The Cost-Effective Choice of Policy Instruments to Cap Aggregate Emissions

with Costly Enforcement^{∗}

Marcelo Caffera^{†} Universidad de Montevideo

Carlos A. Chávez^{‡} Universidad de Concepción

First Draft, June 2009.

Abstract

In this paper we first study the total-cost-effectiveness of inducing expected perfect compliance under a system of emissions standards when not only abating emissions but also monitoring and sanctioning are costly. We find that the cost-effective design of a program that caps aggregate emissions of a given pollutant from a set of firms based on emissions standards is one in which standards are firm-specific and perfectly enforced. We then compare the total (abatement, monitoring and sanctioning) expected costs of such an optimally designed program with that of an optimally designed program based on a perfectly competitive emission permits market, in the context of incomplete information. We find that the expected total costs of capping aggregate emissions to a certain level are minimized by the emission standards program. We conclude that, it is not in the name of cost-effectiveness per se that we economists are to argue in favor of tradable emission permits, but in the name of information advantages for the regulator.

∗We gratefully acknowledge financial support provided by the Agencia Nacional de Investigación e Innovación (ANII) - Fondo Clemente Estable - Uruguay, under Project ANII FCE 2007-351.

†Correspondence address: Departamento de Economía, Universidad de Montevideo, Prudencio de Pena 2440, Montevideo 11600, Uruguay. Email: marcaffera@um.edu.uy.

Tel: (5982) 707 4461. Fax: (5982) 708 3842

‡Departamento de Economía, Universidad de Concepción, Concepción, Chile, E-mail:

cchavez@udec.cl.

1 Introduction

The world witnesses discussions both in the US and abroad about the way and by how much to cap emissions of green-house gases. One of the most important arguments behind these discussions is the costs of the implied emission reductions. We environmental economists have been giving a clear policy recomendation for such an issue for a long time: whenever possible, a regulator should cap emissions by means of a competitive market on emission permits because this policy instrument minimizes the aggregate abatement costs of reaching any chosen cap with minimum information requirements for regulators. Probably based on this policy recommendation, the European Union has adopted an emissions trading scheme (the EU-ETS) as an important instrument to limit its GHG emissions. The Obama administration is also pushing a similar alternative in Congress. The Waxman-Markey´ s American Clean Energy and Security Act has been approved in the House of Representatives and is now being discussed in the Senate. But climate change is not the only environmental issue in which environmental economists seem to have influenced policy decissions: until the appearance of the EU-ETS, the US was home of the major policy experiment with tradable permits; the SO_{2} allowance market to control acid rain. SUBSTANTIATE ALL THIS WITH CITES.

This apparent success of the profession in the recomendation of this policy instrument mainly as a cost-effective way to attain a certain level of environmental quality may be seen as surprising. A tradable emissions permit system minimizes abatement costs, but these are not the only social costs of caping emissions. There are other important costs, such as the cost of monitoring compliance and sanctioning violations. The environmental economics literature has not yet given a definite answer with respect to the relative costeffectiveness of a tradable emission permits system with respect to one based on emission standards when enforcement costs are brought into the picture.^{1} Malik (1992) compares the costs reaching a certain level of aggregate emissions by means of a perfectly enforced program based on emission standards with that of a perfectly enforced program based on tradable permits, for a regulator with perfect information. He concludes that the enforcement costs under tradable permits may be higher than those under emission standards.

1Even more, a recent paper surveying the literature on the choice of policy instruments completely ommits this issue (see Goulder and Parry, 2008).

Therefore, although the program based on tradable permits minimizes the aggregate abatement costs, the total costs of such a program could end up being higher than the total costs of a program based on emission standards. Nevertheless, he does not consider sanctioning costs. Hahn and Axtell (1995) compare the relative costs of a uniform emission standard instrument with that of a tradable permits system allowing non-compliance, but setting the aggregate cap on the total legally allowed level of emissions, not in the actual level of emissions, which cannot be anticipated by the regulator. More importantly, they consider only abatement costs and the fines as the relevant costs to minimize; no monitoring or sanctioning costs are considered. Chávez, et al. (2009) repeat Malik’s exercise for a regulator that, unlike Malik´ s, cannot perfectly observe the abatement costs of the firms, but knows its distribution. With this information, he chooses to inspect all firms with a homogeneous probability that is high enough to assure compliance of the firms with higher abatement costs. With this monitoring strategy, emissions satandards are more costly than tradable permits.

One important aspect that all the above mentioned papers share is that they do not consider the cost-effectiveness of inducing compliance. They simply assume that perfect compliance is the regulator’s objective, as in Malik (1992) and Chavez, et al (2009), or it is simply non-attainable, as in Hahn and Axtell (1995). But inducing compliance is costly for the regulator. Stranlund (2007) seems to be the first to have adressed this issue of whether the regulator can use non-compliance as a way to reduce the costs of a program that cap aggregate emissions. To put it clearly, the question he addresses is the following: if a regulator wants to achieve a certain level of aggregate emissions from a set of firms at the least possible cost using tradable permits, does it have to design the program to allow a certain level of noncompliance or does it have to perfectly enforce such a program? The answer depends on the relative marginal cost of inspecting versus sanctioning, which in turn depends on the form of the fine structure. Taking into account abatement, monitoring and sanctioning costs, Stranlund concludes that the total-cost-effective design of a program based on tradable permits is one in which the marginal penalties are constant and the program is perfectly enforced. Arguedas (2008) replicates Stranlund´ s analysis for the case of an emission standard system, a regulator with perfect information and one firm. She obtains an identical conclusion.

In this paper we first study the cost-effectiveness of inducing compliance in a system of emissions standards, with more than one firm and under the assumption of incomplete information. Considering the total program costs of an emissions standard system (abatement, monitoring, and sanctioning), and allowing the regulator to choose the fine structure to be increasing or linear in the level of violation, we characterize the total-cost-effective design of an emission standard system. Second, we compare the cost of such an optimally designed system of emissions standards with the costs an optimally designed transferable emissions permit system, as in Stranlund (2007), in the context of incomplete information.

We find that the cost-effective design of a program that caps aggregate emissions of a given pollutant from a set of firms based on emissions standards is one in which standards are firm-specific and perfectly enforced. In addition, such a system attains a certain level of aggregate emissions at lower expected costs than an optimally designed system of tradable permits. This is basically because the distribution of emissions generated by the latter differs from the distribution of emissions that minimizes the total costs of the program. We conclude that, it is not in the name of cost-effectiveness per se that we economists are to argue in favor of tradable emission permits, but in the name of information advantages for the regulator.

Our results also produce a clear policy recommendation for the design of environmental policy in developing countries. The environmental policy in these countries have been frequently described as poorly enforced (CITAS). Explanations of this situation frequently mention the budget constraints that regulators suffer in these countries. Our conlcusion suggests that to design a regulation that sets a cap on emissions that is too costly for the regulator to enforce is of little justification in terms of cost-effectiveness. The regulator could attain the same level of emissions (the same cap) with less budget letting the firms to pollute more and perfectly enforcing the laxer regulation.

We conclude that, it is not in the name of cost-effectiveness per se that we economists are to argue in favor of tradable emission permits, but in the name of information advantages for the regulator. In other words, the imperfect information on the actual marginal abatement costs functions of the firms could led the regulator to set a distribution of abatement responsibilities among firms (to set and perfectly enforce emission standards) that may result in lower expected costs but higher actual costs that those of a system of tradable permits. Clearly, more research is needed with respect to the factors affecting the balance of the costs of both instruments.

The paper is organized as follows. In section 2, we present the standard model of compliance behaviour of a risk-neutral polluter firm that faces an emission standard. Using this, in section 3 we give the conditions under which it is cost-effective to induce perfect expected compliance for a regulator that wants to achieve a certain cap on the aggregate emissions of n firms using emissions standards. In Section 4 we characterize the cost-effective design of such a program when the exogenous structure of the penalty is such that it is cost effective to induce expected perfect compliance and when it is not. In Section 5 we let the regulator to choose the appropiate structure of penalties and we characterize the expected-cost-minimizing design based on emissions standards. Finally, in Section 6 we compare the costs of an optimally designed program based on standards and an optimally designed program based on tradable permits.

