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\begin{document}

\title{The Cost-Effective Choice of Policy Instruments to Cap Aggregate
Emissions with Costly Enforcement}
\date{}
\author{}
\maketitle

\begin{abstract}
We study the cost-effectiveness of inducing expected compliance in a program
that caps aggregate emissions of a given pollutant from a set of
heterogeneous firms based on emissions standards. Our analysis considers not
only abatement, but also monitoring and sanctioning costs. We find that the
total cost-effective design of such a program is one in which standards are
firm-specific and perfectly enforced. We also find that the total expected
costs of an optimally designed transferable emission permits system are
always larger than the (minimum) expected costs of an optimally designed
program based on standards, except when the regulator's cost of monitoring a
firm's emissions is the same for all firms. Finally, when it is cost
effective to induce violations, tradable permits minimize expected costs
only under even more special conditions.

JEL Codes: L51, Q28, K32, K42

Keywords: environmental policy, cost-effectiveness, enforcement costs,
monitoring costs.
\end{abstract}

\pagebreak

\section{Introduction}

One of the most important features behind any emissions control policy,
national or international, is the total cost of the implied emissions
reduction. 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 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%
%TCIMACRO{\U{b4}}%
%BeginExpansion
\'{}%
%EndExpansion
s American Clean Energy and Security Act). Until the appearance of the EU -
ETS, the US was home of the major policy experience with tradable permits,
the federal SO$_{2}$ allowance market to control acid rain, as well as other
regional markets such as those for NOx and SOx under the RECLAIM program in
southern California. Other regulatory programs based on transferable
emission permits has been implemented in other regions as well. An example
is Santiago de Chile's Emissions Compensation Program, a market for
emissions capacity of total suspended particles.

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 detected violations. Interestingly,
the 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.\footnote{%
Moreover, a recent paper surveying the literature on the choice of policy
instruments completely ommits this issue (see Goulder and Parry, 2008).}
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. Malik does not consider sanctioning costs because he
focuses on perfectly enforced programs. 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, Ch\'{a}vez, et al. (2009) extend Malik's
contribution for a regulator that, unlike Malik%
%TCIMACRO{\U{b4}}%
%BeginExpansion
\'{}%
%EndExpansion
s, cannot perfectly observe the abatement costs of the firms, but instead
knows its distribution. With this information, the regulator 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 most of the existing work 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 regulator could always decrease the
expected costs of a program that allows non-compliance with an increasing
marginal penalty inducing full compliance with with a constant marginal
penalty.\footnote{%
In a recent work, Stranlund et al (2009) analyze the optimality of perfect
compliance for the case of emission taxes.} Arguedas (2008) addresses the
same question for the case of an emission standard, a regulator with
complete information and one firm. She concludes that "if the regulator is
entitled to choose the structure of the fine, linear penalties are socially
preferred and the optimal policy induces compliance"\ (p. 155). The analysis
of one firm fails nevertheless 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 extend Arguedas (2008) analysis to derive the
condition under which it is cost-effective to induce compliance in a system
of emissions standards with more than one regulated firm, possibly
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 minimizes the total expected costs of attaining a certain level of
aggregate emissions only under very special circumstances. 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 $n$
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 the penalty function 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.

\section{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; 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 firms' 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.

\subsection{A firm compliance behavior under an emission standard}

Assume that reducing emissions of a given pollutant $e$ is costly for a
firm. The (minimum) abatement cost function for this firm, which we will
call firm $i,$ is $c_{i}(e_{i})$, where $e_{i}$ is its level of emissions.%
\footnote{%
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 $e$ [$c_{i}^{\prime }(e_{i})<0$ and $c_{i}^{\prime
\prime }(e_{i})>0$].

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 is audited with probability $\pi _{i}$. 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 $f(v_{i})$
is imposed. Following Stranlund (2007), throughout the paper we assume that
the structure of the penalty function is $f(e_{i}-s_{i})=\phi (e_{i}-s_{i})+%
\frac{\gamma }{2}(e_{i}-s_{i})^{2},$ with $\phi >0$ and $\gamma \geq 0$.

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 $i^{\prime }s$ problem is to
choose the level of emissions to solve%
\begin{gather}
\min_{e_{i}}c_{i}(e_{i})+\pi _{i}f\left( e_{i}-s_{i}\right)
\label{firm i problem} \\
\text{subject to }e_{i}-s_{i}\geq 0  \notag
\end{gather}

The Lagrange equation for this problem is given by $\Gamma
_{i}=c_{i}(e_{i})+\pi _{i}f\left( e_{i}-s_{i}\right) -\eta _{i}\left(
e_{i}-s_{i}\right) $, with $\eta _{i}$ the Lagrange multiplier. The set of
necessary Kuhn-Tucker conditions for a positive level of emissions is: 
\begin{subequations}
\begin{eqnarray}
\frac{\partial \Gamma _{i}}{\partial e_{i}} &=&c_{i}^{\prime }(e_{i})+\pi
_{i}f^{\prime }\left( e_{i}-s_{i}\right) -\eta _{i}=0 \\
\frac{\partial \Gamma _{i}}{\partial \eta _{i}} &=&-e_{i}+s_{i}\leq 0;\eta
_{i}\geq 0;\eta _{i}\left( e_{i}-s_{i}\right) =0
\end{eqnarray}

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, $e_{i}=s_{i}$ if $-c_{i}^{\prime
}(s_{i})\leq \pi _{i}f^{\prime }\left( 0\right) .$ Otherwise, the firm is
going to choose a level of emissions $e_{i}(s_{i},\pi _{i})>s_{i},$ where $%
e_{i}(s_{i},\pi _{i})$ is the solution to $-c_{i}^{\prime }(e_{i})=\pi
_{i}f^{\prime }\left( e_{i}-s_{i}\right) .$ Note that $c_{i}^{\prime
}(s_{i}),$ 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.

