Exam set: Pollution control with non-convex abatement cost Flashcards
Problem 2
Case: Optimal pollution control
Problem: Abatement cost functions of polluting firms are non-convex
- Occur normally when investment in pollution abatement involves a significant fixed cost (K), because you gotta buy new technologies, rethink your company and transport structure. - A relative high emission tax may not be sufficient to motivate firms to engage in pollution abatement.
see math note 2
Question 5.1.1
Use (1) to show that once the firm has incurred the fixed cost K, its MAC will be positive and increasing in the level of abatement
Answer: We want to analyze how the total abatement cost varies with the level of abatement (M^0 - M), assuming the abatement actually takes place.
See math note 2
Hence the marginal cost (MAC) is positive and increasing. Note that we differentiate the abatement cost function with respect to the level of abatement, M^0 - M, not with respect to the level of emissions, M.
Empose a tax
Now suppose, that instead of simply ordering the firm to reduce its pollution, the Enviromental Protection Agency (EPA) imposes an emission tax at the rate t per unit of the pollutant emitted, hoping that this will induce the firm to reduce its emissions tom some target level. The firm´s total pollution-related costs (TC) will then be
see math
Question 5.1.2
Assume that at some t > a. Show that if the firm decides to invest in pollution abatement, its optimal level of abatement will be
see math
Comment briefly on the economic intuition
Answer: To find the optimal abatement effort, we must derive the firm´s optimal emission level. This is the emission level that minimize the firm´s toatal pollution-related costs stated in (2). The first-order condition for minimization of (2) with the respect to M is
see math
Which is identtical to (3).
According to (i), a higher value of a implies a higher MAC for any given level of abatement, and a higher value of b means that MAC rises more steeply with the level of abatement. So, for any given emission tax rate t, the amount of pollution a company choses to reduce is less and less the higher the values of parameters a and b. The higher the values of t, the hihger tax is put upon the company for polluting, whihc necesarialy would lead to a lower level of abatemnt.
Quesiton 5.1.3
Show that if
see meath note 2
The firm will not want to undertake any pollution at all. (Hint. Derive an expression for the firm´s total pollution-related costs if it decides to invest in abatment. Compare this to the total pollution related cost incurred if the fomr does not invest in abatment.)
Answer: if the firm actually chooses to incur the fixed cost K of investing in abatement equipment, its optimal (cost-minimizing) abatement effort will be given by (3). By inserting (3) in (2), we can therfore derive an expression for the firm´s total pollution-related costs TC^a if the it invest in abatement:
see math
If the firm decides not to incur the fixed cost K of investing in abatement equipment, it will not be able to abate, and henve its emissions will be M^0 (initial abatement). Its abatement cost will then be zero, but it will have to pay the total emission tax bill tM^0. Thus the firm´s total pollution-related costs TC^nona if it does not invest in abatement will be
see math
Comparing (iii) and (iv), it is obvious that when the inequality (4) holds, the firm´s total cost will be lower if it does not invest in abetement, so this will be the preffered strategy, assuming that competition forces the firm to focus solely on cost-minimization and it does not consider positive externalities from less pollution.
Question 5.1.4
Suppose EPA´s (environmental protection agency) target is M*. Write down an expression for the emission tax rate that will be needed to attain this target, assuming that the firm actually decides to invest in pollution abatement.
Answer: Define M = M*, insert in (3) and rearrange, hence
See math
Question 5.1.5
For concreteness, we now assume that
a = 10, b = 2, M^0 = 200, M* = 100, K = 1500
Suppose that the EPA can use an emission tax as well as a subsidy to investment in abatement equitment to implement its pollution taget M* = 100. The investment subsidy is granted as a share s of the firm´s fixed cost of pollution abatement. Calculate the emission tax rate t and the minimum value of s that will be necessary to ensure that the pollution target is met, given the parameter values in (5).