2 A firm compliance behavior under an emission standard

In this section we present the standard model of compliance behavior of a risk - neutral polluter firm under an emission standard. (See .....CITAS CLASICAS ACÁ). Reducing emissions of a given pollutant e is costly for this firm. The (minimum) abatement cost function for firm, which we will call firm i, is c_{i}(e_{i}), where e_{i} is the level of emissions of fimr i. Firms’ abatement costs can vary for many reasons, including differences in the type of the good being produced, the techniques and technologies of production and emissions control, input and output prices, and other more specific factors related to the corresponding industrial sector. The abatement cost function is assumed to be strictly decreasing and convex in the firm’s emissions e [c′_{i}(e_{i}) < 0 and c′′_{i}(e_{i}) > 0].

The firm faces an emission standard (a maximum allowable level of emissions) s_{i}. An emissions violation v occurs when the firm’s emissions exceed the emissions standard: v_{i} = e_{i} − s_{i} > 0. The firm is compliant otherwise. The firm faces a random probability of being audited π_{i}. An audit provides the regulator perfect information about firms’ compliance status. If the firm is audited and found in violation, a penalty f(v_{i}) is imposed. For the moment, we just assume that f(v_{i}) = 0 for all e_{i} ≤ s_{i}, and f′(v_{i}) > 0 por all e_{i} > s_{i}.^{2}

2An alternative penalty function could be a two part penalty, i.e. F (v) = F_{0} + f(v), where F_{0} is a fixed fee. Malik (1992), does not consider such type of penalty structure.

Under an emissions standard, a firm i chooses the level of emissions to minimize total expected compliance cost, which consists of its abatement costs plus the expected penalty. Thus, firm i′s problem is to choose the level of emissions to solve

min_{e_{i}} c_{i}(e_{i}) + π_{i}f (e_{i} − s_{i})  (1)
subject to e_{i} − s_{i} ≥ 0

The Lagrange equation for the problem in (1) is given by Γ_{i} = c_{i}(e_{i}) + π_{i}f (e_{i} − s_{i}) − η_{i} (e_{i} − s_{i}), with η_{i} the Lagrange multiplier. The set of necessary Kuhn-Tucker conditions is:

\frac{∂Γ_{i}}{∂e_{i}} = c′_{i}(e_{i}) + π_{i}f′ (e_{i} − s_{i}) − η_{i} = 0
\frac{∂Γ_{i}}{∂η_{i}} = −e_{i} + s_{i} ≤ 0; η_{i} ≥ 0; η_{i} (e_{i} − s_{i}) = 0

Firm i′s choice of emissions: From the Kuhn-Tucker conditions it can be seen that

e_{i} = \begin{matrix}s_{i}, if − c′_{i}(s_{i}) ≤ π_{i}f′ (0) \\ e_{i}(s_{i}, π_{i}) > s_{i}, if − c′_{i}(s_{i}) > π_{i}f′ (0) \end{matrix}

The firm is going to comply with the standard if the expected marginal penalty is not lower than the marginal abatement cost associated with an emissions level equal to the emissions standard. Otherwise, the firm is going to choose a level of emissions e_{i}(s_{i}, π_{i}) > s_{i}, where e_{i}(s_{i}, π_{i}) is the solution to −c′_{i}(e_{i}) = π_{i}f′ (e_{i} − s_{i}) .Note that the marginal abatement costs at the level of the standard can vary among firms, not only because they face a different standard, but also because of the firm’s specific characteristics, possibly not perfectly observable for a regulatory authority.

Arguedas (2008) has already shown that it is not optimal to have a fixed penalty component when inducing compliance with an emissions standard.

3 The Condition under which it is Cost Effective for a Regulator to Induce Perfect Compliance

Now assume a regulator who is in charge of implementing a pollution control program based on emissions standards. The objective of the program is to cap the aggregate level of emissions of a given pollutant to a level E. The regulator wants to achieve this target at the least expected cost, including the abatement costs of the firms and his monitoring and sanctioning costs. Toward this objective he selects the probability of inspection π_{i} and the emission standard s_{i}, for every i. There are n firms that emit this pollutant. The firms differ in their abatement costs, but these are not perfectly observable for the regulator. Nevertheless, he can observe the type of each firm (he can observe whether the firm in question is a pulp and paper mill or a tannery, for example) and has a subjective probability distribution over the possible abatement cost functions of every type of firm. Based on this information, he constructs an expected abatement cost function for every type of firm and uses this as the proxy for the true level of abatement cost. The regulator’s problem is:

min \begin{matrix}(s_{1},s_{2},..,s_{n}) \\ (π_{1},π_{2},..,π_{n}) \end{matrix}E [∑_{i=1}^{n} c_{i}(e_{i}) + μ ∑_{i=1}^{n} π_{i} + β ∑_{i=1}^{n} π_{i}f(e_{i} − s_{i})]

s.t.

1) ē_{i} = e_{i}(s_{i}, π_{i})

2) ∑_{i=1}^{n} ē(s_{i}, π_{i}) = E

3) s_{i} ≤ ē_{i} ∀i = 1, ...n

where the objective function is the total abatement cost of the pollution control program, composed of the expected aggregate abatement, monitoring and sanctioning costs. The expected aggregate abatement costs are

E [∑_{i=1}^{n} c_{i}(e_{i})] . Assuming the cost of an inspection is given by μ, the aggre-

gate monitoring or auditing costs are μ ∑_{i=1}^{n} π_{i}. And assuming that sanctioning

has a cost of β for every dollar, the expected aggregate sanctioning costs are E [β ∑_{i=1}^{n} π_{i}f(e_{i} − s_{i})] .For the moment, we assume that the penalty function

f(e_{i} − s_{i}) is given for the regulator. In other words, he has no ability to change its structure. Although he cannot costlessly observe the emissions, the regulator knows that the firm i will react to a standard s_{i} and a monitoring probability π_{i} according to its reaction function e_{i}(s_{i}, π_{i}).Therefore, he incorporates this incentive compatibility constraint in the problem. Because he cannot observe the abatement cost functions of the firms, the regulator does not know the exact form of this reaction function. Nevertheless, he uses his belief about what the expected abatement cost function for firm i is and the firm’s problem to calculate e_{i}, the level of emissions that he believes the firm will produce as a response to a certain level of the emission standard s_{i} and inspection probability π_{i.} The second constraint summarizes the environmental objective of the program, namely, that the the expected aggregate level of emissions be equal to the target E. Finally, the third constraint ackowledges that it may be in the interest of the firms to violate the emission standard. The Lagrange of the regulator´ s problem can be written as

L =

E [∑_{i=1}^{n} c_{i}(ē_{i}, θ_{i}) + μ ∑_{i=1}^{n} π_{i} + β ∑_{i=1}^{n} π_{i}f(ē_{i} − s_{i})]+λ_{1} [∑_{i=1}^{n} ē_{i} − E]+∑_{i=1}^{n} λ_{2}^{i}(s_{i}−ē_{i})

with λ_{1} and λ_{2}^{i} being the n + 1 multipliers. The n × 2 + n + 1 relevant FOC are:

\frac{∂L}{∂s_{i}} = E [c′_{i}(.)\frac{∂ē_{i}}{∂s_{i}} + βπ_{i}f′(.)(\frac{∂ē_{i}}{∂s_{i}} − 1)] + λ_{1} \frac{∂ē_{i}}{∂s_{i}}  (2)

+λ_{2}^{i}(1 − \frac{∂ē_{i}}{∂s_{i}}) = 0, i = 1, ..., n

\frac{∂L}{∂π_{i}} = E \{c′_{i}(.)\frac{∂ē_{i}}{∂π_{i}} + μ + β [f(ē − s_{i}) + π_{i}f′(ē − s_{i})\frac{∂ē_{i}}{∂π_{i}} ]\}  (3)

+λ_{1} \frac{∂ē_{i}}{∂π_{i}} − λ_{2}^{i} \frac{∂ē_{i}}{∂π_{i}} = 0, i = 1, ..., n

\frac{∂L}{∂λ_{1}} = ∑_{i=1}^{n} ē_{i} − E = 0,λ_{1} ≥ 0  (4)

\frac{∂L}{∂λ_{2}^{i}} = s_{i} − ē_{i} ≤ 0, λ_{2}^{i} ≥ 0, λ_{2}^{i} × (s_{i} − ē_{i}) = 0  (5)

Following Stranlund (2007) and Arguedas (2008) we first identify the conditions under which it is cost-effective for the regulator to induce expected perfect compliance for all firms, that is s_{i} = ē_{i} ∀i = 1, ...n.