\subsection{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.
Towards this objective he selects the probability of inspection $\pi _{i}$
and the emission standard $s_{i},$ for every firm $i.$ There are $n$ 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: 
\end{subequations}
\begin{subequations}
\begin{gather}
\min {}_{\substack{ (s_{1},s_{2},..,s_{n})  \\ (\pi _{1},\pi _{2},..,\pi
_{n}) }}E\left[ \tsum \limits_{i=1}^{n}c_{i}(e_{i})+\tsum
\limits_{i=1}^{n}\mu _{i}\pi _{{\large i}}+\tsum \limits_{i=1}^{n}\beta
_{i}\pi _{{\large i}}f(e_{i}-s_{i})\right] \\
\text{subject to:}  \notag \\
e_{i}=\bar{e}_{i}(s_{i},\pi _{i}) \\
\tsum \limits_{i=1}^{n}\bar{e}(s_{i},\pi _{i})=E \\
s_{i}\leq e_{i}\text{ }\forall i=1,...n
\end{gather}%
where $E\left[ \cdot \right] $ denotes the regulator's subjective expected
value of the program costs. These are comprised of the expected aggregate
abatement costs, $E\left[ \tsum \limits_{i=1}^{n}c_{i}(e_{i})\right] ,$ the
aggregate monitoring costs, $\tsum \limits_{i=1}^{n}\mu _{i}\pi _{{\large i}%
}, $ with $\mu _{i}$ being the cost of inspecting plant $i,$ and the
expected aggregate sanctioning costs, $\tsum \limits_{i=1}^{n}\beta _{i}\pi
_{{\large i}}f(e_{i}-s_{i})$, assuming that sanctioning plant $i$ has a cost
of $\beta _{i}$ per dollar of fine. For the moment, we assume that the
structure of the penalty function $f(e_{i}-s_{i})$ is given for the
environmental regulator$.$ The regulator knows that the firm $i$ will react
to a standard $s_{i}$ and a monitoring probability $\pi _{i}$ according to
its best response function $\bar{e}_{i}(s_{i},\pi _{i}).$ Therefore, he
incorporates this constraint into the problem. Because he cannot observe the
abatement cost functions of the firms, the regulator does not know the best
response function of each particular firm. Nevertheless, he uses his belief
about what the expected abatement cost function for firm $i$ is and the
firm's problem to calculate $\bar{e}_{i},$ 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 $\pi _{i.}$ The second
constraint summarizes the environmental objective of the program, namely,
that the expected aggregate level of emissions must be equal to a
predetermined 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%
%TCIMACRO{\U{b4}}%
%BeginExpansion
\'{}%
%EndExpansion
s problem can be written as

\begin{center}
$L=E\left[ \tsum \limits_{i=1}^{n}c_{i}(\bar{e}_{i})+\tsum
\limits_{i=1}^{n}\mu _{i}\pi _{{\large i}}+\tsum \limits_{i=1}^{n}\beta
_{i}\pi _{{\large i}}f(\bar{e}_{i}-s_{i})\right] +\lambda _{1}\left[ \tsum
\limits_{i=1}^{n}\bar{e}_{i}-{\normalsize E}\right] +\tsum
\limits_{i=1}^{n}\lambda _{2}^{i}(s_{i}-\bar{e}_{i})$
\end{center}

with $\lambda _{1}$ and $\lambda _{2}^{i}$ being the $n+1$ multipliers. The $%
n\times 2+n+1$ necessary Kuhn-Tucker conditions for positive levels of the
standard and the auditing probability are:

\end{subequations}
\begin{gather}
\frac{\partial L}{\partial s_{i}}=E\left[ c_{i}^{\prime }(\bar{e}_{i})\frac{%
\partial \bar{e}_{i}}{\partial s_{i}}+\beta _{i}\pi _{i}f^{\prime }(\bar{e}%
-s_{i})(\frac{\partial \bar{e}_{i}}{\partial s_{i}}-1)\right] +\lambda _{1}%
\frac{\partial \bar{e}_{i}}{\partial s_{i}}  \label{FOC 1} \\
+\lambda _{2}^{i}(1-\frac{\partial \bar{e}_{i}}{\partial s_{i}})=0,i=1,...,n
\notag
\end{gather}

\begin{gather}
\frac{\partial L}{\partial \pi _{i}}=E\left[ c_{i}^{\prime }(\bar{e}_{i})%
\frac{\partial \bar{e}_{i}}{\partial \pi _{i}}+\mu _{i}+\beta _{i}\left( f(%
\bar{e}_{i}-s_{i})+\pi _{i}f^{\prime }(\bar{e}-s_{i})\frac{\partial \bar{e}%
_{i}}{\partial \pi _{i}}\right) \right]  \label{FOC 2} \\
+\lambda _{1}\frac{\partial \bar{e}_{i}}{\partial \pi _{i}}-\lambda _{2}^{i}%
\frac{\partial \bar{e}_{i}}{\partial \pi _{i}}=0,i=1,...,n  \notag
\end{gather}

\begin{equation}
\frac{\partial L}{\partial \lambda _{1}}=\tsum \limits_{i=1}^{n}\bar{e}_{i}-%
{\normalsize E=}0
\end{equation}

\begin{equation}
\frac{\partial L}{\partial \lambda _{2}^{i}}=s_{i}-\bar{e}_{i}\leq 0,\lambda
_{2}^{i}\geq 0,\lambda _{2}^{i}\times \left( s_{i}-\bar{e}_{i}\right) =0
\label{FOC 4}
\end{equation}

We assume that these conditions are necessary and sufficient to characterize
the optimal solution of the problem. Using these conditions, we derive the
following Proposition:

\textbf{Proposition 1 }\textit{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 all firms
to comply with the standards if and only if }%
\begin{equation}
\mu _{i}\frac{f^{\prime \prime }(0)}{f^{\prime }(0)}\leq \beta _{i}f^{\prime
}(0)  \label{con1'}
\end{equation}%
\textit{for all }$i.$\textit{\ If this condition is not met and the
regulator wants to achieve the cap cost-effectively, it should induce those
plants for which }$\mu _{i}\frac{f^{\prime \prime }(0)}{f^{\prime }(0)}%
>\beta _{i}f^{\prime }(0)$ \textit{to} \textit{violate the emission
standards.}

\textbf{Proof of Proposition 1: }see Appendix.

The right-hand side of (\ref{con1'}) is the marginal increase in the
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 $\pi _{i}$ 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.

Our Proposition 1 is an extension of Arguedas' (2008) Proposition 1 to the
case of $n$ firms and heterogeneous monitoring and sanctioning costs ($\mu
_{i}\neq $ $\mu _{j}$ and $\beta _{i}\neq {\normalsize \beta }_{{\normalsize %
j}}$, for at least some $i\neq j,$ $i,j=1,..n).$ It 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. Proposition 1 is
also telling 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
monitoring and sanctioning costs are the same for all firms, in which case
the regulator induces all firms to comply or to violate. Nevertheless, there
are several reasons why the regulator's monitoring and sanctioning costs may
be different for different firms. Stranlund et al (2009) mention the
distance between the firm and the enforcing agency, the variation in the
production technologies within and between industry sectors and the number
of discharge points per plant. The latter could be an example of a firm
investment to conceal noncompliance (Heyes 2000). At the same time, the
imposition of penalties can motivate firms to engage in costly activities to
contest enforcement actions (Jost 1997a, 1997b). Consequently, sanctioning
costs may differ between firms because of their differing propensity to
litigate sanctions and challenge the legislation (Kambhu 1989).

\section{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.

