# Does Abatement Cost Mean Cost of Reducing Emissions with Output Held Constant?

In the economics of pollution control, frequent reference is made to 'abatement cost' and/or 'marginal abatement cost'. Is it normally implicit in the use of these terms in economics that output is held constant? In other words, that the cost referred to is the cost of reducing emissions while producing the same output?

Common & Stagl (1) offer an explanation of abatement cost which explicitly states that abatement may be achieved by a reduction in the activity producing the emissions. However, most presentations of abatement cost that I have seen do not make such a statement. So would that perhaps be a non-standard usage from an ecological economics perspective?

Please note this is a question about correct or normal usage of 'abatement cost'. There is of course a real issue about whether we should be producing and consuming less so as to reduce damage to the environment, but I'm not asking about that here.

Reference: (1) Common M & Stagl S (2005) Ecological Economics: An Introduction Cambridge University Press p 415

To offer a theoretical micro-economic perspective, "abatement" can indeed be achieved by reducing the level of output of a company, since in most cases pollution is positively associated with the level of production.

But in such an approach, there is no firm-level hard cost, at least not in the mid- to long-term (i.e. abstracting for transitory scale-down costs): The firm does not engage in any abatement activity (which would create direct costs for it) - it simply produces less, and by that alone, abatement happens.

A cost-benefit analysis at the social level regarding the trade-off "lower production - less pollution" is another matter. At the micro-economic level, "abatement cost" usually refers to cost born by private businesses to "clean-up".

But in economics, it is more than that: it is more realistic to wonder, what would it take (in terms of incurred costs at firm-level) to not sacrifice output while achieving abatement? And optimally what could be the minimum cost to achieve abatement without sacrificing output?
But this question already has inherently the concept of a constant level of output in it, or more properly, a given level of output, exactly as any other cost-minimization problem analyzed in economics.

More formally, assume a production function $q = f(\mathbf x)$ and a "pollution production" function $e = h(\mathbf x, z)$ where $\mathbf x$ are factors used in production, while $z$ is resources used for abatement, i.e. we have $\partial e / \partial z <0$.

Then, the enhanced cost-minimization problem of the firm can be written with respect to a given level of production and a given level of pollution (given, not constant).

$$\min TC = \mathbf p'\mathbf x + p_zz$$

$$\text {s.t.} f(\mathbf x) = \bar q, \;\;\; \bar e = h(\mathbf x, z)$$

The Lagrangean of the problem is

$$L = \mathbf p'\mathbf x + p_zz + \lambda[\bar q - f(\mathbf x)] + \mu [\bar e -h(\mathbf x, z) ]$$

which will lead to the first-order conditions

$$\mathbf p -\lambda \nabla_x f(\mathbf x) =\mu \nabla_x h(\mathbf x, z)$$

and

$$p_z = \mu \frac {\partial h(\mathbf x, z)}{\partial z}$$

These will give us some cost-minimizing relations

$$\mathbf x^* = g_1(z^*, \bar q, \bar e, \mathbf p, p_z);\;\; z^* = g_2(\mathbf x^*, \bar q, \bar e, \mathbf p, p_z)$$

and eventually

$$\mathbf x^* = \tilde g_1(\bar q, \bar e, \mathbf p, p_z);\;\; z^* = \tilde g_2(\bar q, \bar e, \mathbf p, p_z)$$

where also present are understood to be the various parameters of $f$ and of $h$.

These determine input absorption for given level of output and pollution, and so also total such cost

$$TC^* = \mathbf p'\mathbf x^* + p_zz^*$$

Then the marginal abatement cost for given level of output is (the negative of)

$$\frac {\partial TC^*}{\partial \bar e} = \mathbf p'\nabla_{\bar e}\tilde g_1(\bar q, \bar e, \mathbf p, p_z) + p_z\frac {\partial \tilde g_2(\bar q, \bar e, \mathbf p, p_z)}{\partial \bar e}$$

Note that here the concept takes into account not only how costs may change due to a different level of employment for the abatement resource $z$, but also how this will affect the input-factor mix for output production, and so the output production cost.

No, there's no assumption that output is constant.

Indeed, in most cases, it's expected that output will change, if only a little.

For example, there are abatement costs that involve a one-off increase in output which brings about a long-term decrease in output, such as insulating buildings to abate greenhouse-gas emissions from fossil-fuel powered heating.

Let's take the case of an Australian dwelling being insulated late in the year, during the start of summer, in November 2014 (I'll take the accouting year to be the same as the calendar year, rolling over on 1 Jan). So for 2014, we have had full heating costs from the winter that took place May-August, as well as the manufacture and installation of the insulation materials; so 2014 sees an increase in output over 2013. But in 2015, that insulation won't be manufactured or installed, and the dwelling's heating demand will be lower, so 2015's output is below that of both 2014 and 2013.

To produce output-constant costs would require putting the abatement measures through a macroeconomic model, as part of their cost-estimation. Abatement cost calculations are messy and uncertain enough as it is (before we even get onto the problems with marginal abatement cost curves, one of their more common presentations), without adding a whole extra layer of messy uncertainty that is macroeconomic modelling.

• +1 A perfectly reasonable answer to the question as I worded it. I was thinking of output at firm level, not aggregate output, but didn't make that clear - a lesson for me to be more careful with wording in future. Dec 23 '14 at 12:15
• Heh, I'm so used to only thinking of the macro level, that firm-level never even occurred to me ... Dec 23 '14 at 16:22