If ē_{i} = s_{i}, from (5) we know that λ_{2}^{i} ≥ 0.Because we have also that λ_{1} ≥ 0, we can re-write the firts order conditions of the regulator’s problem as:

\frac{∂L}{∂s_{i}} = \{E [c′_{i} (s_{i})] + βπ_{i}f′(0) + (λ_{1} − λ_{2}^{i})\} \frac{∂ē_{i}}{∂s_{i}} − βπ_{i}f′(0) + λ_{2}^{i} = 0
\frac{∂L}{∂π_{i}} = \{E [c′_{i} (s_{i})] + βπ_{i}f′(0) + (λ_{1} − λ_{2}^{i})\} \frac{∂ē_{i}}{∂π_{i}} + μ = 0

Re-arranging the expressions and dividing:

\frac{∂ē_{i}/∂s_{i}}{∂ē_{i}/∂π_{i}} = \frac{βπ_{i}f′(0) − λ_{2}^{i}}{−μ}

From the firm’s optimal choice of emissions, we know that

−c′_{i}(e_{i}) = π_{i}f′ (e_{i} − s_{i})

From where,

∂ē_{i}/∂π_{i} = \frac{−f′}{c′′_{i} + π_{i}f′′} < 0

and

∂ē_{i}/∂s_{i} = \frac{π_{i}f′′}{c′′_{i} + π_{i}f′′} > 0  (6)

Because a cost-minimizing regulator that wants to achieve ē_{i} = s_{i} wil set π_{i} such that E [−c′_{i}(s_{i})] = π_{i}f′ (0) in order not to waste monitoring resources, we can write

\frac{∂ē_{i}/∂s_{i}}{∂ē_{i}/∂π_{i}} = \frac{π_{i}f′′(0)}{c′′_{i}(s_{i}) + π_{i}f′′(0)} × \frac{c′′_{i}(s_{i}) + π_{i}f′′(0)}{−f′(0)} = \frac{π_{i}f′′(0)}{−f′(0)} = \frac{βπ_{i}f′(0) − λ_{2}^{i}}{−μ}

or

μ\frac{π_{i}f′′(0)}{f′(0)} = π_{i}βf′(0) − λ_{2}^{i}

From where, using λ_{2}^{i} ≥ 0,

μ\frac{π_{i}f′′(0)}{f′(0)} ≤ π_{i}βf′(0)  (7)

We have proved that when the regulator induces (expected) compliance, this condition is met. The reverse is also true. When this condition is met, it is cost effective for the regulator to induce firm i to comply with the emission standard. Why? The right-hand side of (7) is the marginal increase in the expected sanctioning costs if the regulator decreases the standard. The left hand side is the marginal decrease in monitoring costs that the regulator can afford when he decreases the monitoring probability accordingly so as to leave the level of emissions unchanged. What the condition is saying is if the firm is complying with the standard and moving the standard and the monitoring probability so as to make the firm to marginally violate the standard increases the sanctioning costs more than it decreases the monitoring costs, it is not a good idea to do so. The regulator should leave things as they are: set π_{i} and s_{i} so as to induce the firm to comply with the standard. Otherwise, allowing the firm to violate the standard will increase the costs of the program.

Dividing both sides of equation (7) by π_{i} we obtain

μ\frac{f′′(0)}{f′(0)} ≤ βf′(0)  (8)

This new expression allows us to note that the condition under which it is cost effective for a regulator to induce perfect compliance does not depend on any characteristic of the individual firms, but only on the penalty structure and the unit costs of monitoring and sanctioning, which we have assumed are the same for all firms. (See below for a discussion about this assumption).

The above discussion is summarized in the following proposition:

Proposition 1 The cost-effective policy induces expected compliance for all i if and only if μ\frac{f′′(0)}{f′(0)} ≤ βf′(0). If this condition is not met, the cost effective policy induces positive violations of the emission standards for all i.

This Proposition is the same as Proposition 1 in Arguedas (2008). Therefore, we can conclude that Arguedas’ (2008) Proposition 1 is robust to n firms and to imperfect information when we assume that monitoring and sanctioning costs (μ and β) do not differ among firms. Nevertheless, there may be good reasons to assume the contrary. Monitoring costs may vary among firms because of the distance between the firm and the enforcing agency or the number of discharge points may cause μ to vary among firms.^{3} Similarly, sanctioning costs may differ between firms because of their differing propensity to litigate sanctions and challenge the legislation, which may be a function of their budget, their visibility, their environmental strategy, or other characteristics. Therefore, we could have that μ = μ_{i} and β = β_{i}, i = 1, ..n, and μ_{i} ≠ μ_{j} and β_{i} ≠ β_{j}, for at least some i ≠ j. In this case, the condition in Proposition 1 may be valid for some firms and not be valid for other ones. In other words, it could be cost-effective for the regulator to induce violations for some firms and compliance for the rest. We address this issue below.

Stranlund (2007) reached the same result (exactly the same condition) for the case of transferable permits. Therefore, the condition under which it is cost-effective for a cost-minimizing regulator to induce compliance is not instrument-dependent, eventhough the cost-effective monitoring strategy is.

We now turn to characterize the cost effective program to control emissions with standards when it is optimal to induce compliance and when it is not. These two possibilities arise because we still assume that the penalty structure is exogenously given to the environmental regulator.

4 Characterization of the cost-effective program to control emissions with standards when penalties are given

When the structure of the penalty function f(.) is outside the box of tools that the regulator has to design a cost-effective program based on emissions standards, condition 8 eithers holds or not. In other words, when the penalty structure is given, according to condition 8 it is cost-effective to induce perfect compliance or it is not. In the first case, it is easy to show that the optimal policy (π_{1}^{∗}, π_{2}^{∗}, ...π_{n}^{∗}, s_{1}^{∗}, s_{2}^{∗}, ...s_{n}^{∗}) that induces expected compliance is characterized by (1) E [c′_{i}(s_{i}^{∗}, θ)] + μ\frac{dπ_{i}^{∗}}{ds_{i}} + λ_{1} = 0 and (2) π_{i}^{∗} = \frac{E[−c′_{i}(s_{i}^{∗},θ)]}{f′(0)} (See Proof 1 in the Appendix). Note that from (1) we can conclude that E [c′_{i}(s_{i}^{∗}, θ_{i})] + μ\frac{dπ_{i}^{∗}}{ds_{i}} = E [c′_{j}(s_{j}^{∗}, θ_{j})] + μ\frac{dπ_{j}^{∗}}{ds_{j}} for all i ≠ j, (i, j) = 1, ..., n. When it is cost-effective to induce expected compliance, the regulator has to set s_{i}, s_{j} such that the sum of marginal expected abatement and monitoring costs are equal between firms. This result was already obtained by Chávez, et. al (2009). Malik (1992) also reached the same result but in a different context; with perfect information on abtement costs and an exogenously given objective of perfect compliance. Note also that allocating emissions responsibilities in this way does not imply perfect compliance with certainty. In the presence of imperfect information about abatement costs, perfect compliance would require the regulator to set π_{i}^{∗} = \frac{−c′_{i}(s_{i}^{∗},θ_{L}^{i})}{f′(0)}, with c′_{i}(s_{i}^{∗}, θ_{L}^{i}) being the largest possible value of the marginal abatement cost of complying with the standard for firm i. It follows then that the monitoring probability that the regulator has to choose to induce perfect compliance with certainty is larger than the one that it has to choose to induce expected compliance. An immediate corollary that follows from this conclusion is that a program designed to induce perfect compliance with certainty does not minimize the expected costs of the program.