\subsection{A given penalty function}

When the penalty structure is exogenously given to the regulator, condition (%
\ref{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 $\left( \pi _{1}^{\ast },\pi _{2}^{\ast },...\pi _{n}^{\ast
},s_{1}^{\ast },s_{2}^{\ast },...s_{n}^{\ast }\right) $ that induces
expected compliance is characterized by: 
\begin{gather}
E\left[ c_{i}^{\prime }(s_{i}^{\ast })\right] +{\large \mu }_{i}\frac{d\pi
_{i}^{\ast }}{ds_{i}}=E\left[ c_{j}^{\prime }(s_{j}^{\ast })\right] +{\large %
\mu }_{j}\frac{d\pi _{j}^{\ast }}{ds_{j}},\text{ for all }i\neq
j,(i,j)=1,...,n,  \label{Caracterizacion Programa compliance for all i} \\
\text{and }\pi _{i}^{\ast }=\frac{E\left[ -c_{i}^{\prime }(s_{{\large i}}^{%
{\large \ast }})\right] }{f^{\prime }(0)},\text{for all }i=1,...,n.  \notag
\end{gather}%
(See Proof of equation (9) in the Appendix). When it is cost-effective to
induce expected compliance, the regulator has to set emission standards such
that the \textit{sum} of marginal expected abatement and monitoring costs
are equal between firms, a result obtained by Ch\'{a}vez, 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 inspecting all firms with a
probability $\pi =\max \left[ \pi _{i}^{\ast }\right] =\max \left[ \frac{E%
\left[ -c_{i}^{\prime }(s_{{\large i}}^{{\large \ast }})\right] }{f^{\prime
}(0)}\right] $ for all $i=1,..,n$, as in Ch\'{a}vez, et al. 2009. An
immediate corollary is that a program designed to induce perfect compliance
with certainty in this fashion does not minimize the expected costs of the
program.

When (\ref{con1'}) does not hold, a regulator interested in minimizing the
social costs of 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 other words, the expected cost-minimizing standards must
be set such that $\bar{e}_{i}>s_{i}^{\ast }$ for all $i.$ From Kuhn-Tucker
condition (\ref{FOC 4}), this implies that $\lambda _{2}^{i}=0.$ From this
follows that the relevant Kuhn Tucker conditions in this case are 
\begin{equation*}
\frac{\partial L}{\partial s_{i}}=E\left[ c_{i}^{\prime }(\bar{e}_{i})\frac{%
\partial \bar{e}_{i}}{\partial s_{i}}+\beta _{i}\pi _{i}f^{\prime }(\bar{e}%
_{i}-s_{i})(\frac{\partial \bar{e}_{i}}{\partial s_{i}}-1)\right] +\lambda
_{1}\frac{\partial \bar{e}_{i}}{\partial s_{i}}=0
\end{equation*}%
\begin{gather*}
\frac{\partial L}{\partial \pi _{i}}=E\left \{ c_{i}^{\prime }(\bar{e}_{i})%
\frac{\partial \bar{e}_{i}}{\partial \pi _{i}}+\mu _{i}+\beta _{i}\left[ f(%
\bar{e}_{i}-s_{i})+\pi _{i}f^{\prime }(\bar{e}-s_{i})\frac{\partial \bar{e}%
_{i}}{\partial \pi _{i}}\right] \right \} \\
+\lambda _{1}\frac{\partial \bar{e}_{i}}{\partial \pi _{i}}=0
\end{gather*}

both for $i=1,..,n.$ Dividing the above two equations by $\frac{\partial 
\bar{e}_{i}}{\partial s_{i}}$ and $\frac{\partial \bar{e}_{i}}{\partial \pi
_{i}}$ respectively, we obtain: 
\begin{subequations}
\begin{gather}
E\left[ c_{i}^{\prime }(\bar{e}_{i})\right] +\beta _{i}\pi _{i}f^{\prime }(%
\bar{e}_{i}-s_{i})\left( \frac{\partial \bar{e}_{i}/\partial s_{i}-1}{%
\partial \bar{e}_{i}/\partial s_{i}}\right) =-\lambda _{1}\text{ }
\label{Prop2vieja1} \\
E\left[ c_{i}^{\prime }(\bar{e}_{i})\right] +\frac{\mu _{i}}{\partial \bar{e}%
_{i}/\partial \pi _{i}}+\frac{\beta _{i}f(\bar{e}_{i}-s_{i})}{\partial \bar{e%
}_{i}/\partial \pi _{i}}+\beta \pi _{{\large i}}f^{\prime }(\bar{e}%
_{i}-s_{i})=-\lambda _{1}\text{ }  \label{Prop2vieja2}
\end{gather}%
for all $i=1,...,n.$ Based on these, we can characterize the expected cost
minimizing program to control emissions with standards when it is
cost-effective to induce non-compliance in the context of incomplete
information and given penalties. This is done in Proposition 2 below.

\textbf{Proposition 2 }\textit{If the optimal policy }$\left( \pi _{1}^{\ast
},\pi _{2}^{\ast },...\pi _{n}^{\ast },s_{1}^{\ast },s_{2}^{\ast
},...s_{n}^{\ast }\right) $\textit{\ induces non compliance for all firms,
it is characterized by} 
\end{subequations}
\begin{gather}
E\left[ c_{i}^{\prime }(\bar{e}_{i})\right] +\beta _{i}\pi _{i}^{\ast
}f^{\prime }(\bar{e}_{i}-s_{i}^{\ast })\left( \frac{\partial \bar{e}%
_{i}/\partial s_{i}-1}{\partial \bar{e}_{i}/\partial s_{i}}\right) =
\label{Prop21} \\
E\left[ c_{j}^{\prime }(\bar{e}_{j})\right] +\beta _{j}\pi _{j}^{\ast
}f^{\prime }(\bar{e}_{j}-s_{j}^{\ast })\left( \frac{\partial \bar{e}%
_{j}/\partial s_{j}-1}{\partial \bar{e}_{j}/\partial s_{j}}\right)  \notag \\
E\left[ c_{i}^{\prime }(\bar{e}_{i})\right] +\frac{\mu _{i}}{\partial \bar{e}%
_{i}/\partial \pi _{i}}+\frac{\beta _{i}f(\bar{e}_{i}-s_{i}^{\ast })}{%
\partial \bar{e}_{i}/\partial \pi _{i}}+\beta \pi _{{\large i}}^{\ast
}f^{\prime }(\bar{e}_{i}-s_{i}^{\ast })=  \label{Prop22} \\
E\left[ c_{j}^{\prime }(\bar{e}_{j})\right] +\frac{\mu _{j}}{\partial \bar{e}%
_{j}/\partial \pi _{j}}+\frac{\beta _{j}f(\bar{e}_{j}-s_{j}^{\ast })}{%
\partial \bar{e}_{j}/\partial \pi _{j}}+\beta \pi _{j}^{\ast }f^{\prime }(%
\bar{e}_{j}-s_{j}^{\ast })  \notag
\end{gather}%
\textit{for all }$i\neq j$\textit{, }$\left( i,j\right) =1,...,n$\textit{. }

\textbf{Proof of Proposition 2: }it follows from the previous discussion.

