Marcelo Caffera
Universidad de Montevideo
Carlos A. Chávez
Universidad de Concepción
The Cost-Effective Choice of Policy Instruments to Cap Aggregate Emissions with Costly Enforcement

We study the cost-effectiveness of inducing expected perfect compliance considering the abatement, monitoring and sanctioning costs in a program that caps aggregate emissions of a given pollutant from a set of heterogeneous firms based on emissions standards. We find that the total cost-effective design of such a program is one in which standards are firm-specific and perfectly enforced. We then compare the expected costs of such an optimally designed program with that of an optimally designed program based on a perfectly competitive emission permits market, which also calls for perfect enforcement according to Stranlund (2007). We find that the latter never minimizes the total expected costs of attaining a certain level of aggregate emissions. This result holds also in the case when it is cost effective to induce violations.

Introduction
One of the most important features behind any emissions control policy, national or international, is the total cost of the implied emissions reduction. 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. This policy recomendation has had its impact: the European Union adopted an emissions trading scheme, the European Union Emissions Trading Scheme (EU-ETS), as an important instrument to limit its emissions of greenhouse gases. The Obama administration is also pushing a similar alternative in the U.S. Congress (The Waxman-Markey´s American Clean Energy and Security Act). Until the appearance of the EU - ETS, the US was home of the major policy experiment with tradable permits; the SO   allowance market to control acid rain.
The apparent success of this policy recommendation may be seen as surprising, though, because abatement costs are not the only social costs of caping emissions. There are other relevant costs, such as the cost of monitoring compliance and sanctioning violations. Interestingly, the environmental economics literature has not yet given a definite answer on the relative cost-effectiveness of a tradable emission permits system with respect to one based on emission standards when enforcement costs are brought into the picture. Malik (1992) compares the costs of reaching a given level of aggregate emissions by means of a perfectly enforced program based on uniform 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. 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, because he focuses on perfectly enforced programs, 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 considering only abatement costs and fines. These authors do not consider monitoring or sanctioning costs. More recently, Chávez, et al. (2009) extend Malik's contribution for a regulator that, unlike Malik´s, cannot perfectly observe the abatement costs of the firms, but instead 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. The authors prove that emissions standards are more costly than tradable permits with this monitoring strategy.
One important aspect that all of the above 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). Stranlund (2007) seems to be the first to have adressed the issue of whether the regulator can use non-compliance as a way to reduce the costs of a program that caps 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 structure of the penalty function. 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, a regulator with complete information and one firm. She obtains an identical conclusion. We think, nevertheless, that the analysis with one firm fails to illustrate a central aspect of the design of cost-effective regulation in the real world; namely, how does the regulator have to allocate emissions responsibilities and monitoring and sanctioning efforts among different firms in order to minimize the total cost of the pollution control program.
In this paper we first derive the condition under which it is cost-effectivene to induce compliance in a system of emissions standards, with more than one firm, firm-specific monitoring and sanctioning costs and incomplete information. Considering the total costs of the program (abatement, monitoring, and sanctioning), we then characterize the total expected cost effective design of an emission standard system and compare it to the costs an optimally designed transferable emissions permit system, as in Stranlund (2007), under different assumptions of the penalty structure.
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, we find that an optimally designed system of tradable permits never minimizes the total expected costs of attaining a certain level of aggregate emissions. This is basically because the distribution of emissions generated by the market for permits and its corresponding cost-effective monitoring differ from the distribution of emissions and monitoring efforts that minimizes the total costs of the program. This result holds both in the case when it is cost-effective to induce compliance and when it is cost-effective to induce violations.
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. We use this model to derive the condition under which it is cost-effective for a regulator to induce perfect expected compliance in a system of emissions standards that caps the aggregate emissions of    firms. In Section 3 we characterize the cost-effective design of such a program both when it is cost effective to induce expected perfect compliance, and when the opposite is true. We then let the regulator to choose the structure of penalties and we characterize the expected-cost-minimizing design of a program based on emissions standards in this case. In Section 4 we compare the costs of a program based on standards with that of a program based on tradable permits. Finally, in section 5 we present our conclusions.
The Cost-Effectiveness of Inducing Perfect Compliance
In this section we answer the following question: when it is cost-effective for a regulator to induce perfect compliance? In order to do it, we first present the standard model of compliance behavior of a risk - neutral polluter firm under an emission standard (See Malik, 1992 or Harford, 1978). From this model we derive the emissions level with which the firm responds to the regulation. We then present the problem that a total cost minimizing regulator solves, taking into account the firm`s best reponses, when designing a program that caps aggregate emissions setting standards. From this model we derive the condition under which it is cost-effective for the regulator to induce perfect compliance. The model we present here extends Arguedas' (2008) by including more than one firm and Stranlund's (2007) by differentiating monitoring and sanctioning costs among firms.
A firm compliance behavior under an emission standard
Assume that reducing emissions of a given pollutant    is costly for a firm. The (minimum) abatement cost function for this firm, which we will call firm    is   , where    is the level of emissions of firm   . The abatement cost function is assumed to be strictly decreasing and convex in the firm's emissions    [   and   ].
The firm faces an emission standard (a maximum allowable level of emissions)   . An emissions violation    occurs when the firm's emissions exceed the emissions standard:   . The firm is compliant otherwise. The firm faces a random probability of being audited   . An audit provides the regulator with perfect information about the firm's compliance status. If the firm is audited and found in violation, a penalty    is imposed. For the moment, we just assume that    for all    and    por all   
Under an emissions standard, a firm    chooses the level of emissions to minimize total expected compliance cost, which consists of its abatement costs plus the expected penalty. Thus, firm    problem is to choose the level of emissions to solve