3It is also true that on-line monitoring, as it is the the case in the US SO2 program based on tradable allowances, may work in the opposite direction: it could make monitoring costs more homogeneous among firms than classic monitoring.

When 8 does not hold, it is not cost-effective for the regulator ot induce compliance. In other words, a regulator interested in implementing a program that caps aggregate emissions up to a certain level, has to design such program, meaning to choose the auditing probability and the emission standard for each firm, so as to allow a certain level of non-compliance. In the context of imperfect information, the characterization of the cost-effective program to control emissions with standards when penalties are given and it is cost-effective not to induce expected compliance is given by Proposition 2.

Proposition 2 if the optimal policy (π_{1}^{∗}, π_{2}^{∗}, ...π_{n}^{∗}, s_{1}^{∗}, s_{2}^{∗}, ...s_{n}^{∗}) induces noncompliance in expected terms, it is characterized by

(1) E [c′_{i}(ē_{i}, θ_{i})] + μ\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}}

+β (\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} f(ē_{i} − s_{i}) + π_{i}^{∗}f′(ē_{i} − s_{i}) (\frac{∂v̄_{i}/∂s_{i}}{∂ē_{i}/∂s_{i}} ))

= E [c′_{j}(ē_{j}, θ_{j})] + μ\frac{∂π_{j}^{∗}/∂s_{j}}{∂ē_{j}/∂s_{j}}

+β (\frac{∂π_{j}^{∗}/∂s_{j}}{∂ē_{j}/∂s_{j}} f(ē_{j} − s_{j}) + π_{j}^{∗}f′(ē_{j} − s_{j}) (\frac{∂v̄_{j}/∂s_{j}}{∂ē_{j}/∂s_{j}} ))

for all i ≠ j

and

(2) π_{i}^{∗} = \frac{E [−c′_{i}(ē_{i})]}{f′(ē_{i} − s_{i})} for all i = 1, ...n.

The Proof is in the Appendix.

The first observation from Proposition 3 is that when it is cost-effective to induce a positive expected level of noncompliance, the cost-effective design of a program based on emission standards requires the regulator to set these standards such that the sum of the expected marginal abatement, monitoring and sanctioning costs are equal for all firms.

Another observation is that the effect of moving the standard on the cost-effective level of monitoring π_{i}^{∗}, which is negative, is larger in the case of non-complaince than in the case of expected compliance. It can be seen that the second term in the left-hand side of (1), μ\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} , is the marginal monitoring cost. When ē_{i} = s_{i}, the marginal monitoring costs of moving the standard is ∂π_{i}^{∗}/∂s_{i}, because ∂ē_{i}/∂s_{i} = 1. But with expected violations, moving the standard has an additional effect on expected emissions ē. We know from (6) that 0 < ∂ē_{i}/∂s_{i} < 1. That is, if the regulator laxes the standard, expected emissions increase proportionally less than the standard (the expected violation decreases). Therefore the corresponding decrease in π_{i}^{∗} that is neccessary to leave the expected emissions unchanged is larger in the case of non-compliance than in the case of compliance, |\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} | > |∂π_{i}^{∗}/∂s_{i}| .

The last term of the left-hand side is the marginal expected sanctioning costs. It can be seen that moving the standard has two effects on the sanctioning costs. First, the decrease in the costs-effective level of monitoring (π_{i}^{∗}) caused by an increase in the standard affects the sanctioning costs, because less sanctions are discovered, in the amount β \frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} f(ē_{i} − s_{i}). Second, the

change in the standard affects the level of violations in \frac{∂\overline{v}_{i}/∂s_{i}}{∂ē_{i}/∂s_{i}} . The numerator of this expression is the direct change in violations due to the change in the standard. The denominator introduces the fact that a change in the standards put pressure on the aggregate cap of emissions by an amount ∂ē_{i}/∂s_{i}, and therefore requires the level of violations to decrease decrease even more (0 < ∂ē_{i}/∂s_{i} < 1).

Lastly, assuming μ and β to be the same for all firms, we can conclude from Proposition 2 that the cost-effective level of emission standards are firmspecific. The only reason behind this result is the heterogeneity in marginal abatement costs c′_{i}(ē_{i}), because these costs generate the variation \frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} in the required monitoring and in the optimal size of the violation, and ultimately in the marginal cost of imposing sanctions (last term of the left-hand side). The expression also suggests that even if marginal abatement costs were the same for all firms, differences in monitoring costs and sanctioning costs among firms (μ_{i} ≠ μ_{j,} β_{i} ≠ β_{j}) could also call for differences in the cost-minimizing standards.

5 The expected-cost-minimizing design of a program based on standards when the regulator can choose the fine structure

Having characterized the optimal program when it is optimum to induce compliance and when it is optimum to induce non-compliance, we now allow the regulator to choose the fine structure and therefore the optimality of inducing expected compliance or not. We consider only two fine structure: linear and increasing. Consequently, the regulator has basically to compare four possible alternatives and choose the one tha minimizes the expected cost of reaching the cap E on emissions. The four alternatives are (1) to induce expected compliance with linear penalties, (2) to induce expected complaince with increasing penalties, (3) to induce an expected level of violations with linear penalties, and (4) to induce an expected level of violations with increasing penalties. In equilibrium, to induce expected compliance with linear or increasing penaties has the same expected costs because under complaince there are no sanctioning costs. Also, to induce non-compliance with linear penalties is ruled out by Proposition 1: it is never cost-effective to induce non-compliance when the marginal fine is linear. Therefore, the choice for the regulator boils down to a comparison between the costs of two alternatives: to induce expected compliance or not to induce expected compliance with increasing penalties. The result of this comparison is given in the next Proposition:

Proposition 3 The optimal policy (s_{1}^{∗}, s_{2}^{∗}, ..., s_{n}^{∗}, π_{1}^{∗}, π_{2}^{∗}, ...π_{n}^{∗}, f^{∗}) induces compliance and it is characterized by (1) E [c′_{i}(s_{i}^{∗})] + μ\frac{dπ_{i}^{∗}}{ds_{i}} = E [c′_{j}(s_{j}^{∗})] + μ\frac{dπ_{j}^{∗}}{ds_{j}}

for all i = 1, ...n, i ≠ j, (2) π_{i}^{∗} = \frac{E[−c′_{i}(s_{i}^{∗})]}{f′(0)}, and (3) f(e_{i} − s_{i}) = ϕ(e_{i} − s_{i}) + \frac{γ}{2}(e_{i} − s_{i})^{2} for all i, with ϕ set as high as neccessary to induce all firms to comply and γ is set at any value as long as μγ ≤ βϕ^{2}.

See the proof of Proposition 3 in the Appendix.

The proof is illustrated in the following graph with n = 2:

Illustration of Proposition 3

In the above Figure emissions of firm 1 are measured from left to right and emissions of firm 2 from right to left. The initial situation is assumed to be one in which is optimum to induce violations and f^{n′} = γ(e_{i} − s_{i}). The regulators sets s_{1}^{n} and s_{2}^{n} and the firms expected level of emissions are ē_{1}^{n} and ē_{2}^{n}, such that ē_{1}^{n} + ē_{2}^{n} = E, the size of the box. This is policy P^{n}. But if the regulator changes the fine structure and sets a constant marginal penalty ϕ for both firms equal to the larger marginal penalty in P^{n}, which is that of firm 1, and increases both emission standards up to s_{1}^{c} = ē_{1}^{n} and s_{2}^{c} = ē_{2}^{n}, respectively, without changing the probabilities of inspection, the result is another policy P^{c} that induces expected compliance with constant marginal penalties ϕ = γ × (ē_{1}^{n} − s_{1}^{n}) and that meets the policy objective E with lower expected costs: expected abatement costs and monitoring costs are the same and expected sanctioning costs are zero.