Proposition 2 is telling that when it is cost-effective to induce
non-compliance for every firm, the regulator has to choose $\pi _{i}$ and $%
s_{i}$ such that: (a) the sum of the expected marginal abatement plus
sanctioning costs of moving $s_{i}$ is the same accross firms (from (\ref%
{Prop21})), and (b) the sum of the expected marginal abatement, monitoring
and sanctioning costs of changing $\pi _{i}$ is the same accross firms (from
(\ref{Prop22})). Condition (\ref{Prop21}) is quite intuitive. The firm
reacts to a change in $s_{i}$ by adjusting $e_{i}$ by the amount $\partial 
\bar{e}_{i}/\partial s_{i},$ in expected terms. This change in $\bar{e}_{i}$
has an effect on the abatement costs of the firm $i,$ but also an effect on
the sanctioning costs of the regulator. We know that $0<\partial \bar{e}%
_{i}/\partial s_{i}<1.\footnote{%
This result was obtained as part of the Proof of Proposition 1; see Appendix
for details.}$ Thus, a change in $s_{i}$ causes the level of violation to
change, and therefore the level of the expected fines that the regulator is
going to charge firm $i$ with. This in turn means a change in the expected
sanctioning costs for the regulator. The regulator sets $s_{i}$ equating
these two marginal costs among firms. It does a similar thing when adjusting 
$\pi _{i}$ (condition \ref{Prop22}). A marginal change in the inspection
probability affects all costs of the program: it affects firm's $i$
abatement costs \textit{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 $i.$ The regulator sets $\pi _{i}^{\ast }$
such that the sum of these three marginal costs, measured in units of
expected emissions, are the same among all firms.

Furthermore, from (\ref{Prop2vieja1}) and (\ref{Prop2vieja2}), we can obtain
the following%
\begin{equation}
\frac{\mu _{i}}{\partial \bar{e}_{i}/\partial \pi _{i}}+\frac{\beta _{i}f(%
\bar{e}_{i}-s_{i}^{\ast })}{\partial \bar{e}_{i}/\partial \pi _{i}}=-\frac{%
\beta _{i}\pi _{i}^{\ast }f^{\prime }(\bar{e}_{i}-s_{i}^{\ast })}{\partial 
\bar{e}_{i}/\partial s_{i}}  \label{Prop23}
\end{equation}%
for all $i=1,...,n.$ 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 $\pi _{i}$ is equal to the marginal sanctioning costs of moving $%
s_{i} $ for every firm $i.$

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 $\mu _{i}$ and $\beta _{i}$ to be the same for
all firms, condition $\left( \ref{con1'}\right) $ either holds or not 
\textit{for every firm}. Thus, the regulator must induce compliance or
non-compliance for every firm in the program. In this case, it would be the
heterogeneity in marginal abatement costs $c_{i}^{\prime }(\bar{e}_{i})$
that would call for firm-specific standards. Similarly, if marginal
abatement costs were the same for all firms, but monitoring and sanctioning
costs differ among firms ($\mu _{i}\neq \mu _{j\text{, }}\beta _{i}\neq
\beta _{j})$ the cost-minimizing standards could also differ among firms.

Finally, in the case when monitoring and sanctioning costs differ between
firms and condition $\left( \ref{con1'}\right) $ holds for a group of firms
while it does not hold for another group of firms, the conditions
characterizing the expected cost minimizing design of the program would be
given by $\left( \ref{Caracterizacion Programa compliance for all i}\right)
, $ for the group of firms for which condition $\left( \ref{con1'}\right) $
holds, plus conditions (\ref{Prop21}) and (\ref{Prop22}), for group of firms
for which it does not hold.

\subsection{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 $f$, and therefore
the optimality of inducing expected compliance or not. We consider only two
marginal fine structures: linear and increasing. The general fine structure
can be writen as $f(e-s)=\phi (e-s)+\frac{\gamma }{2}(e-s)^{2},$ where $\phi 
$ is a positive constant and $\gamma \geq 0$. Consequently, the regulator
has basically to compare four possible alternatives and choose the one that
minimizes the expected total costs of reaching the cap $E$ 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 to induce an
expected level of violations with increasing penalties. The result of this
comparison is given in the next Proposition:

\textbf{Proposition 3 }\textit{The optimal policy }$(s_{1}^{\ast
},s_{2}^{\ast },...,s_{n}^{\ast },\pi _{1}^{\ast },\pi _{2}^{\ast },...\pi
_{n}^{\ast },f^{\ast })$\textit{\ induces compliance and it is characterized
by (1) }$E\left[ c_{i}^{\prime }(s_{i}^{\ast })\right] +\mu _{i}\frac{d\pi
_{i}^{\ast }}{ds_{i}}=E\left[ c_{j}^{\prime }(s_{j}^{\ast })\right] +\mu _{j}%
\frac{d\pi _{j}^{\ast }}{ds_{j}}$\textit{\ for all }$i,j=1,...,n,$\textit{\ }%
$i\neq j,$\textit{\ (2) }$\pi _{i}^{\ast }=\frac{E\left[ -c_{i}^{\prime }(s_{%
{\large i}}^{\ast })\right] }{f^{\prime }(0)}$ for all $i=1,..,n,$\textit{\
and (3) }$f^{\ast }=\phi (e_{i}-s_{i})+\frac{\gamma }{2}(e_{i}-s_{i})^{2}$%
\textit{\ for all }$i,$\textit{\ with }$\phi \ $\textit{set as high as
possible and \ }$0\leq \gamma \leq \min \left[ \frac{\beta _{i}}{\mu _{i}}%
\right] \times $\textit{\ }$\phi ^{2}$\textit{.}

\textbf{Proof of Proposition 3}: see the Appendix.

The expected cost minimizing policy when a regulator wants to cap aggregate
emissions of a given pollutant to a certain level $E$ through emission
standards will be one that induces expected compliance with a constant
marginal penalty or an increasing marginal penalty, as long as $\mu
_{i}\gamma \leq \beta _{i}\phi ^{2}$ for all $i$ (otherwise the regulator
mistakenly increases the cost of the program by making cost-effective not to
induce perfect compliance). Because there are no sanctioning costs in
equilibrium, the penalty structure affects the program's costs only through
the monitoring costs: the larger the value of $\phi $ $(f^{\prime }(0))$,
the lower $\pi _{i}^{\ast },$ for all $i.$ Nevertheless, precisely because $%
\gamma $ does not affect $\pi _{i}^{\ast }$, a penalty function with a
positive value of $\gamma $ such that $0\leq \gamma \leq \min \left[ \frac{%
\beta _{i}}{\mu _{i}}\right] \times \bar{\phi}^{2}$ is also optimum because
it satisfies $\left( \ref{con1'}\right) $ and does not affect the minimum
costs of the program. Our conclusions in this respect differ from Arguedas'
(2008).

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 the environmental objective (the aggregate level of emissions 
$E$ in our case) and the abatement responsibilities among firms (the
standards) first, while another agency or office may be in charge of
designing the monitoring and enforcing strategy afterward, 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 probably be
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 $\phi $ "as high as
possible". In the real world $\phi $ will 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 $\phi $ is combined with a binding monitoring budget, the
environmental regulator may not be capable of assuring expected compliance
for all $i$ and by this way minimize the total expected costs of the
emissions control program.