Answer: Equation (v) gives us the emission tax rate needed to implement the pollution target, assuming that the firm aquires the necessary equitment for pollution abetment (goes through with the fixed cost). Inserting the first four numbers form (5) in (v) gives,
math
If EPA pays a fraction s of the expenditure K that must be incurred to acquire the equitment for pollution abatement, it follows from (2) and (3) that the firm´s total pollution-related costs will be
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If the firm does not invest in abatement, its total pollution-related cost will simply be tM^0. On the other hand, if the firm will invest in abatement if this does not increase its total cost, it follows that the minimum necessary rate of investment subsidy must satisfy the condition
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Substituting the values of a, b and K from (v) into (viii) along wiht the solution for t stated in (vi), we obtain
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Hence the EPA must grant a subsidy of at least on third of the total investment expenditure to induce the firm to invest in abatement.
Question 5.1.6
What is the social cost of attaining the pollution target M* = 100, given the parameter values in (5)? How is the social coist shared between the government and the firm? (Hint to answering the latter question: What is the total cost imposen on the firm, and what is the net revenue gain for the government?)
Answer: In general, the social cost of pollution abatement is simply the total abatement cost given by equation (1). Inserting the numbers in (5) into (1), we have that
math
In broader terms, we understand the social cost of abatement to be the difference between the total cost imposed on the firm, TC^a, and the net revenue gain by the government, R. In this way we have a better understanding of how the cost are divided in the society.
The total cost to the firm is given by equation (vii). When the subsidy rate satisfies (viii), it cancels the fixed cost out, it follow from (5), (vi) and (vii) that
math
The government´s net revenue R is
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Substracting the government´s gain from the cost to the firm, we get 42.000 - 16.000 = 26.000 which is exactly the social cost of pollution control.
Question 5.1.7
Although it is not necessarily a rational policy, the revenue from “green” taxes is sometimes earmarked to finance public expenditure on environmental policy initiatives. Suppose the government would like to use its net revenue from pollution taxes to pave the way for a more ambitious future enviromental policies. Discuss breiefly how the governement could spend the revenue in a useful way in the context of the model above.
Answer: The government might government might use the revenue from pollution taxes to support R&D (research and development) in “green” technologies. In the context of our model, such R&D could bring down the future fixed investment cost K of investing in abatement equipment. This would reduce the future investment subsidies needed to induce firms to invest on abatement. In this way the R&D financed by pollution taxes could help ensure that more firms will engange in abatement. (comment. This question is speculative, so there is no single objectively correct answer).
Question 5.1.8
The model above assumes that no abatement is possible unless the firm incurs some fixed invesment cost. Is that a reasonable assumption?
Answer to Question 5.1.8: In the real world there are many forms of pollution abatement that involve a significant initial investment cost. For example, think of plant and equipment for treatment of waste water and garbage, instalment of filters on top of factory chimneys, equipment for carbon-capture-and-storage, investment in wind turbines to replace fossil fuels, etc. More generally, firms will often have to reorganize production in some way if they wish to reduce their emissions. Such reorganization may also involve a one-time cost that can be seen as an initial cost of investing in abatement. Hence it is important to account for fixed costs when analyzing what is needed to induce private economic agents to reduce their pollution. However, firms may also be able to reduce their emissions by substituting towards less polluting inputs (for example, by switching from a highly energy-intensive towards a more labour-intensive production technology) or by simply cutting their volume of output. Such forms of abatement do not necessarily involve any initial investment cost.
(Comment: An acceptable answer to Question 5.1.8 does not have to be as elaborate as stated here).
Indeed, if we stick to the assumption that abatement costs can be approximated by quadratic functions, it may be that a firm faces two different abatement cost functions of the form
c=α(M0−M)+β2(M0−M)2 (xiii)
C=a(M0−M)+b2(M0−M)2+K (xiv)
Where
α>a, β>b. (xv)
Here the function (xiii) could represent the firm’s abatement costs if it does not invest in new equipment that facilitates abatement. In that case the marginal abatement cost captured by the parameters αα and ββ may be quite high. However, if the firm is willing to incur a fixed cost KK of investment in modern abatement technology, it may instead face the abatement cost function (xv) which involves a lower marginal abatement cost. The firm’s optimal cost-minimizing choice between the two options will depend on the values of the policy instruments tt and ss chosen by the EPA.
