Exam set: Pollution control via taxation versus Command-and-control Flashcards
Problem 4
Case: Optimal pollution control with taxation versus command-and-control
Problem: Study the potential efficiency gain from EPA uses an emission tax as its policy instrument versus if they uses command-and-control.
Firms:
- Two firms (A and B)
- Uniformly mixing pollutants, therefore we look at the total emission from the two firms.
- The firms get the higher benefit if they only abate a little. Abatement in the view of the firm is considered to be a cost, because they somtimes have to finance it.
EPA:
- EPA want to max the net benefit from pollution defined in (2) with respect to individual emission levels M_A and M_B.
- Subject to the M_A + M_B = M* and M_A + M_B = M* < inital pollution from both fimrs.
Question 6.1.1: Discus breifly what kind of costs could be included in the category “abatement cost”.
Answer: There are several ways for a firm to cut emissions.
One way is to cut the level of output: another is to substitute towards less polluting (but typically more expensive) inputs. In these cases the cost of abatement takes the form of the profit lost when output is cut and the firm switches to more expensive inputs.
Another way of abating pollution is to invest in “end of pipe” equipment such as a filter on a chimney that captures the pollutant before they have reached the sourrunding environment.
Yet another way to abate is to invest in new an cleaner production technologies, thereby in effect switching to a new production function. Both of these two later abatement methods will involve cost seen as a fixed amount.
Question 6.1.2
Set up the Lagrangian for the EPA´s maximization problem (denote the Lagrange multiplier associated with the constraint (3) by lambda).
Answer: The Lagrangian implied by equation (2) ans (3) takes the form
see math note 2
Question 6.1.3
Derive the first-order condition for the EPA´s optimal choice of M_A and M_B. Give an economic interpretation of the first order conditions. What is the economic interpretation of the Lagrange multiplier lambda? Should both firrms undertake the same amount of abatement, or should they both emit the same quantity of the pollutant.
Answer: From (i) we get the first order conditions
see math note 2
The lagrange multiplier lambda is the shadow price of pollution: it measures the price that society (represented by the EPA) is willing to pay to reduce emissions by an extra unit (MAC). In broader terms, the shadow price represent the marginal cost or benefit of adjusting emissions level.
Specifically, lambda meassures the cost that EPA is willing to impose on firms to reduce emissions by an extra unit (see Sørensen (2020) for further interpretation of the Lagrange multiplier). Thus conditions (ii) and (iii) imply that each firm should abate its emissions up to the point where its marginal abatement cost (lefthand side of the equations) equals th shadow price of pollution i.e. the price that society is willing to pay for reducing pollution by one more unit.
A key implication of (ii) and (iii) is that
See math note 2
In other words , an optimal allocation of abatement requires that the marginal cost be equalized across firms since this is a necessary condition for minimization of society´s total abatement costs. If we do not equalize the marginal cost across firms and instead tell the companies to reduce the same amount of pollution, we will not get the optiaml lowest cost for society, assuming the firms MAC is different from the start. If one firm has a low MAC, due to implemented green technologies in its production chain, and the other firm has a high MAC, due to few or zero implemented green technologies, then we will not get the social optimum if government ordered every firm to abate the same amount of pollution units.
Since the functions in (iv) will generally have different quantiative properties, condition (iv) implies that the optimal abatement effort will typically differ across firms, just as their optimal emission levels will generally differ.
Why should we equalize the MACs? E.g. chatten
E.g (chatten)
Scenario:
Imagine there are two factories, Factory A and Factory B, that both emit pollution. The government has set a target to reduce total pollution by 100 units. Each factory has different costs associated with reducing their emissions.
* Factory A: Has older technology and thus higher costs for reducing emissions.
* Factory B: Has newer technology and can reduce emissions more cheaply.
Cost Structure:
* Factory A:
○ Cost to reduce 1 unit of pollution: $30
○ Cost to reduce 2 units of pollution: $60 (and so on, $30 per unit)
* Factory B:
○ Cost to reduce 1 unit of pollution: $10
○ Cost to reduce 2 units of pollution: $20 (and so on, $10 per unit)
Without Equalizing Marginal Costs:
Suppose the government initially requires both factories to reduce pollution by 50 units each without considering the differences in their abatement costs.