\section{Comparing costs of emission standards and tradable permits}

\subsection{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 a fine structure that can be
linear or increasing in the margin, as long as $\phi $ is set as high as
possible and condition $\left( \ref{con1'}\right) $ holds. We know from
Stranlund (2007) that the optimal design of a program based on tradable
permits is one in which the program is perfectly enforced, where every firm
is audited with a homogeneous probability $\pi ^{\ast }=\frac{\bar{p}}{\phi }
$ for all $i$, whith $\bar{p}$ being the expected full-compliance
equilibrium price of the permits market and $\phi =f^{\prime }(0).$
Therefore, as in the case of emission standards, the structure of the
penalty function (whether it is increasing at a constant or an increasing
rate) does not affect the equilibrium (minimimum) costs of the program, as
long as $\mu _{i}\gamma \leq \beta _{i}\phi ^{2}$ (it is cost-effective to
induce perfect compliance). What affects the program's cost is $\phi .$ 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:

\textbf{Proposition 4 }\textit{A regulator that wants to cap the aggregate
level of emissions of a given pollutant from a set of firms will minimize
the total expected costs of doing so by implementing firm-specific emissions
standards and perfectly enforcing this program according to Proposition 3. A
system of tradable permits minimizes the total expected costs of such a
pollution control program only if }$\mu _{i}=\mu _{j}$\textit{\ for all\ }$%
i\neq j,$ $(i,j)=1,..,n.$

\textbf{Proof of Proposition 4: }The proof that the expected total costs of
an emission standards program is lower than the expected total costs of a
transferable emission permits system is trivial. By definition, in the
optimally designed emission standards 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 $E.$ Therefore, the total expected
costs of the emission standards 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 minimize the expected total costs of capping aggregate
emissions at a certain level $E.$ 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 $v_{i}$ and the level of monitoring $\pi _{i}$
for each firm $i,$ $i=1,...,n,$ where $v_{i}=e_{i}-l_{i},$ and $l_{i}$ is
the quantity of permits demanded by firm $i.$ More formally, the regulator's
problem is:\ 
\begin{subequations}
\label{ProofProp2prima}
\begin{equation*}
\min_{\substack{ \left( v_{1},...,v_{n}\right)  \\ \left( \pi _{1},...,\pi
_{n}\right) }}E\left[ \tsum \limits_{i=1}^{n}c_{i}\left( v_{i}+l_{i}\left( 
\bar{p},\pi _{i}\right) \right) \right] +\tsum \limits_{i=1}^{n}\mu _{i}\pi
_{i}+\tsum \limits_{i=1}^{n}\pi _{i}\beta _{i}f(v_{i})
\end{equation*}

subjet to%
\begin{equation*}
\tsum \limits_{i=1}^{n}v_{i}+l_{i}\left( \bar{p},\pi _{i}\right) =E
\end{equation*}

and 
\begin{equation*}
v_{i}\geq 0
\end{equation*}

where $l_{i}\left( \bar{p}\right) $ is firm's $i$ demand function for
permits, with $\bar{p}$ the expected equilibrium price of permits, and $L$
the total number of permits issued, such that $\tsum
\limits_{i=1}^{n}l_{i}\left( \bar{p},\pi _{i}\right) \equiv L.$

The Lagreangean of this problem is%
\begin{equation*}
\Lambda =E\left[ \tsum \limits_{i=1}^{n}c_{i}\left( v_{i}+l_{i}\left( \bar{p}%
,\pi _{i}\right) \right) \right] +\tsum \limits_{i=1}^{n}\mu _{i}\pi
_{i}+\tsum \limits_{i=1}^{n}\pi _{i}\beta _{i}f(v_{i})+\lambda \left( \tsum
\limits_{i=1}^{n}v_{i}+l_{i}\left( \bar{p},\pi _{i}\right) -E\right)
\end{equation*}

The Kuhn - Tucker conditions of this problem are:%
\begin{gather}
\frac{\partial \Lambda }{\partial \pi _{i}}=E\left[ c_{i}^{\prime }\left(
\cdot \right) \right] \left( \frac{\partial l_{i}}{\partial \bar{p}}\frac{%
\partial \bar{p}}{\partial \pi _{i}}+\frac{\partial l_{i}}{\partial \pi _{i}}%
\right) +\mu _{i}+\beta _{i}f(v_{i})+\lambda \left( \frac{\partial l_{i}}{%
\partial \bar{p}}\frac{\partial \bar{p}}{\partial \pi _{i}}+\frac{\partial
l_{i}}{\partial \pi _{i}}\right) \geq 0;  \label{KT1Proof4} \\
\pi _{i}\geq 0;\frac{\partial \Lambda }{\partial \pi _{i}}\pi _{i}=0\text{, }%
i=1,..,n  \notag
\end{gather}%
\begin{equation}
\frac{\partial \Lambda }{\partial v_{i}}=E\left[ c_{i}^{\prime }\left( \cdot
\right) \right] +\pi _{i}\beta _{i}f^{\prime }(v_{i})+\lambda \geq
0;v_{i}\geq 0;\frac{\partial \Lambda }{\partial v_{i}}v_{i}=0,\text{ }%
i=1,..,n  \label{KT2Proof4}
\end{equation}%
\begin{equation*}
\frac{\partial \Lambda }{\partial \lambda }=\tsum%
\limits_{i=1}^{n}v_{i}+l_{i}\left( \bar{p},L\right) -E=0
\end{equation*}

When it is optimum to induce perfect compliance for all $i$ $(v_{i}=0)$, (%
\ref{KT1Proof4}) can be re-written, assuming $\pi _{i}>0$ for all $i,$ as:%
\begin{equation}
\frac{\partial \Lambda }{\partial \pi _{i}}=E\left[ c_{i}^{\prime }\left(
\cdot \right) \right] +\frac{\mu _{i}}{\frac{\partial l_{i}}{\partial \bar{p}%
}\frac{\partial \bar{p}}{\partial \pi _{i}}+\frac{\partial l_{i}}{\partial
\pi _{i}}}+\lambda =0;\text{ }i=1,..,n  \label{FOC1Proof4}
\end{equation}

We know from Stranlund and Dhanda (1999) that, independently of its
compliance status, in a competitive permits market, every firm $i$ decides
its level of emissions such that $-c_{i}^{\prime }\left( \cdot \right) =\bar{%
p}.$ Using this, and assuming $\frac{\partial \bar{p}}{\partial \pi _{i}}=0$
(perfect competition in the permits market), (\ref{FOC1Proof4}) can be
written as%
\begin{equation*}
\bar{p}+\frac{\mu _{i}}{\partial l_{i}/\partial \pi _{i}}=-\lambda \text{
for all }i=1,..,n
\end{equation*}