The Lagrange equation for this problem is given by   , with    the Lagrange multiplier. The set of necessary Kuhn-Tucker conditions for a positive level of the standard and emissions is: 

From the above Kuhn-Tucker conditions it can be seen that 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. That is,    if    Otherwise, the firm is going to choose a level of emissions    where    is the solution to    Note that    the marginal abatement costs evaluated at 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 completely observable for a regulator.
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    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. Towards this objective he selects the probability of inspection    and the emission standard    for every firm    There are    firms that emit this pollutant. The firms differ in their abatement costs, but these are not completely 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: 

where    denotes the regulator's subjective expected value of the program costs. These are comprised of the expected aggregate abatement costs, the total monitoring costs and the expected total sanctioning costs. The expected aggregate abatement costs are    Assuming the cost of inspecting plant    is given by    the aggregate monitoring or auditing costs are    Assuming that sanctioning plant    has a cost of    per dollar of fine, the expected aggregate sanctioning costs are    For the moment, we assume that the structure of the penalty function    is given for the environmental regulator   The regulator knows that the firm    will react to a standard    and a monitoring probability    according to its reaction function    Therefore, he incorporates this constraint in the problem. Because he cannot observe the abatement cost functions of the firms, the regulator does not know the reaction function of each particular firm. Nevertheless, he uses his belief about what the expected abatement cost function for firm    is and the firm's problem to calculate    the level of emissions that he believes the firm will produce as a response to a certain level of the emission standard    and inspection probability    The second constraint summarizes the environmental objective of the program, namely, that the expected aggregate level of emissions cannot exceed a predetermined target    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
  
with    and    being the    multipliers. The    necessary Kuhn-Tucker for positive levels of the standard and the auditing probability are:

We assume that these conditions are sufficient to characterize the optimal solution of the problem. From the above conditions we derive the following Proposition:
Proposition 1 When the penalty structure is given, the cost-effective design of a pollution control program that caps aggregate emissions using emissions standards, calls the regulator to induce compliance with the standards for all    if and only if   for all    If this condition is not met and the regulator wants to achieve the cap cost-effectively, it should induce violations of the emission standards for all those plants for which   .
Proof: See Appendix.
Our Proposition 1 is an extension of Arguedas' (2008) Proposition 1 to the case of    firms and heterogeneous monitoring and sanctioning costs (      and   , for at least some       Our Proposition 1 is also analogous to the condition derived by Stranlund (2007) for the case of transferable permits, but with homogeneous monitoring and sanctioning costs. Therefore, a first conclusion is that the condition under which it is cost-effective for a regulator to induce compliance is not instrument-dependent. Nevertheless, one important thing that our Proposition 1 is telling is that when monitoring and sanctioning costs differ among firms, it could be cost-effective for the regulator to induce violations for some firms and compliance for the rest. This result cannot be observed when one assumes that monitoring and sanctioning costs are the same for all firms. In this case, the condition under which it is cost effective for a regulator to induce compliance does not depend on any individual characteristic of the firms, only on the penalty structure and the homogeneous costs of monitoring and sanctioning. Nevertheless, there are several reasons why auditing different firms may imply different costs for the regulator. One of these reasons may be the distance between the firm and the enforcing agency. Another may be the number of discharge points per plant. At the same time, 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.
The cost minimizing design of a program based on emission standards
We now turn to characterize the expected cost minimizing design of a program that controls pollution with emission standards. We do this for the cases in which the penalty structure is out of the control of the environmental regulator, and when it is not.
A given penalty function
When the penalty structure is exogenously given to the regulator, condition (con1') dictates him whether it is cost-effective to induce perfect compliance or not. In the first case, it is easy to show that the optimal policy    that induces expected compliance is characterized by: 

(See Proof 2 in the Appendix). When it is cost-effective to induce expected compliance, the regulator has to set emission standards such that the sum of marginal expected abatement and monitoring costs are equal between firms, a result obtained by Chávez, et. al (2009) and Malik (1992) in the context of complete information on abtement costs and a given objective of perfect compliance. Note that allocating emissions responsibilities in this way does not imply perfect compliance with certainty. In the presence of incomplete information, the regulator could attain perfect compliance with certainty setting    with    being the largest possible value of the marginal abatement cost of complying with the standard among all firms   It is easy to see that this monitoring probability 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 in this fashion (as in Chávez, et. al (2009)) does not minimize the expected costs of the program.
When (con1') does not hold, it is not cost-effective for the regulator to induce compliance for all firms. In other words, a regulator interested in implementing a program that caps aggregate emissions 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 incomplete information and given penalties, the characterization of the expected cost-effective program to control emissions with standards when it is cost-effective to induce non-compliance for all firms is given by Proposition 2.
Proposition 2 If the optimal policy    induces non compliance for all firms, it is characterized by

for all   ,   . 
See Proof in the Appendix.
Proposition 2 is telling that when it is cost-effective to induce non-compliance for every firm, the regulator has to choose    and    such that: (1) the sum of the expected marginal abatement plus sanctioning costs of moving    is the same accross firms, and (2) the sum of the expected marginal abatement, monitoring and sanctioning costs of changing    is the same accross firms. Condition (Prop21) is quite intuitive. The firm reacts to a change in    by adjusting    by the amount    in expected terms. This change in    has an effect on the abatement costs of the firm    but also an effect on the sanctioning costs of the regulator. We know by (de/ds) that    Thus, a change in    causes the level of violation to change, and therefore the level of the expected fines that the regulator is going to charge firm    with. This in turn means a change in the expected sanctioning costs for the regulator. The regulator sets    equating these two marginal costs among firms, and it does a similar thing when adjusting    (condition Prop22). A marginal change in the inspection probability affects all costs of the program: it affects firm's    abatement costs via a change in the level of emissions, it affects the auditing costs directly, and also affects the sanctioning costs because it changes the number of violations being discovered and because it changes the amount of violation by firm    The regulator sets    such that the sum of these three marginal costs, measured in units of expected emissions, are the same among all firms.
Furthermore, from (ProofProp2), we can obtain the following

