[자료] Optimal flow-pollution level (in Environment economics)

자료: Environmental Economics

4.3.1 Optimal flow-pollution level

From an economic perspective, problems of environmental damages arise because the allocation of resources are not optimal. As seen in chapter 3 markets fail because the costs of polluting activities do not fully reflect its harmful impact on the environment. The upper panel in Abb. 26 shows a simple static version for a flow pollutant, where E stands for the quantity of pollution emission per period, B(E) for the total benefit from emitting the quantity of pollution emission E and D(E) for the total damage done by polluting the quantity of pollution emission E.

Note that benefits as well as damages are depending only on the magnitude of pollution flows. Hence, damage and benefit functions are simply:

Formel (14) Damages of pollution: D=D(E)

Formel (15) Benefits of pollution: B=B(E)

[Abb. 26]

The slopes of both functions B(E) and D(E) as shown in Abb. 26 have the general form often assumed for economic analyses. The damage or costs of pollution will rise with the amount of emissions. Further it assumed that the damage or costs are rising with an increasing rate.

Mathematically: ΔD/ΔE＞ 0 and Δ^2D/ΔE^2 ＞0

ΔD/ΔE=D'(E) is the marginal damage(or cost) function, labelled MD(E) in the lower panel of Abb. 26. The Marginal damage is increasing with increasing emissions. As the level of pollution increases, the damages associated with the marginal unit of pollution becomes larger and the rate at which they become larger is increasing. The marginal damage function is upward sloping: Quantity matters! An intuitive example of daily life will be helpful to demonstrate this: Drinking a glass of whiskey a day may have no harmful effect on your body, drinking 1 bottle a day may result to liver damage, drinking 2 bottles of whiskey you will become addicted to alcohol und will be unable make your living by working, and drinking 3 bottles you might die due to alcohol poisoning.

The benefit function even will be upward sloping, but it will rise with a decreasing rate as emissions increase.

Mathematically: ΔB/ΔE＞0 and Δ^2B/ΔE^2 ＜0

Again, ΔB/ΔE=D'(E) is the marginal benefit function, labelled MB(E) in the lower pabel of Abb. 26, and the marginal benefit curve is downward sloping. The reason for this is that pollution abatement costs depends on the emission level. If there are no restrictions of emissions, that is, it is costless to pollute, a firms will produce at the cost minimizing output level, emissions will be at maximum. Assume a power plant, if there are no restrictions on sulphur, the sulphur emissions will be maximal. To reduce a small amount of sulphur emissions a simple and cheap desulphurisation plant will be sufficient. But to reduce the sulphur emission at a larger amount a more sophisticated and therefore more expensive desulphurisation plant is needed. That is, abatement costs rise with increasing level of emissions reductions.

The principle of diminishing marginal benefits (utility) is well known from the consumer preferences, the {more} of something we consume, the less satisfaction we got from additional consumption.

However, for some pollutants the damage and benefits functions might have different properties. The optimal pollution level is where net benefit, i.e. NB=B(E)-D(E), will be maximal. This is at emission level E* in Abb. 26. The simple unconstrained

optimization problem is given by:

Formel (16) max(E) NB = B(E) - D(E)

from the first order condition we get:

Formel (17) ∂NB/∂E=∂B/∂E - ∂D/∂E = 0 ⇔  ∂B/∂E = ∂D/∂E

Hence, the optimal level of pollution is given where marginal benefits of pollution B(E) equals marginal damage of pollution D(E), i.e. graphically the intersection of the marginal benefit curve with the marginal damage curve. This determines the optimal level of pollution: E*.

• If society produces more emissions than E* {then} marginal damages are larger {than} marginal benefits. A reduction by one unit of E reduces the environmental damages at an larger amount (measured in money units) {than} the reduction in the benefits, i.e. marginal damages decreases, marginal benefits increase. Net benefits increase while reducing emissions.
• Conversely, if the emission level is less than E* , marginal benefits are higher than marginal damages, increasing emissions results in higher net benefit.

At the intersection of the marginal damage and the marginal benefit curve, the optimal level of emission E* is determined. But even the value of marginal damage and marginal benefit is determined. This value is given by λ * . Since in our model there is no market, and hence, no market price for emissions, λ * is called a ‘shadow price’. This price will become important discussing environmental policy instruments in chapter 5.

While explaining the slope of the marginal benefit curve by abatement costs, an alternative diagram is presented in Abb. 27. The abatement cost reflect the cost of reducing pollution to a lower level, so that there are fewer damages. This cost includes labour, capital and energy necessary to reduce emissions.

[Abb. 27] 

Note that in general the marginal abatement cost curve could be identical to the marginal benefit curve. But the marginal benefit curve might reflect other aspects, not mentioned in the MAC, in particular opportunity cost from reducing the levels

of production or consumption.

[Abb. 28] 

The economical{ly} optimal level of pollution again will be E*, where the marginal damage (cost) curve MD intersects with the marginal abatement cost curve MAC. It is the level that minimises the total costs of pollution which is the sum of total abatement cost and total damages. 

• Total abatement costs are represented by the area B and total damages are represented by area A in Abb. 27, at the optimal emission level E*. 
• {The area under the marginal damage, respectively the marginal abatement cost function shows the total damages, respectively the total abatement costs.}

That the sum of total abatement cost and total damage is minimal at the level E*, can be demonstrated using Abb. 28. At emission level E’, total damage are represented by the area C. Total abatement cost are represented by the areas A+B+D+G. Compared to the sum of total cost in the case of emission level E*, which is represented by the areas A+B+C+G, total costs will rise by the amount represented by area D. This is the efficient {efficiency} loss due to excessive abatement. 

An efficient{efficiency} loss arise from too moderate abatement, for example at emission level E". Total abatement cost are represented by the area G and total damages are represented by the areas A+B+C+F. Compared to the optimal level E*, this results in higher total costs, characterized by the area F. In Tab 8 these results are summarized. (...)