This implies that the following identity must hold in the cost-minimizing
design of perfectly enforced tradable permits market: $\bar{p}+\frac{\mu _{i}%
}{\partial l_{i}/\partial \pi _{i}}=\bar{p}+\frac{\mu _{j}}{\partial
l_{j}/\partial \pi _{j}}$ for all $i\neq j,\left( i,j\right) =1,..,n.$ Now,
we also know from Stranlund and Dhanda (1999) that every firm is demanding
permits so that $\bar{p}=\pi _{i}f^{\prime }(v_{i})$. Using this condition,
we can see that $\frac{\partial l_{i}}{\partial \pi _{i}}=\frac{f^{\prime
}(v_{i})}{\pi _{i}f^{\prime \prime }(v_{i})}$ for all $i=1,..,n.$ So, when $%
v_{i}=0,$ we can write $\bar{p}+\mu _{i}\frac{\pi _{i}f^{\prime \prime }(0)}{%
f^{\prime }(0)}=\bar{p}+\mu _{j}\frac{\pi _{j}f^{\prime \prime }(0)}{%
f^{\prime }(0)}$ for all $i\neq j,\left( i,j\right) =1,..,n.$ Cost-effective
monitoring requires $\pi _{i}=\bar{p}/f^{\prime }(0)$ for all $i=$ $1,..,n.$
Substituting this expression for $\pi _{i}$ and $\pi _{j}:$%
\begin{equation*}
\bar{p}+\mu _{i}\frac{\bar{p}f^{\prime \prime }(0)}{\left( f^{\prime
}(0)\right) ^{2}}=\bar{p}+\mu _{j}\frac{\bar{p}f^{\prime \prime }(0)}{\left(
f^{\prime }(0)\right) ^{2}}\text{ for all }i\neq j,\left( i,j\right) =1,..,n
\end{equation*}

In a competitive market for emission permits (i.e: one that generates a
unique equilibrium price $\bar{p}),$ the above equality holds if and only if 
$\mu _{i}=\mu _{j}$. Thus, we can conclude that, if $\mu _{i}\neq \mu _{j}$
for any two firms $i$ and $j,$ $i\neq j,$ a competitive system of tradable
permits will not minimize the total costs of program that caps aggregate
emissions to a certain level, Q.E.D.

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 environmental economists have been
advocating for over the last forty years. But monitoring and enforcement
costs were not taken into account in the analysis that led to this policy
recomendation; only aggregate abatement costs, which tradable permits
certainly minimize. Also, environmental economists have been advocating
tradable permits as cost-effective policy instrument when compared to 
\textit{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 \textit{and }%
monitoring costs. This conclusion can be extended to the setting of
incomplete information if we consider \textit{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 why we are cautious about deriving policy
recomendations from Proposition 4. More research is needed in this area
before this can be done. In spite of this cautiousness, we do want to
emphasize that, according to Proposition 4, it is not in the name of
cost-effectiveness that we are to argue in favor of tradable emission
permits. Moreover, tradable permits do not emerge from this analysis either
with an advantage over emission standards as clear as in the case of
costless and perfect enforcement with respect to the amount of information
needed by the regulator to design the program: in order to set the
appropriate inspection probability the regulator has to predict the
equilibrium price of the permits market, which depends on the unknown
abatement costs of the firms.

\subsection{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 outside the control the environmental authority. Assume
that this is the case and that $\gamma >0.$ 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 $%
\mu _{i}\gamma $ $\leq \beta _{i}\phi ^{2}$ for all $i$ or not). Assume that 
$\mu _{i}\gamma $ $>\beta _{i}\phi ^{2}\ $for all $i.$ Then it is
cost-effective to design a program that induce a given expected level of
non-compliance for all $i$. 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 $E.$

\subsubsection{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 $i$ $(v_{i}>0$
for all $i),$ equations (\ref{KT1Proof4}) and (\ref{KT2Proof4}) can be
re-written, assuming $\pi _{i}>0$ for all $i,$ as: 
\end{subequations}
\begin{equation}
\frac{\partial \Lambda }{\partial \pi _{i}}=E\left[ c_{i}^{\prime }\left(
\cdot \right) \right] +\frac{\mu _{i}+\beta _{i}f(v_{i})}{\frac{\partial
l_{i}}{\partial \bar{p}}\frac{\partial \bar{p}}{\partial \pi _{i}}+\frac{%
\partial l_{i}}{\partial \pi _{i}}}+\lambda =0;\text{ }i=1,..,n
\label{KT1Proof5}
\end{equation}%
\begin{equation}
\frac{\partial \Lambda }{\partial v_{i}}=E\left[ c_{i}^{\prime }\left( \cdot
\right) \right] +\pi _{i}\beta _{i}f^{\prime }(v_{i})+\lambda =0,\text{ }%
i=1,..,n  \label{KT2Proof5}
\end{equation}

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 $(e_{i}-l_{i}>0).$ In a similar fashion to the emission
standards program, in the optimally designed tradable permits program the
regulator sets $\pi _{i}$ and $v_{i}$ for all $i$ such that: (a) the sum the
marginal abatement, monitoring and sanctioning costs of changing $\pi _{i}$
are equal across firms (equation \ref{KT1Proof5}) and (b) the sum of
marginal abatement and sanctioning costs of changing $v_{i}$ are equal
across firms (equation \ref{KT2Proof5}). From equations (\ref{KT1Proof5})
and (\ref{KT2Proof5}) we can also obtain 
\begin{equation}
\frac{\mu _{i}+\beta _{i}f(v_{i})}{\frac{\partial l_{i}}{\partial \bar{p}}%
\frac{\partial \bar{p}}{\partial \pi _{i}}+\frac{\partial l_{i}}{\partial
\pi _{i}}}=\pi _{i}\beta _{i}f^{\prime }(v_{i}),\text{ }i=1,..,n
\label{KT3Proof5}
\end{equation}

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 sets the sum of the marginal monitoring and sanctioning costs of
changing $\pi _{i}$ equal to the marginal sanctioning costs of moving $v_{i}$
for every firm $i.$

\subsubsection{Comparison of Costs}

Having characterized the optimal emissions trading program, we now show that
this program minimizes the total expected costs of capping aggregate
emissions to $E$ only under even more special conditions. In order to do
this, we recall from the proof of Proposition 4 that every firm $i$ that
violates their permits holdings in a competitive emission permits market
chooses its level of emissions such that $-c_{i}^{\prime }(\cdot )=\bar{p}$
and the quantity of permits to demand such that $\bar{p}=\pi _{i}f^{\prime
}(v_{i}).$ Using both expressions, we can write (\ref{KT2Proof5}) as%
\begin{equation*}
\left( -1+\beta _{i}\right) \bar{p}=-\lambda ,\text{ for all }i=1,..,n
\end{equation*}

or 
\begin{equation*}
\beta _{i}=1-\frac{\lambda }{\bar{p}},\text{ for all }i=1,..,n
\end{equation*}

It is clear from the above equation that if sanctioning costs differ among
firms $(\beta _{i}\neq \beta _{j}$ for some $i\neq j,$ $\left( i,j\right)
=1,..,n)$, a competitive permits market (one that generates a unique
equilibrium price $\bar{p}$ for all firms) will not minimize the total
expected costs of capping aggregate emissions to a level $E,$ while allowing
some degree of noncompliance. We express this result more formally in the
Proposition below.