Note that this reasoning is robust to differing monitoring and sanctioning costs among firms. With given increasing penalties (γ > 0), Proposition 1 tells us that it could be the case that the regulator has to allow some firms to violate the standard, while inducing compliance to others in order to minimize the total costs of the program. But when the regulator has the ability to choose the penalty structure, NO HEMOS PROBADO ESTO. A SU VEZ, ME PARECE QUE NO ES CORRECTO. DE ACUERDO A LA PROOF DE ARRIBA, EL REGULADOR PODRÍA SUBIR LOS ESTÁNDARES DE LAS MÁS COSTOSAS, BAJAR EL DE LAS MENOS COSTOSAS Y HACER CUMPLIR A TODAS.

In conclusion, the expected cost minimizing policy when a regulator wants to achieve a certain level of aggregate emissions E with emission standards will be one that induces expected compliance. The structure of the fine does not play any role in equilibrium. Expected compliance could be induced with a constant marginal penalty or an increasing marginal penalty.

Proposition 3 has important implications for the real-world policy design. It is not difficult to think of emission control programs in the real world that were designed or are being designed by different agencies or offices inside a regulatory agency. Think for example of ....PONER EJEMPLOS. If this is the case, one agency or office may set first the environmental objective (the aggregate level of emissions E in our case) and the abatement responsibilities among firms (the standards) while another agency or office may be in charge of designing the monitoring and enforcing strategy, for which it could be using fine structures defined by the general civil or criminal law. Proposition 3 suggests that the resulting regulatory design will be probably sub-optimum, except for the cases in which the penalty structure is the appropriate to induce expected perfect compliance and the offices are coordinated so as to set standards and monitoring probabilities according to Proposition 3.

Proposition 3 does not give a clear rule for setting ϕ "as high as possible" or "as high as neccessary". In the real world ϕ will be given be bounded upward by things such as the possibility that firms may go bankrupt, ... VER LITERATURE Wasserman (1992), Segerson and Tietenberg (1991), Becker (1968).

Finally, if marginal penalties are constant the result we obtained for the case of emissions standards is the same Stranlund (2007) obtained for tradable permits: both programs need to be designed so as to achieve expected compliance and this is done using a constant marginal penalty. Arguedas (2008) proved the same result for the case of one firm.

6 Comparing Costs Between an optimally designed program based on standards and an optimally designed program based on tradable permits

We have seen that the optimal design of a program based on emissions standards is one in which standards are firm-specific (set according to Proposition 2) and perfectly enforced (with the fine structure playing no role in equilibrium). We know from Stranlund (2007) that the optimal design of a program based on tradable permits is one in which the marginal fine is constant and the program is perfectly enforced. The question remains whether a regulator interested in controlling emissions of a given pollutant by setting a cap on aggregate emissions in an expected cost minimizing manner should implement a perfectly enforced program based on firm-specific standards as in Proposition 2 above or a perfectly enforced program based on tradable permits as in Stranlund (2007). That is, once we know the optimal design of the programs based on tte two instruments, what instrument should a regulator use if it wants to minimze the total expected costs of the program? The answer is given in the following Proposition:

Proposition 4 If a regulator wants to control the emissions of a given pollutant by setting a cap on the aggregate level of emissions of this pollutant it will minimize the expected total costs by implementing a firm-specific emissions standards and perfectly enforcing this program.

Proof. The total expected costs of a program that sets a cap on aggregate emissions is given by the expected abatement costs of the regulated firms and the expected monitoring and sanctioning costs of the regulator. That is,

ECP^{k} = EAC^{k} + EMC^{k} + ESC^{k}

where ECP^{k} is the total expected costs of the program k, EAC^{k} is the expected abatement costs of the program k, EMC^{k} is the expected monitoring costs of the program k, ESC^{k} is the expected sanctioning costs of the program k and k = emission standards or tradable permits. We know from Proposition 5 that the optimally designed program based on emission standards must induce expected compliance. We also know from Stranlund (2007) that an optimally designed program based on tradable permits must also induce expected compliance. As a result, our comparison of the programs does not need to take into account ESC because these are zero in both programs when optimally designed. Taking this into account, and assuming that the emission standards program is enforced with a constant marginal

penalty function, we know from Proposition 2 that π_{i}^{∗} = \frac{E[−c′_{i}(s_{i}^{∗},θ_{i})]}{ϕ} in the optimally designed program based on standards and from Stranlund (2007) that π_{i}^{∗} = π^{∗} = \frac{p̄}{ϕ} for all i in the case of the optimally designed program based on tradable permits. Consequently we can write

ECP^{T EP} = E (∑_{i=1}^{n} c_{i} (e_{i} \binom{}{p̄} , θ_{i})) + μn\frac{p̄}{ϕ}
ECP^{ES} = E (∑_{i=1}^{n} c_{i} (s_{i}^{∗}, θ_{i})) + \frac{μ}{ϕ}E (∑_{i=1}^{n} −c′_{i} (ē_{i} (s_{i}^{∗}, θ_{i})))

where ECP^{T EP} is the expected cost of an optimally designed program based on tradable emission permits and ECP^{ES} is the expected cost of an optimally designed program based on emission standards. The proof that ECP^{ES} < ECP^{T EP} is trivial because, by definition, the emission standards and monitoring probabilities in the optimally designed ES program are allocated so as to minimize the total expected costs of the program when this is perfectly enforced. That is, when the costs of the program consist only of abatement and monitoring costs. Therefore, the total expected costs of this program are lower than the total expected costs of a program based on tradable permits, which consists of a different allocation of emissions and monitoring probablities. QED. (¿ES SUFICIENTE ESTO PARA LA

Figure 1:

DEMOSTRACIÓN? NO DEBERIAMOS DEMOSTRAR FORMAL-MENTE QUE COMO ILUSTRA EL GRAFICO LA ASIGAN-CION DE EMISIONES DEL tep ES DISTINTA A LA DEL es PROGRAM?

Moreover, we know that abatement costs are minimized with a trad-

able permits program. Therefore, E (∑_{i=1}^{n} c_{i} (e_{i} \binom{}{p̄} , θ_{i})) < E (∑_{i=1}^{n} c_{i} (s_{i}^{∗}, θ_{i})) .

Therefore, it must be the case that the larger expected monitoring costs of the tradable emission permits program more than compensate its lower abatement costs when compared to the (optimally designed) emission standards program.