for all    This condition says that in the cost minimizing solution the regulator equates the marginal costs of moving the standard with that of moving the monitoring probability for every firm. More specifically, the sum of the marginal monitoring and sanctioning costs of moving    is equal to the marginal sanctioning costs of moving    for every firm   
We have assumed that the unit cost of an inspection (   and the per dollar cost of a fine (   can differ between firms. In the particular case when    and    are the same for all firms, the conditions characterizing an expected cost-minimizing design of a regulatory program that controls emissions with standards are essentially the same, except that in this case the condition    either holds or not for every firm. Thus, the regulator must induce compliance or non-compliance for every firm in the program. On the other hand, if the monitoring and sanctioning costs differ between firms it could be the case that condition    holds for a group of firms and does not hold for another group of firms. In this case, the conditions caracterizing the expected cost minimizing design of the program would be a combination of conditions    (Prop21) and (Prop22).
We can conclude from Proposition 2 that the cost-effective level of emission standards are firm-specific whenever abatement and/or enforcement costs differ among firms. Assuming    and    to be the same for all firms, it would be the heterogeneity in marginal abatement costs    that would call for firm-specific standards. Similarly, if marginal abatement costs were the same for all firms, differences in monitoring costs and sanctioning costs among firms (   could also call for differences in the cost-minimizing standards.
The regulator can choose the structure of the penalty function
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 structure of the penalty function, and therefore the optimality of inducing expected compliance or not. We consider only two fine structures: linear and increasing in the level of the violation. Consequently, the regulator has basically to compare four possible alternatives and choose the one that minimizes the expected cost of reaching the cap    on emissions. The four alternatives are (1) to induce expected compliance with linear penalties, (2) to induce expected compliance 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. To induce expected compliance with linear or increasing penaties has the same minimum expected costs because under compliance 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 (with linear or increasing marginal penalty) 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    induces compliance and it is characterized by (1)    for all       (2)    and (3)    for all    with   set as high as possible and       .
See the proof of Proposition 3 in the Appendix.
The expected cost minimizing policy when a regulator wants to cap aggregate emissions of a given pollutant to a certain level    through 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, as long as    for all    (otherwise the regulator mistakenly increases the cost of the program by making cost-effective not to induce perfect compliance).
Proposition 3 has important implications for the real-world policy design. The first and most obvious one is that there is no justification in terms of the costs of the program to design it to allow violations if the fine structure is under the control of the environmetal policy administrator. It is not difficult though 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. If this is the case, one agency or office may set first the environmental objective (the aggregate level of emissions    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-optimal, 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    "as high as possible". In the real world    will be given be bounded upward by things such as the possibility that firms may have insufficient assets to cover the fines (Segerson and Tietenberg 1991) or the unwillingness of judges or juries to impose very high penalties (Becker 1968). Note that if this upper bound of    is combined with a binding monitoring budget, the environmental regulator may not be capable of assuring expected compliance for all    and by this way minimize the total expected costs of the emissions control program.
Comparing costs of emission standards and tradable permits
Optimally designed programs
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 3) 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 also one in which the program is perfectly enforced. Stranlund (2007) concludes that this has to be done using a constant marginal penalty. Instead, we argue that, as in the case of emission standards, the structure of the penalty does not play any role in equilibrium. According to Stranlund (2007), a cost minimizer regulator who wants to achieve expected perfect compliance in a system of tradable permits must set the monitoring probability for firm    (   such that    for all   , where    is the expected full-compliance equilibrium price of the permits market and    It is easy to see from this condition that the structure of the penalty function (whether it is increasing at a constant or an increasing rate) plays no role in the (minimimum) costs of the program. As in the case of emission standards, what affects these costs is    The value of    could take any positive value as long as    (it is cost-effective to induce perfect compliance).
Notwithstanding, 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 3 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 the two instruments, what instrument should a regulator use if it wants to minimize 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 not minimize the total costs of doing so by implementing a system of tradable permits. On the contrary, expected total costs of such a pollution control program will be minimized by Implementing firm-specific emissions standards and perfectly enforcing this program according to Proposition 3.
Proof: See the Appendix.
Proposition (4) states that an optimally designed program based on firm-specific emissions standards, not one based on tradable permits, minimizes the expected total costs of a pollution control program that caps aggregate emissions to a certain level. This result may be surprising because it seems to contradict what we environmental economists have been advocating for over the last forty years. Nevertheless, the reason why we have been advocating tradable permits is because they minimize aggregate abatement costs. Monitoring and enforcement costs were not part of the analysis that led to this policy recomendation. Also, we have been advocating tradable permits as cost-effective policy instrument when compared to uniform (i.e: not firm-specific) emission standards. We know that in a world of perfect information there is no relative advantage of one instrument over the other in terms of abatement cost-effectiveness (Weitzman, 1974). Proposition (4) tells that when enforcement costs are brought into the picture this conclusion changes: firm specific standards are to be implemented because the functioning of a tradable permits market cannot by itself exploit the differences in abatement and monitoring costs. This conclusion can be extended to the setting of incomplete information if we talk about expected costs, not actual costs. Of course, when the regulator cannot observe firms' marginal abatement costs, it may commit relevant mistakes in the estimation of the abatement costs functions. If this is the case, the realized social costs of setting and enforcing a global cap on emissions via firm-specific standards could end up being more expensive than doing it via an emissions trading scheme. This is the reason we are cautious about deriving policy recomendations from Proposition (4). More research is needed in this area before this can be done. (The same caveat is valid for Proposition (5) below). What Proposition (4) tells is that it is not in the name of cost-effectiveness 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.
Comparing costs when it is cost - effective to induce non-compliance
As discussed above, it may be a common situation in the real world that the fine structure is given to the environmental authority. Assume that this is the case and that    In this setting, 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       for all    or not). Assume that      for all    Then it is cost-effective to design a program that induce a given expected level of non-compliance for all   . In this case, how do the cost of a program based on emission standards compare with one based on tradable permits?
In order to answer this question, we first characterize the cost-effective design of a pollution capping program based on tradable permits when it is cost-effective to induce a given expected level of aggregate non-compliance. Then we see if this optimally design program minimizes the total expected costs of reaching the cap   
Characterization of the cost-effective design of a program based on tradable permits when is is cost-effective to induce non-compliance
When it is optimum not to induce perfect compliance for all       for all    equations (KT1Proof4) and (KT2Proof4) can be re-written, assuming    for all    as:

These equations characterize the optimal design of a tradable permits program when it is cost - effective to induce all firms to violate their permit holdings    In a similar fashion to the emission standards program, in the optimally designed tradable permits program the regulator sets    and    for all    such that: (a) the sum the marginal abatement, monitoring and sanctioning costs of changing    are equal across firms (equation KT1Proof5) and (b) the sum of marginal abetement and sanctioning costs of changing    are equal across firms (equation KT2Proof5). From equations (KT1Proof5) and (KT2Proof5) we can obtain 

Therefore, in the optimal design of a tradable permits program when it is cost - effective to induce all firms to violate their permit holdings the regulator also has to set the sum of the marginal monitoring and sanctioning costs of changing    equal to the marginal sanctioning costs of moving    for every firm   
Comparison of Costs
Having characterized the optimal emissions trading program, we now show that this program does not minimize the total expected costs of capping aggregate emissions to    In order to do this, we recall from the proof of Proposition (4) that every firm    that violates their permits holdings in a competitive emission permits market chooses its level of emissions such that    and the quantity of permits to demand such that    Using both expressions, we can write (KT2Proof5) as

or 

It is clear from the above equation that if sanctioning costs differ among firms    for some      , a competitive permits market (one that generates a unique equilibrium price    for all firms) will not minimize the total expected costs of capping aggregate emissions to a level    while allowing some degree of noncompliance. We express this result more formally in the Proposition below.
Proposition 5 If a regulator wants set a cap on the aggregate level of emissions of a pollutant and it is cost-effective to induce all firms to violate the regulation (      for all   , it will minimize the total expected costs of such a regulatory program by implementing a system of firm-specific emissions standards as characterized by Proposition 2, not a system of tradable permits.
Proposition (5) is robust to the case when    and    do not differ between firms. If    and    for all    and we assume that the permits market is perfectly competitive, so that   then equation (KT3Proof5) can be written as