\textbf{Proposition 5 }\textit{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 (}$\mu _{i}\gamma $\textit{\ }$%
>\beta _{i}\phi ^{2}$\textit{\ for all }$i)$\textit{, 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.}

\textbf{Proof of Proposition 5: }It follows from the previous discussion.

Proposition (5) is robust to the case when $\mu $ and $\beta $ do not differ
between firms. If $\mu _{i}=\mu _{j}$ and $\beta _{i}=\beta _{j}$ for all $%
i\neq j,$ and we assume that the permits market is perfectly competitive, so
that $\frac{\partial \bar{p}}{\partial \pi _{i}}=0,\ $then equation (\ref%
{KT3Proof5}) can be written as%
\begin{equation*}
\frac{\mu +\beta f(v_{i})}{\partial l_{i}/\partial \pi _{i}}=\pi _{i}\beta
f^{\prime }(v_{i})\text{ for all }i=1,..,n
\end{equation*}

But we know from Stranlund (2007) that if $\mu $ and $\beta $ 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%
\begin{equation*}
\frac{\mu +\beta f(v)}{\partial l_{i}/\partial \pi }=\pi \beta f^{\prime }(v)%
\text{ for all }i=1,..,n
\end{equation*}

Using $\bar{p}=\pi f^{\prime }(v)$ and $\partial l_{i}/\partial \pi
=f^{\prime }(v)/\pi f^{\prime \prime }(v),$%
\begin{equation*}
\left( \mu +\beta f(v)\right) \frac{f^{\prime \prime }(v)}{\left( f^{\prime
}(v)\right) ^{2}}=\beta \text{ for all }i=1,..,n
\end{equation*}

This condition will not be met except in the special case where $\mu =0$ and 
$f(v)\frac{f^{\prime \prime }(v)}{\left( f^{\prime }(v)\right) ^{2}}=1.$
Therefore, in the general case where $\mu $ and $\beta $ 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.

\section{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
firms, as well as on the structure of the penalty for violations. 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. 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 total
expected costs of the latter are always larger than the (minimum) expected
costs of the former, except when the regulator's cost of monitoring a firm's
emissions are the same for all firms. Moreover, when it is cost effective to
induce violations, tradable permits minimize expected costs only under even
more special conditions. The reason behind these results is that a tradable
permits market cannot by itself exploit the differences in abatement \textit{%
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.

Because the distribution of emissions and monitoring efforts in a
cost-effective design of a tradable permits system does not reproduce the
distribution of emissions and monitoring efforts in the cost-effective
design of a program that caps aggregate emissions of a pollutant, we argue
that it is not in the name of cost-effectiveness that we are to argue in
favor of tradable emission permits. Nevertheless, we are cautious in
deriving policiy recommendations. 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.

Finally, 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.

{\LARGE Appendix}

\textbf{Proof of Proposition 1 }If $\bar{e}_{{\normalsize i}}=s_{i},$ from (%
\ref{FOC 4}) we know that $\lambda _{2}^{i}\geq 0.$ Because we have also
that $\lambda _{1}\geq 0,$ we can re-write the first order conditions (\ref%
{FOC 1}) and (\ref{FOC 2}) of the regulator's problem as:%
\begin{eqnarray*}
\frac{\partial L}{\partial s_{i}} &=&\left \{ E\left[ c_{i}^{\prime }\left(
s_{i}\right) \right] +\beta _{i}\pi _{i}f^{\prime }(0)+\left( \lambda
_{1}-\lambda _{2}^{i}\right) \right \} \frac{\partial \bar{e}_{i}}{\partial
s_{i}}-\beta _{i}\pi _{i}f^{\prime }(0)+\lambda _{2}^{i}=0 \\
\frac{\partial L}{\partial \pi _{i}} &=&\left \{ E\left[ c_{i}^{\prime
}\left( s_{i}\right) \right] +\beta _{i}\pi _{i}f^{\prime }(0)+\left(
\lambda _{1}-\lambda _{2}^{i}\right) \right \} \frac{\partial \bar{e}_{i}}{%
\partial \pi _{i}}+\mu _{i}=0
\end{eqnarray*}%
Re-arranging the expressions and dividing:%
\begin{equation*}
\frac{\partial \bar{e}_{i}/\partial s_{i}}{\partial \bar{e}_{i}/\partial \pi
_{i}}=\frac{\beta _{i}\pi _{i}f^{\prime }(0)-\lambda _{2}^{i}}{-\mu _{i}}
\end{equation*}%
From the firm's optimal choice of emissions, we know that%
\begin{equation*}
-c_{i}^{\prime }(e_{i})=\pi _{i}f^{\prime }\left( e_{i}-s_{i}\right)
\end{equation*}%
From where,%
\begin{equation*}
\partial \bar{e}_{i}/\partial \pi _{i}=\frac{-f^{\prime }}{c_{i}^{\prime
\prime }+\pi _{i}f^{\prime \prime }}<0
\end{equation*}%
\qquad and%
\begin{equation}
0<\partial \bar{e}_{i}/\partial s_{i}=\frac{\pi _{i}f^{\prime \prime }}{%
c_{i}^{\prime \prime }+\pi _{i}f^{\prime \prime }}<1  \label{de/ds}
\end{equation}%
Because a cost-minimizing regulator that wants to achieve $\bar{e}_{%
{\normalsize i}}=s_{i}$ will set $\pi _{i}$ such that $E\left[
-c_{i}^{\prime }(s_{i})\right] =\pi _{i}f^{\prime }\left( 0\right) $ in
order not to waste monitoring resources, we can write%
\begin{equation*}
\frac{\partial \bar{e}_{i}/\partial s_{i}}{\partial \bar{e}_{i}/\partial \pi
_{i}}_{\bar{e}_{{\normalsize i}}=s_{i}}=\frac{\pi _{i}f^{\prime \prime }(0)}{%
c_{i}^{\prime \prime }(s_{i})+\pi _{i}f^{\prime \prime }(0)}\times \frac{%
c_{i}^{\prime \prime }(s_{i})+\pi _{i}f^{\prime \prime }(0)}{-f^{\prime }(0)}%
=\frac{\pi _{i}f^{\prime \prime }(0)}{-f^{\prime }(0)}=\frac{\beta _{i}\pi
_{i}f^{\prime }(0)-\lambda _{2}^{i}}{-\mu _{i}}
\end{equation*}%
or%
\begin{equation*}
\mu _{i}\frac{\pi _{i}f^{\prime \prime }(0)}{f^{\prime }(0)}=\pi _{i}\beta
_{i}f^{\prime }(0)-\lambda _{2}^{i}
\end{equation*}%
From where, using $\lambda _{2}^{i}\geq 0,$ 
\begin{equation}
\mu _{i}\frac{\pi _{i}f^{\prime \prime }(0)}{f^{\prime }(0)}\leq \pi
_{i}\beta _{i}f^{\prime }(0)  \label{cond 1}
\end{equation}%
Dividing both sides of equation (\ref{cond 1}) by $\pi _{i}$ we obtain $\mu
_{i}\frac{f^{\prime \prime }(0)}{f^{\prime }(0)}\leq \beta _{i}f^{\prime
}(0)\ $for all $i.$ We have proved that when a cost - minimizing regulator
induces (expected) compliance, this condition is met. The reverse is also
true. Assume to the contrary that $\mu _{i}\frac{f^{\prime \prime }(0)}{%
f^{\prime }(0)}\leq \beta _{i}f^{\prime }(0)$ holds but $\bar{e}_{%
{\normalsize i}}>s_{i}.$ If $\bar{e}_{{\normalsize i}}>s_{i},$ we know from (%
\ref{FOC 4}) that $\lambda _{2}^{i}=0$ and 
\begin{eqnarray*}
\frac{\partial L}{\partial s_{i}} &=&\left \{ E\left[ c_{i}^{\prime }\left( 
\bar{e}_{{\normalsize i}}\right) \right] +\beta _{i}\pi _{i}f^{\prime }(\bar{%
e}_{{\normalsize i}}-s_{i})+\lambda _{1}\right \} \frac{\partial \bar{e}_{i}%
}{\partial s_{i}}-\beta _{i}\pi _{i}f^{\prime }(\bar{e}_{{\normalsize i}%
}-s_{i})=0 \\
\frac{\partial L}{\partial \pi _{i}} &=&\left \{ E\left[ c_{i}^{\prime
}\left( \bar{e}_{{\normalsize i}}\right) \right] +\beta _{i}\pi
_{i}f^{\prime }(\bar{e}_{{\normalsize i}}-s_{i})+\lambda _{1}\right \} \frac{%
\partial \bar{e}_{i}}{\partial \pi _{i}}+\mu _{i}+\beta _{i}f(\bar{e}_{%
{\normalsize i}}-s_{i})=0
\end{eqnarray*}%
From these, and the firm's optimal choice of emissions:%
\begin{equation*}
\frac{\partial \bar{e}_{i}/\partial s_{i}}{\partial \bar{e}_{i}/\partial \pi
_{i}}=\frac{\pi _{i}f^{\prime \prime }(\bar{e}_{{\normalsize i}}-s_{i})}{%
-f^{\prime }(\bar{e}_{{\normalsize i}}-s_{i})}=\frac{\beta _{i}\pi
_{i}f^{\prime }(\bar{e}_{{\normalsize i}}-s_{i})}{-\mu _{i}-\beta _{i}f(\bar{%
e}_{{\normalsize i}}-s_{i})}
\end{equation*}%
After substituting for the functional form of $f,$ operating and
rearranging, we can write%
\begin{equation*}
\mu _{i}\gamma -\beta _{i}\phi ^{2}=\mu _{i}f^{\prime \prime }(0)-\beta
_{i}\left( f^{\prime }(0)\right) ^{2}=\beta _{i}\gamma \left( -\phi
(e_{i}-s_{i})-\frac{\gamma }{2}(e_{i}-s_{i})^{2}+2\phi (e_{i}-s_{i})+\gamma
(e_{i}-s_{i})^{2}\right) >0
\end{equation*}%
which is a contradiction. Hence, when $\mu _{i}\frac{f^{\prime \prime }(0)}{%
f^{\prime }(0)}\leq \beta _{i}f^{\prime }(0)$ is met, it is cost effective
for the regulator to induce firm $i$ to comply with the emission standard,
Q.E.D.