* Total Cost for Factory A:
50 units×$30/unit=$150050 units×$30/unit=$1500
* Total Cost for Factory B:
50 units×$10/unit=$50050 units×$10/unit=$500
* Total Cost for Society:
$1500+$500=$2000$1500+$500=$2000
With Equalizing Marginal Costs:
Now, let’s consider the optimal scenario where the government uses a policy to equalize the marginal cost of abatement across both factories. The policy could be in the form of tradable permits, where factories can trade emission reductions.
Factory A can reduce emissions by 20 units at a cost of $600 (since 20 units ×× $30/unit = $600), and Factory B can reduce the remaining 80 units at a cost of $800 (since 80 units ×× $10/unit = $800).
* Total Cost for Factory A:
20 units×$30/unit=$60020 units×$30/unit=$600
* Total Cost for Factory B:
80 units×$10/unit=$80080 units×$10/unit=$800
* Total Cost for Society:
$600+$800=$1400$600+$800=$1400
Intuitive Understanding:
1. Efficiency: By allowing Factory B to handle more of the abatement, we utilize its lower abatement cost more effectively. Factory A, with its higher costs, does less abatement.
2. Cost Minimization: The total cost to society is minimized when we let the factory with the cheaper abatement options do more of the work. This results in a lower overall cost than if each factory were required to abate an equal amount of pollution.
Conclusion:
In the second scenario, the total cost of reducing 100 units of pollution is $1400, which is significantly less than the $2000 in the first scenario. This demonstrates why equalizing the marginal cost of abatement across firms results in the most cost-effective way to achieve the pollution reduction target.
Now suppose the EPA aims to implement the optimal allocation of pollution abatement by imposing an emission tax at the rate t per unti of pollution emitted. For each firm the total cost TC_i, related to it pollution will then be
Math See note 2
Question 6.1.4
As a necessary condition for max of profits, each firm must minmize its total pollution related cost given by (4). Derive the first order condition for cost-minimization by the indiviual firm and give an economic interpretation of this condtion. Can EPA implement the optimal allocation of via the emiision tax? If so, how?
Answer: From equation (4) we find the first order condition
see math
Thus each firm will abate up to the point where its marginal abatment cost (MAC) equals the emission tax rate. If the MAC is higher than t, the firm can reduce its total cost by reducing its abatement effort and allowing its emission tax bill to increase (higher M_i –> tM_i increases). If the MAC is lower than t, the firm will increase the total cost by increasing its abatment effort, decreasing the amount of pollution M_i so the tax bill wil fall.
From (ii), (iii) and (iv) we see that EPA can implement the socially optimal allocation of abatement by choosing the emission tax rate
see math
The value fo lambda can be calculated, but it is not necessary. Only complicates things.
For concreteness, let us now assume that the abatement cost functions take the following form where a, b_A and b_B are constants, and where we assume that M^0_A are not equal to M^0_B and b_A are not equal to b_B. Hence our two fimrs have different initital pollution and different marginal abatement costs,
see math
Question 6.1.5
Verify that the abatement cost functions (5) and (6) have hte same general properties as the abatement cost functions stated in (1). Meaning check the properties of the first and second derivative,
Answer: According to (5) and (6), when M^0_i - M_i = 0 we have that C_i(0) = 0, i = A,B, as assumed in (1). In the relevant range of emissions where M_i is less or equal to M^0_i, it follows from (5) and (6) that
see math
Thus the first and second deratives of the abatement functions (5) and (6) have the signs assumed in (1).
Question 6.1.6
Show that when the abatement cost functions are given by (5) and (6), the optimal allication of abatement efforts will be given by the following expressions:
see math
Explain the economic intuition behind these results
Answer: From (iv, equalized MAC across firms) and (vii, new MAC) it follows that an optimal social allocation abatement requires that
see math
At the same time, the pollution target (3) requries
see math
Inserting (x) int0 (ix) and rearanging, we get
see math
As stated in (7).
Since M^0_A + M^0_B - M* is the total abatement mandated by the EPA and alpha is the share of total abatement undertaken by firm A, it follows that the remaining share of abatement undertaken by firm B given by formula (8). From the definition of alpha we see that the share of abatement allocated to firm A is higher the larger the value of b_B realtive to b_A. The parameters b_A and b_B are the slopes of the marginal abatement cost curves of the two firms, so it is intuitive that it is optimal to allow firm B to carry out a smaller share of the total abatement if the MAC rises faster with the ammount of abatement in firm B than in firm A. following this firms A´s MAC is lower than firm B´s, due to e.g. better green technologies in the production chain. Indeed, given our assumption in (6) that b_B > b_A, it follows from the definition of alpha that it is socially optimal to allocate more than half of the total abatement effort to firm A.