But we know from Stranlund (2007) that if    and    do not differ between firms, the regulator must induce a uniform violation across firms and monitor all firms with a uniform probability. Thus, the above equation can be written as

Using    and   

This condition will not be met except in the special case where    and    Therefore, in the general case where    and    do not differ between firms it is also true that a system of tradable emission permits does not minimize the expected costs of capping aggregate emissions when it is cost-effective to induce violations.
Conclusion
In this paper we first derive the condition under which it is cost effective for a regulator to induce perfect compliance in an emissions control program. This condition depends on the cost of monitoring and sanctioning a firm, as well as on the structure of the penalty for violations. Therefore, it is not instrument-dependent. If the condition is met, the regulator has to induce perfect compliance independently of whether it is implementing emission standards or transferable permits. Nevertheless, because we assume that the regulator's monitoring and sanctioning costs are firm-specific, the condition itself is firm-specific. In other words, it is possible that cost-effectiveness calls the regulator to induce some firms to comply with the legislation while at the same time let others violate the legislation. This cannot happen when one assumes that the regulator's monitoring and sanctioning costs are the same for all firms. In this case, the regulator has either to induce compliance on all firms or to induce violations on all firms.
Second, we characterize the total-cost minimizing design of a program that caps aggregate emissions of a given pollutant from a set of heterogeneous firms based on emissions standards when it is cost effective to induce perfect compliance and when it is not. We then allow the regulator to choose the optimality of inducing compliance or not assuming that it can choose the structure of the penalty function. Doing this we find that the total cost-effective design of such a program is one in which standards are firm-specific and perfectly enforced.
Third, we compare the expected costs of such an optimally designed program with that of an optimally designed program based on a perfectly competitive emission permits market, which also calls for perfect enforcement according to Stranlund (2007). This comparison allows us to conclude that the latter never minimizes the total expected costs of attaining a certain level of aggregate emissions. Moreover, this result holds also in the case when it is cost effective to induce violations. The reason behind these results is that a tradable permits market cannot by itself exploit the differences in abatement and monitoring costs, only the former. Consequently, the allocation of emission responsibilities that results from a tradable permits market and its corresponding cost-effective monitoring differ from the ones that minimize the total expected costs; namely, that of the optimally designed emission standards program.
Given that the distribution of emissions and monitoring efforts in the cost-effective design requires information that is private, and that the distribution of emissions and monitoring efforts in a cost-effective design of a tradable permits system does not reproduce the former, we conclude that it is not in the name of cost-effectiveness that we economists are to argue in favor of tradable emission permits. Nevertheless, we are cautious in deriving policiy recomendations. The incomplete 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 than those of a system of tradable permits. Clearly, more research is needed concerning this issue.
Nevertheless, our results produce a clear policy recommendation for the design of environmental policy in developing countries, as our own. The environmental policy in these countries has been frequently described as poorly enforced (see, for example, Russell and Powell 1996; Eskeland and Jimenez 1992; O'Connor 1998; Seroa da Motta et al 1999). Explanations of this situation frequently mention the budget constraints that regulators suffer in these countries. Our conclusion 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 the overall cost-effectiveness of the program. The regulator could attain the same level of aggregate emissions with less budget relaxing the non-enforced cap and perfectly enforcing the laxer regulation.
Appendix
Proof of Proposition 1 If    from (FOC 4) we know that    Because we have also that    we can re-write the first order conditions of the regulator's problem as:

Re-arranging the expressions and dividing:

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

From where,

  and

Because a cost-minimizing regulator that wants to achieve    will set    such that    in order not to waste monitoring resources, we can write

or

From where, using    

We have proved that when a cost - minimizing 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    to comply with the emission standard. Why? The right-hand side of (cond 1) is the marginal increase in the expected sanctioning costs when the regulator marginally decreases the standard. The left hand side is the marginal decrease in monitoring costs that the regulator can attain when he decreases the monitoring probability accordingly so as to leave the level of emissions unchanged. Therefore, what the condition is saying is the following: if the firm is complying with the standard and moving the standard and the monitoring probability so as to make the firm marginally violate the standard increases the sanctioning costs more than it decreases the monitoring costs, it is not cost-effective to do so. The regulator should leave things as they are: set    and    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 (cond 1) by    we obtain