The intuition behind this result can be illustrated with the aid of Figure XXX below:

In the Figure, (ē_{1}^{T EP} , ē_{2}^{T EP}) is the expected equilibrium allocation of emissions resulting from a perfectly enforced program based on tradable permits. We call p̄ the equilibrium price of this market, assumed to be perfectly competitive. We know from Proposition 2 that the total expected costs of a program based on emission standards are minimized when the standards and the monitoring probabilities are chosen such that

E [c′_{1}(s_{i}^{∗}, θ_{1})] + μ\frac{dπ_{1}^{∗}}{ds_{1}} = E [c′_{2}(s_{2}^{∗}, θ_{2})] + μ\frac{dπ_{2}^{∗}}{ds_{2}}

with π_{i}^{∗} = \frac{E[−c′_{i}(s_{i}^{∗},θ_{i})]}{f′(0)}, i = 1, 2. In the case of linear marginal penalties,

\frac{dπ_{i}^{∗}}{ds_{i}} = \frac{E[−c′′_{i}(s_{i}^{∗},θ_{i})]}{ϕ} ≤ 0, i = 1, 2, and

E [−c′_{1}(s_{i}^{∗}, θ_{1})] + μ\frac{E [c′′_{1}(.)]}{ϕ} = E [−c′_{2}(s_{2}^{∗}, θ_{2})] + μ\frac{E [c′′_{2}(.)]}{ϕ}

If we assume that E [c′′_{2}] > E [c′′_{1}] > 0 as in Figure XXX, \frac{E[c′′_{1}(.)]}{ϕ} < \frac{E[c′′_{2}(.)]}{ϕ}. Therefore, because E [−c′_{1}(ē_{1}^{T EP} , θ_{1})] = E [−c′_{2}(ē_{2}^{T EP} , θ_{2})] at (ē_{1}^{T EP} , ē_{2}^{T EP}), the optimal emission standards allocation must be to the left of (ē_{1}^{T EP} , ē_{2}^{T EP}), the expected allocation resulting from a tradable permits program. This allocation is drawn as (s_{1}^{∗}, s_{2}^{∗}). We know that (ē_{1}^{T EP} , ē_{2}^{T EP}) minimizes the expected total abatement costs (curve EAC) of all possible allocations responsibilities among firm 1 and 2 such that e_{1} + e_{2} = E. We also know that, by definition, the total expected costs of an optimally designed emission standards program such that e_{1}+e_{2} = E are minimized at (s_{1}^{∗}, s_{2}^{∗}). These costs are represented by the curve ECP, consisting of the sum of expected abatement (EAC) and minimum expected monitoring costs (EMC).

Remark 5 If the regulator could observe marginal abatement costs, the costs effective solution to control emissions would call for a system of perfectly enforced firm-specific emissions standards.

Of course, it is not the case that a regulator can observe firms’ marginal abatement costs. In fact, it may commit relevant mistakes in the estimation of the abatement costs functions. (PONER EJEMPLOS DE ESTIMA-CIONES DE COSTOS DE ABATIMIENTO VIA PRECIO DE EQUILIB-RIO EN EL SO2 MARKET DE EEUU Y EN EL EUETS). If this is the case, the realized social costs of setting and enforcing a global cap on emissions via standards could end up being more expensive than doing it via an emissions trading scheme. In any case, it is not in the name of cost-effectiveness per se that we economists are to argue in favor of tradable emission permits, but in the name of information advantages: the regulator needs to know nothing about abatement costs when designing and enforcing an emissions trading scheme, and by this way it may be a cheaper instrument than emissions standards in terms of the realized social costs of setting a global cap on emissions. (Comparar con Weitzman (1974) y Montero (2002)?)

NOS QUEDA ESTUDIAR O COMENTAR SOBRE LOS FACTORES DE LOS QUE DEPENDE QUE EL REGULADOR TERMINE COME-TIENDO ERRORES TAL QUE LA ASIGNACIÓN DE ESTANDARES SEA TAL QUE (EL COSTO """REAL""" del P^{ES} termine siendo superior a al costo del P^{T EP}.

7 Comparing costs when it is cost-effective to induce non-compliance

RESTARIA COMPARAR AMBOS PROGRAMAS CUANDO ES OPTIMO IDUCIR VIOLACIONES EN TERMINOS ESPERADOS. mOTIVATION: as discussed above, it may be common that the fine structure is given to the environmental authority. Assume that γ > 0. In this case, whether the regulator has to perfectly enforce the program or not depends on the relative size of the monitoring and sanctioning parameters (i.e: whether μγ ⪋ βϕ^{2}). Assume that μγ > βϕ^{2}, then it is cost-effective to design a program that induce a given expected level of non-compliance. How do the cost of such a program based on emission standards compare with one based on tradable permits?

8 Appendix

Proof 1. When ē_{i} = s_{i}, expected violations are zero and therefore there are only two types of expected costs; monitoring and abatement. Moreover,

ē_{i} = s_{i} implies: E [−c′_{i}(s_{i}^{∗})] ≤ π_{i}^{∗}f′(0), or π_{i}^{∗} ≥ \frac{E[−c′_{i}(s_{i}^{∗})]}{f′(0)}. But if the regulator

can induce ē_{i} = s_{i} with π_{i}^{∗} = \frac{E[−c′_{i}(s_{i}^{∗})]}{f′(0)} it would not be cost-effective to

select π_{i}^{∗} > \frac{E[−c′_{i}(s_{i}^{∗})]}{f′(0)} because it would increase monitoring costs. Therefore,

π_{i}^{∗} = \frac{E[−c′_{i}(s_{i}^{∗},θ)]}{f′(0)}. Moreover, the Lagrangean of the problem when ē_{i} = s_{i} and π_{i}^{∗} = \frac{E[−c′_{i}(s_{i}^{∗})]}{f′(0)} is

L = E {∑_{ι=1}^{n} c_{i}(s_{i}) + μ ∑_{ι=1}^{n} π_{i}^{∗}} + λ_{1} [∑_{ι=1}^{n} s_{i} − E]

and therefore

\frac{dL}{ds_{i}} = E [c′_{i}(s_{i})] + μ\frac{dπ_{i}^{∗}}{ds_{i}} + λ_{1} = 0 i = 1, 2, ...n

Proof of Proposition 2. Define the expected level of violation of firm i as v̄_{i} = ē_{i} −s_{i}. From 5, we know that if ē_{i} > s_{i}, λ_{2}^{i} = 0 ∀i = 1, ...n. Using this

and π_{i}^{∗} = \frac{E[−c′_{i}(ē_{i})]}{f′(ē_{i}−s_{i})} we can write the Lagrangean of the regultor’s problem as

L = E [∑_{i=1}^{n} c_{i}(ē_{i}) + μ ∑_{i=1}^{n} π_{i}^{∗} + β ∑_{i=1}^{n} π_{i}^{∗}f(ē_{i} − s_{i})] + λ_{1} [∑_{i=1}^{n} ē_{i} − E]

and the relevant FOC for the problem of choosing s_{i} is

\frac{∂L}{∂s_{i}} = E [c′_{i}(ē_{i})\frac{∂ē_{i}}{∂s_{i}} ] + μ\frac{∂π_{i}^{∗}}{∂s_{i}}

+β (\frac{∂π_{i}^{∗}}{∂s_{i}} f(ē_{i} − s_{i}) + π_{i}^{∗}f′(ē_{i} − s_{i}) (\frac{∂ē_{i}}{∂s_{i}} − 1)) + λ_{1} \frac{∂ē_{i}}{∂s_{i}} = 0

\frac{∂L}{∂s_{i}} = E [c′_{i}(ē_{i})] + μ\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} +

β (\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} f(ē_{i} − s_{i}) + π_{i}^{∗}f′(ē_{i} − s_{i}) (\frac{∂ē_{i}/∂s_{i} − 1}{∂ē_{i}/∂s_{i}})) = −λ_{1}

\frac{∂L}{∂s_{i}} = E [c′_{i}(ē_{i})] + μ\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} +

β (\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} f(ē_{i} − s_{i}) + π_{i}^{∗}f′(ē_{i} − s_{i}) (\frac{∂v̄_{i}/∂s_{i}}{∂ē_{i}/∂s_{i}} )) = −λ_{1}

A similar FOC results from doing ∂L/∂s_{j}, where i ≠ j,

\frac{∂L}{∂s_{j}} = E [c′_{j}(ē_{j}, θ_{j})] + μ\frac{∂π_{j}^{∗}/∂s_{j}}{∂ē_{j}/∂s_{j}}

+β (\frac{∂π_{j}^{∗}/∂s_{j}}{∂ē_{j}/∂s_{j}} f(ē_{j} − s_{j}) + π_{j}^{∗}f′(ē_{j} − s_{j}) (\frac{∂v̄_{j}/∂s_{j}}{∂ē_{j}/∂s_{j}} )) = −λ_{1}