Question 6.1.7
Show that in order to implement the optimal allocation of abatement, the EPA must choose an emission tax rate equal to
see math
Explain the economic intuition for this result (how does the emission tax rate depend on the varirous parameters of the abatement cost functions?)
Answer: From (vi) we know that t = lambda. It then follows from (ii) and (vii) that
see math
Inserting (7) into (xi) and using the dfinition of alpha, we get (9)
see math
The intuitive implicataions. First, the emissions tax rate has to be higher the greater the total amount of abatement M^0_A + M^0_A - M* that the EPA requires. Second, the higher the value of a - that is, the higher the MAC at any given level of abatement - the harder it is to get firms to abate, because it is expensiveeeee, so the higher is the necessary emission tax rate. Third, a steeper slope, b, of the MAC curves (a faster increase in MAC as abatement increases) also lead to a higher emission tax to attain the pollution target. Both a and b therefore decreases the level of abatement, due to the cost inflicted by them on the firms, which will lead the EPA to increase the level of emission tax rate to reach the social optimal abatement target.
Now suppose that, instead of relying on an emission tax, the EPA decides to use non-transferable emission quotas to attain its pollution target M. Suppose further that the EPA allows each firm to emit the same quantity of the pollutant while keeping total emission equal to M. We have
See math
Question 6.1.8
Discuss briefly whether the allocation of emission rights in the command-and-control regime (10) can be said to be fair.
Answer: At first glance it might seem fair both firms must live up to the same pollution standard. However, given our assumption that b_B > b_A, it will be more costly for firm B than for firm A to obey the pollution standard. Furthermore, emission form firm B are no more harmfull then emissions from firm A since we are dealing with a uniformly mixing pollutant. Against this background, one can argue that it would not be fair to impose a stricter pollution target on firm A than on firm B, since abatement is cheaper for firm A. Assuming we want to reach a social optimal pollution target, which benefits all.
Assume for concreteness that
M^0_A = 100, M^0_B = 150, M* = 100, a = 10, b_A = 1, b_B = 2, (11)
Question 6.1.9
Denote the sum of the abatement costs of the two firms under the emission tax regime and under the command-and-control regime by TAC^t and TAC^c, repectively. Calculate TAC^t and TAC^c , using the parameter values in (11). By how many percent does the total abatement cost under command-and-control exceed the total abatement cos tunder the emisson tax regime? Asumming that the parameter values (11) are in fact realistic, does this type of calculation give a balanced picture of the relative merits of emission taxes versus command-and-control regimes? Discuss
Anwer: Using (5) and (6), we obtain the following general formula for the aggregate abatement costs under the tax regime
Under the command-and-control regime using M_A = M_B = M*/2 (10), so the aggregate abatement costs under this regime will be
see math
Under the tax regime the common tax will ensure an equalization of the marginal abatement cost across the two firms. Provided the tax rate is set at the “right” level which ensures that pollution is reduced to the target level M*, the tax regime will ensure fulfilment of the conditions (7) and (8) for an optimal cost-effictive pollution control. Inserting (7) and (8) in in (xii), we get
see math
Using the parameter values (11), we have
see math
Instering the numbers in (11) and (xv) into (xiv), we find
see math
For comparisson, we insert the parameter values in (11) into (xiii), we find that the aggregate abatement cost in the command-and-control regime is
see math
Hence the additional relative cost of choosing command-and-control rather thena emission tax is
see math
In this particular numertical example.
However, even if the real-world abtement cost differ as much as assumed here, this exposition overstates the advantage of the tax regime over the command-and-control aproach. Specifically, we have assumed that the EPA knows the abatement cost functions (5) and (6) so that it is able to set the tax rate t at the right level which ensures the total emissions will be exactly equal to M* right from the beginning of the regulation. But if EPA actually has this information about the abatment cost of induvidual firms, it could use this information to impose firm specific comand-and-control pollution targets satisfying the optimum conditions (7) and (8, thereby securing the same cost-effective allocation of abatement as the allocation emerging under the tax regime. In other words, there is no reason why EPA should follow the simplistic rule (10) when setting pollution targets for induvidual firms if they have the fulle info about the firms abatement cost functions.