, Q.E.D.
Proof 2 When   , expected violations are zero and therefore there are only two types of expected costs; monitoring and abatement. Moreover, if the regulator wants to achieve    it has to set    such that    or    Furthermore, if the regulator can induce    with    it would not be cost-effective to select    Therefore,    In this case, the Lagrange of the regulator's problem can be re-written as

Therefore, the       necessary conditions defining the    interior solutions for the standards are

It follows directly from (KT1givenpenaltyfullcompl) that when it is cost-effective to induce expected compliance for all   , the optimal policy is to set    such that    for all       Q.E.D.
Proof of Proposition 2 When it is cost-effective to induce non-compliance for all firms, the expected cost-minimizing standards must be set such that    From Kuhn-Tucker condition (FOC 4), this implies that    It is easy to see that the relevant Kuhn Tucker conditions in this case are 

both for    Dividing (ProofProp21nueva) and (ProofProp22) by    and    respectively, we obtain:

for all   Q.E.D.
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 structures: linear and increasing. The general fine structure can be writen as    where    is a positive constant and   .
If the optimal policy is going to induce compliance for all   , condition (con1') requires that

We also know from Section 3 that in this case the characterization of the cost-effective design of a program based on standards calls for the following monitoring probability: 

From here we can conclude:
(1) The regulator must choose the linear component    of the fine structure as high as possible because this will decrease the optimum level of the inspection probability   and by this way the monitoring costs. Conceptually, this calls for    because this will make the monitoring costs equal to zero, but in the real world there may be limits to the upper value of   , of course.  
(2) If we call    the highest possible value of    any value of      , will still make cost-efective to induce compliance for every firm and will not hav an effect on the minimum expected costs of the program, namely   
Our conclusions in this respect differ from Arguedas' (2008). She concludes: "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    is as high as possible and    is as low as possible, since only the first component affects the probability." On the contrary, we conclude that       plays no role (it does not affect the minimum costs of the program). The penalty function can be linear    or increasing    as long as    This is because there are no sanctioning costs in equilibrium and all that the penalty function affects are the monitoring costs, through    Therefore, our conclusion: If the optimal policy induces compliance for all   , the cost-minimizing shape of the fine must be such that the linear component    is set as high as possible. The value of the progressive component    is irrelevant in equilibrium as long as    where    is the chosen level of   .
If the regulator is going to induce non-compliance, how does it have to choose    and    in order to minimize the costs of a program that produces    In other words, can the regulator decrease the expected costs of the program by altering the fine structure (the value of    and    once the optimal standards, inspections probabilities and emissions have been chosen? Notice that to choose the appropriate fine structure the regulator should optimize in the values of    and    keeping violations, and fines, constant. If    changing    and    so as to keep    constant requires    But with    firms, it is impossible to move    and    such that    for all    Keeping    contant for all    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    and    for each    Now we ask how to choose    and    in order to minimize the costs of a program that produces    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    and   ) while choosing    optimally. In order to answer this question, we evaluate the Lagrangean of the regulator's problem at    when    and    and change    and    such that    that is   