From both FOCs, it is straightforward to see that the regulator chooses s_{i} and s_{j} such that

E [c′_{i}(ē_{i})] + μ\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}}

+β (\frac{∂π_{i}^{∗}/∂s_{i}}{∂ē_{i}/∂s_{i}} f(ē_{i} − s_{i}) + π_{i}^{∗}f′(ē_{i} − s_{i}) (\frac{∂v̄_{i}/∂s_{i}}{∂ē_{i}/∂s_{i}} ))

= E [c′_{j}(ē_{j})] + μ\frac{∂π_{j}^{∗}/∂s_{j}}{∂ē_{j}/∂s_{j}}

+β (\frac{∂π_{j}^{∗}/∂s_{j}}{∂ē_{j}/∂s_{j}} f(ē_{j} − s_{j}) + π_{j}^{∗}f′(ē_{j} − s_{j}) (\frac{∂v̄_{j}/∂s_{j}}{∂ē_{j}/∂s_{j}} ))

for all i ≠ j

The second part of Proposition 2, the choice of π_{i} such that π_{i}^{∗} = \frac{E[−c′_{i}(ē_{i})]}{f′(ē_{i}−s_{i})} for all i = 1, ...n, follows directly from the assumption of a cost-minimizing regulator, and the non-compliant firm’s choice of emissions. QED Proof of Proposition 3. In order to prove Proposition 3, we need first to answer a previous question: what is the cost-minimizing structure of the fine when it is optimum to induce compliance and when it is not. We consider only two fine structure: linear and increasing. The general fine structure can be writen as

f(e − s) = ϕ(e − s) + \frac{γ}{2} (e − s)^{2}

where ϕ is a positive constant and γ ≥ 0.

If the optimal policy induces compliance, sanctioning costs are zero. We also know from Section 3 that in this case the characterization of the costeffective design of a program based on standards calls for the following monitoring probability:

π_{i}^{∗} = \frac{E [−c′_{i}(s_{i}^{∗})]}{f′(0)} = \frac{E [−c′_{i}(s_{i}^{∗})]}{ϕ}

From here we can conclude:

(1) The regulator must choose the linear component ϕ of the fine structure as high as possible because this will decrease the optimum level of the inspection probability, π_{i}^{∗}, and by this way the monitoring costs. Conceptually, this calls for ϕ = ∞ because this will make the monitoring costs equal to zero. But in the real world there may be limits to the upper value of ϕ. These limits may be given by...CITATIONS FROM THE LITERATURE.

(2) The size or value of γ does not matter. The program has the same

minimum expected costs = ∑_{i=1}^{n} c_{i}(ē_{i}, θ_{i}) + μ ∑_{i=1}^{n} π_{i}^{∗} for all γ ≥ 0, with π_{i}^{∗} =

\frac{E[−c′_{i}(s_{i}^{∗},θ)]}{ϕ}.

(3) The structure of the fine does not matter as long as μ\frac{γ}{ϕ} ≤ βϕ, Our conclusions in this respect differ from Arguedas’ (2008). She con-

cludes: "the larger the linear gravity component the lower the minimum probability to achieve compliance and therefore the social costs. Therefore, the optimal fine is one on which f′(0) is as high as possible and f′′(0) is as low as possible, since only the first component affects the probability." On the contrary, we conclude that γ (f′′(0)) plays no role (it does not affect the costs of the program). The penalty function can be linear (γ = 0) or increasing (γ > 0), as long as μ\frac{γ}{ϕ} ≤ βϕ. This is because there are no sanctioning costs and all that the penalty function affects are the monitoring costs, through ϕ. Therefore, our conclusion: If the optimal policy induces expected compliance, the cost-minimizing shape of the fine is such that the linear component is set as high as possible (the progressive component is irrelevant in equilibrium).

If the optimal policy induces non-compliance, how to choose ϕ and γ in order to minimize the costs of a program that produces E?

To answer this question, first note that in the n-firm scenario, it is impossible to keep fines constant for all firms for different fine structures if ϕ and γ are common for all firms. If f(e − s) = ϕ(e − s) + \frac{γ}{2}(e − s)^{2}, changing ϕ and γ so as to keep f constant requires \frac{e−s}{2} = −\frac{dγ}{dϕ}. But with n firms, it is impossible to move ϕ and γ such that \frac{e_{i}−s_{i}}{2} = −\frac{dγ}{dϕ} for all i. Keeping f contant for all i requires a firm-specific fine parameters. We assume that this is the case and we show that the optimal design of the program calls for a uniform fine structure.

If the fine structure is firm-specific, we have f_{i}(ē_{i} − s_{i}) = ϕ_{i}(ē_{i} − s_{i}) + \frac{γ_{i}}{2} (ē_{i} − s_{i})^{2}, and f′_{i}(ē_{i} − s_{i}) = ϕ_{i} + \frac{γ_{i}}{2} (ē_{i} − s_{i}) for each i. Then we ask how to choose ϕ_{i} and γ_{i} in order to minimize the costs of a program that produces E when it is optimal to induce expected violations. Following Arguedas (2008), we ask ourselves whether we can decrease the costs of a program that induces a certain expected level of violation for each firm changing the fine structure (changing the values of ϕ_{i} and γ_{i}) while choosing π_{i} optimally. In order to answer this question, we evaluate the Lagrangean of the regulator’s problem

at π_{i} = π_{i}^{∗} = \frac{E[−c′_{i}(ē_{i})]}{f′(ē_{i}−s_{i})} when ē_{i} > s_{i} and ∑_{i} ē_{i} = E and change ϕ_{i} and γ_{i} such

that df_{i} = 0, that is −\frac{dϕ_{i}}{dγ_{i}} = \frac{ē_{i}−s_{i}}{2} .

L = E [∑_{i=1}^{n} c_{i}(ē_{i})] + μ ∑_{i=1}^{n} π_{i}^{∗} + β ∑_{i=1}^{n} π_{i}^{∗}f_{i}(ē_{i} − s_{i})
dL = \frac{∂L}{∂ϕ_{i}} dϕ_{i} + \frac{∂L}{∂γ_{i}} dγ_{i}
dL = [μ\frac{∂π_{i}^{∗}}{∂ϕ_{i}} + β [\frac{∂π_{i}^{∗}}{∂ϕ_{i}} f_{i}(ē_{i} − s_{i}) + π_{i}^{∗}(ē_{i} − s_{i})]] dϕ_{i}

+ [μ\frac{∂π_{i}^{∗}}{∂γ_{i}} + β [\frac{∂π_{i}^{∗}}{∂γ_{i}} f_{i}(ē_{i} − s_{i}) + π_{i}^{∗} \frac{(ē_{i} − s_{i})^{2}}{2} ]] dγ_{i}

\frac{dL}{dϕ_{i}} = [μ\frac{∂π_{i}^{∗}}{∂ϕ_{i}} + β [\frac{∂π_{i}^{∗}}{∂ϕ_{i}} f_{i}(ē_{i} − s_{i}) + π_{i}^{∗}(ē_{i} − s_{i})]]

+ [μ\frac{∂π_{i}^{∗}}{∂γ_{i}} + β [\frac{∂π_{i}^{∗}}{∂γ_{i}} f_{i}(ē_{i} − s_{i}) + π_{i}^{∗} \frac{(ē_{i} − s_{i})^{2}}{2} ]] \frac{dγ_{i}}{dϕ_{i}}

\frac{dL}{dϕ_{i}} = [μ\frac{∂π_{i}^{∗}}{∂ϕ_{i}} + β [\frac{∂π_{i}^{∗}}{∂ϕ_{i}} f_{i}(ē_{i} − s_{i}) + π_{i}^{∗}(ē_{i} − s_{i})]]

− [μ\frac{∂π_{i}^{∗}}{∂γ_{i}} + β [\frac{∂π_{i}^{∗}}{∂γ_{i}} f_{i}(ē_{i} − s_{i}) + π_{i}^{∗} \frac{(ē_{i} − s_{i})^{2}}{2} ]] \frac{2}{ē_{i} − s_{i}}

