 x2532
Fax: (519) 763-8497
January 1999
Acknowledgments
I thank, without implicating, John Livernois, two anonymous referees and an Associate Editor for detailed and helpful comments.
Journal of Economic Literature Classification H2, Q2.
Keywords: Marginal abatement costs, cost functions, emissions functions.
Proposed running head: Marginal Abatement Cost Curve
Abstract
The relationship between a firms 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 non-differentiability 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.
A DERIVATION OF THE MARGINAL ABATEMENT COST CURVE
1. Introduction.
The marginal abatement cost curve (hereafter the MAC) links a firms 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 paper 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 non-obvious 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 1974, Cropper and Oates 1992 and countless others), identifying this condition facilitates theoretical consistency. Some analytical and policy implications are also discussed. For instance, a non-differentiable 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, McDermott and Caulkins (1996) discuss the case of electric utilities which use scrubbers, fuel switching, reallocation of production among st does not generally hold when emissions are constrained: see below.
I am grateful to an Associate Editor for this point.
See their Figure 1, page 43.
See, for example, Morgenstern, Pizer and Shih (1997), who estimate that true economic costs to firms average only 13 cents for every dollar of reported abatement expenditures.
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