Dividing both sides by   and substituting    for    we obtain 

We know that    and    Substituting, 

And after some operations we obtain 

This means that the regulator can decrease the costs of a program that induces a violation    for each firm by decreasing    (and increasing    accordingly so as to keep the equilibrium fine constant). The intuition behind this result follows form two observations. First, by increasing the marginal equilibrium penalty the regulator decreases the equilibrium inspection probability    needed to induce a given expected level of violation    This decreases monitoring costs while keeps the rest of the costs constant. Second, the marginal equilibrium penalty increases more if the regulator increases    than if it increases    The first term in the right-hand side of (dL/dfi) is the marginal effect of a change in    on the expected costs of the program. The second term is the marginal effect of a change in    These two effects act in opposed directions because keeping the fine constant requires increasing one parameter and decreasing the other. Decreasing    increases the expected monitoring costs by    and by this way increases also the expected sanctioning costs by    Ii is easy to see from (dL/dfi) that increasing    by the quantity that keeps    constant decreases both costs by more than this (The second term is larger that the first term). Therefore the final effect is to decrease the total expected costs of the program (expected abatement costs do not change).
Now, decreasing    has a limit and this limit is    Under a negative value of    it will always exist a (sufficiently small) level of violation that makes the fine negative. But a negative fine violates our assumption that    for all levels of violations. On the other hand, there is no theoretical maximum value for    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:    for all    The regulator always decreases monitoring costs by increasing    for the same level of violation. This is true for all firms and therefore it must set    as high as possible for all firms. Because we are in the case where the regulator induces non-compliance, condition    for all    must hold. And because we have just said that the cost minimizing shape of the penalty function requires    for all    the above condition only requires    Therefore, there is no positive lower limit to    In conclusion, if the optimal policy induces expected non-compliance, the best shape of the penalty function is one in which the linear component    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    with    for all    (with    as high as possible following the results above),    and    Now consider an alternative policy    such that    and    for all    and    for all    with    By construction, this policy induces expected compliance because    for all    Moreover,    is cheaper than    in expected terms because expected abatement costs are the same under both programs (   for all    expected monitoring costs are the same under both programs (   for all    but under policy    there are no expected sanctioning costs because there are no expected violations, Q.E.D.
Proof of Proposition 4 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, 

where    is the total expected costs of the program       is the expected abatement costs of the program       is the expected monitoring costs of the program   ,    is the expected sanctioning costs of the program    and    emission standards or tradable permits. We know from Proposition 3 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    because these are zero in both programs when they are 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    in the optimally designed program based on standards and from Stranlund (2007) that    for all    in the case of the optimally designed program based on tradable permits. Consequently we can write

where    is the expected total cost of an optimally designed program based on tradable emission permits and    is the expected total cost of an optimally designed program based on emission standards.
The proof that    is trivial. By definition, in the optimally designed    program, which has to induce perfect compliance, the emission responsibilities (standards) and monitoring probabilities are allocated so as to minimize the total expected costs of a program that caps aggregate emissions at    Therefore, the total expected costs of the    program must be lower than the total expected costs of an optimally designed program based on tradable permits, which produces a different allocation of emissions and monitoring probablities. Put it differently, an optimally designed tradable permits program does not minimze the expected total costs of capping aggregate emissions at a certain level    We provide a proof of this latter assertion below.
In order to make the regulator's problem under a system of tradable permits comparable to the regulator's problem under a system of emission standards, assume that under a system of tradable permits, a cost minimizing regulator chooses the level of violation    and the level of monitoring    for each firm       where    and    is the quantity of permits demanded by firm    More formally, the regulator's problem is: 

subjet to

and 

where    is firm's    demand function for permits, with    the equilibrium price of permits, and    the total number of permits issued, such that   
The Lagreangean of this problem is

The Kuhn - Tucker conditions of this problem are:

When it is optimum to induce perfect compliance for all      , (KT1Proof4) and (KT2Proof4) can be re-written, assuming    for all    as:

We know from Stranlund and Dhanda (1999) that, independently of its compliance status, in a competitive permits market, every firm    decides its level of emissions such that    Using this, and assuming    (perfect competition in the permits market), (FOC1Proof4) can be written as

This implies, for any given two firms    and       that the following identity must hold in the cost-minimizing design of perfectly enforced tradable permits market:

Now, we also know from Stranlund and Dhanda (1999) that every firm is demanding permits so that   . Using this condition, we can see that

So, when    we can write

Cost-effective monitoring requires    for all       Substituting this expression for    and   

It is easy to see that, in a competitive market for emission permits (i.e: one that generates a unique equilibrium price    the above equality holds if and only if   . Thus, we can conclude that, if    for any two firms    and       a competitive system of tradable permits will not minimize the total costs of program that caps aggregate emissions to a certain level, QED.
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