\frac{dL}{dϕ_{i}} = μ\frac{∂π_{i}^{∗}}{∂ϕ_{i}} + β [\frac{∂π_{i}^{∗}}{∂ϕ_{i}} (ϕ_{i}(ē_{i} − s_{i}) + \frac{γ_{i}}{2} (ē_{i} − s_{i})^{2})]

−\frac{2μ}{ē_{i} − s_{i}} \frac{∂π_{i}^{∗}}{∂γ_{i}} − β [\frac{∂π_{i}^{∗}}{∂γ_{i}} (2ϕ_{i} + γ_{i}(ē_{i} − s_{i}))]

We know that \frac{∂π_{i}^{∗}}{∂ϕ_{i}} = \frac{−E[−c′_{i}(ē_{i})]}{[ϕ_{i}+γ_{i}(ē_{i}−s_{i})]^{2}} and \frac{∂π_{i}^{∗}}{∂γ_{i}} = \frac{−E[−c′_{i}(ē_{i})]}{[ϕ_{i}+γ_{i}(ē_{i}−s_{i})]^{2}} × (ē_{i} − s_{i}).

Therefore,

\frac{dL}{dϕ_{i}} = −\frac{E [−c′_{i}(ē_{i})]}{[ϕ_{i} + γ_{i}(ē_{i} − s_{i})]^{2}} [μ + β (ϕ_{i}(ē_{i} − s_{i}) + \frac{γ_{i}}{2} (ē_{i} − s_{i})^{2})]  (9)

+\frac{E [−c′_{i}(ē_{i})]}{[ϕ_{i} + γ_{i}(ē_{i} − s_{i})]^{2}} × (ē_{i} − s_{i}) [\frac{2μ}{ē_{i} − s_{i}} + β (2ϕ_{i} + γ_{i}(ē_{i} − s_{i}))]

\frac{dL}{dϕ_{i}} = −\frac{E [−c′_{i}(ē_{i})]}{[ϕ_{i} + γ_{i}(ē_{i} − s_{i})]^{2}} [μ + β (ϕ_{i}(ē_{i} − s_{i}) + \frac{γ_{i}}{2} (ē_{i} − s_{i})^{2})]

+\frac{E [−c′_{i}(ē_{i})]}{[ϕ_{i} + γ_{i}(ē_{i} − s_{i})]^{2}} [2μ + β (2ϕ_{i}(ē_{i} − s_{i}) + γ_{i}(ē_{i} − s_{i})^{2})]

\frac{dL}{dϕ_{i}} = \frac{E [−c′_{i}(ē_{i})]}{[ϕ_{i} + γ_{i}(ē_{i} − s_{i})]^{2}}

× [2μ − μ − βϕ_{i}(ē_{i} − s_{i}) − β \frac{γ_{i}}{2} (ē_{i} − s_{i})^{2} + 2βϕ_{i}(ē_{i} − s_{i}) + βγ_{i}(ē_{i} − s_{i})^{2}] (10)

\frac{dL}{dϕ_{i}} = \frac{E [−c′_{i}(ē_{i})]}{[ϕ_{i} + γ_{i}(ē_{i} − s_{i})]^{2}} [μ + β (ϕ_{i}(ē_{i} − s_{i}) + \frac{γ_{i}}{2} (ē_{i} − s_{i})^{2})] > 0

This means that the regulator can decrease the costs of a program that induces a violation (ē_{i} − s_{i}) for each firm by decreasing ϕ_{i} and increasing γ_{i} (if ϕ_{i} decreases γ_{i}hs to increase so as to keep the equilibrium fine constant). The intuition behind this result follows form two observations. First, is that by increasing the marginal equilibrium penalty the regulator decreases the equilibrium inspection probability π_{i}^{∗} needed to induce a given expected level of violation (ē_{i} − s_{i}). This decreases monitoring costs while keeps the rest of the costs constant. Second, the marginal equilibrium penalty increases more if the regulator increases γ_{i} than if it increases ϕ_{i}. The first term in the righthand side of 9 is the marginal effect of a change in ϕ_{i} on the expected costs of the program. The second term is the marginal effect of a change in γ_{i}. These two effects act in opposed directions because keeping the fine constant requires increasing one parameter and decreasing the other. Decreasing ϕ_{i}

increases the expected monitoring costs by \frac{−E[−c′_{i}(ē_{i},θ_{i})]}{[f(ē_{i}−s_{i})]^{2}} × μ and increases the

expected sanctioning costs by \frac{E[−c′_{i}(ē_{i},θ_{i})]}{[f(ē_{i}−s_{i})]^{2}} [βf(ē_{i} − s_{i})] . Increasing γ_{i} by the quantity that keeps f(ē_{i} − s_{i}) constant decreases both costs by more than this. Therefore the final effect is to decrease the total expected costs of the program (expected abatement costs do not change).

Now, decreasing ϕ_{i} has a limit and this limit is ϕ_{i} = 0. Under a negative value of ϕ_{i} it will always exist a (sufficiently small) level of violation that makes the fine negative. But a negative fine violates our assumption that f ≥ 0 for all levels of violations. On the other hand, there is no theoretical maximum value for γ_{i}. In theory this value is infinite, and therefore it is not firm-specific. Therefore, the expected cost minimizing design of a program based on standards calls for a uniform penalty structure for all firms: f(ē_{i} − s_{i}) = \frac{γ}{2}(ē_{i} − s_{i})^{2} for all i.The regulator always decreases monitoring costs by increasing γ, for the same level of violation. This is true for all firms and therefore it must set γ as high as possible for all firms. Therefore, if the optimal policy induces expected non-compliance, the best shape of the penalty function is one in which the linear component ϕ = 0 and the progressive component is set "as high as possible" for all firms.

Having answered what is the cost-minimizing structure of the fine when it is optimum to induce compliance and when it is not, we now prove Proposition 3. Following Arguedas (2008), assume that it is optimum to induce expected non-compliance, and call the optimal policy P^{n} = (s_{1}^{n}, s_{2}^{n}, ..., s_{n}^{n}, π_{1}^{n}, π_{2}^{n}, ...π_{n}^{n}, f^{n}), with f^{n} = \frac{γ}{2}(e_{i} − s_{i})^{2} for all i (with γ as high as possible following the re-

sults above), π_{i}^{n} = \frac{E[−c′_{i}(ē_{i}^{n})]}{γ(ē_{i}^{n}−s_{i}^{n})} and ∑_{i=1}^{n} ē_{i}^{n} = E. Now consider an alternative

policy P^{c} = (s_{1}^{c}, s_{2}^{c}, ..., s_{n}^{c}, π_{1}^{c}, π_{2}^{c}, ...π_{n}^{c}, f^{c}) such that s_{i}^{c} = ē_{i}^{n} and π_{i}^{c} = π_{i}^{n} for all i, and f^{c} = ϕ(e_{i} − s_{i}) for all i with ϕ = γ × max_{i} [ē_{i}^{n} − s_{i}^{n}] . By construction, this policy induces expected compliance because π_{i}^{c}f^{c′} = π_{i}^{c}ϕ = π_{i}^{c}γ × max_{i} [ē_{i}^{n} − s_{i}^{n}] ≥ E [−c′_{i}(ē_{i}^{n})] = E [−c′_{i}(s_{i}^{c})] for all i. Moreover, P^{c} is cheaper than P^{n} in expected terms because expected abatement costs are the same under both programs (s_{i}^{c} = ē_{i}^{n} for all i), expected monitoring costs are the same under both programs (π_{i}^{c} = π_{i}^{n} for all i), but under policy P^{c} there are no expected sanctioning costs because there are no expected violations. QED.

References

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[3] Malik, Arun. (1992). “Enforcement Cost and the Choice of Policy Instruments for Controlling Pollution”, Economic Inquiry XXX: 714-721.