\textbf{Proof of equation (9) }When $\bar{e}_{{\normalsize i}}=s_{i}$,
expected violations are zero and therefore there are no sanctioning costs.
Moreover, if the regulator wants to achieve $\bar{e}_{{\normalsize i}}=s_{i}$
it has to set $\pi _{i}$ such that $E\left[ -c_{i}^{\prime }(s_{{\large i}}^{%
{\large \ast }})\right] \leq \pi _{i}^{\ast }f^{\prime }(0),$ or $\pi
_{i}^{\ast }\geq \frac{E\left[ -c_{i}^{\prime }(s_{{\large i}}^{{\large \ast 
}})\right] }{f^{\prime }(0)}.$ But, if the regulator can induce $\bar{e}_{%
{\normalsize i}}=s_{i}$ with $\pi _{i}^{\ast }=\frac{E\left[ -c_{i}^{\prime
}(s_{{\large i}}^{{\large \ast }})\right] }{f^{\prime }(0)}$ it would not be
cost-effective to select $\pi _{i}^{\ast }>\frac{E\left[ -c_{i}^{\prime }(s_{%
{\large i}}^{{\large \ast }})\right] }{f^{\prime }(0)}.$ Therefore, $\pi
_{i}^{\ast }=\frac{E\left[ -c_{i}^{\prime }(s_{{\large i}}^{{\large \ast }})%
\right] }{f^{\prime }(0)}.$ In this case, the Lagrange of the regulator's
problem can be re-written as%
\begin{equation*}
L=E\left[ \tsum \limits_{i=1}^{n}c_{i}(s_{i})+\tsum \limits_{i=1}^{n}\mu
_{i}\pi _{{\large i}}^{\ast }\right] +\lambda _{1}\left[ \tsum%
\limits_{i=1}^{n}s_{i}-{\normalsize E}\right] 
\end{equation*}

Deriving $L$ with respect to $s_{i},$ $i=1,..,n,$ we obtain (\ref%
{Caracterizacion Programa compliance for all i}), Q.E.D.

\textbf{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.

\textit{If the optimal policy is going to induce compliance for all }$i$%
\textit{, }condition (\ref{con1'}) requires that $\mu _{i}\gamma \leq \beta
_{i}\phi ^{2}$ for all $i=1,...,n.$ We also know from Section 3 that in this
case the characterization of the cost-effective design of a program based on
standards calls $\pi _{i}^{\ast }=\frac{E\left[ -c_{i}^{\prime }(s_{{\large i%
}}^{{\large \ast }})\right] }{f^{\prime }(0)}=\frac{E\left[ -c_{i}^{\prime
}(s_{{\large i}}^{{\large \ast }})\right] }{\phi }.$ From this we can
conclude that the regulator must choose the linear component $\phi $ of the
fine structure as high as possible because this will decrease the optimum
level of the inspection probability$,\pi _{i}^{\ast },$ and by this way the
monitoring costs. Conceptually, this calls for $\phi =\infty $ because this
will make the monitoring costs equal to zero, but in the real world there
may be limits to the upper value of $\phi $. If we call $\bar{\phi}$ the
highest possible value of $\phi ,$ any value of $\gamma :$ $0\leq \gamma
\leq \min \left[ \frac{\beta _{i}}{\mu _{i}}\right] \times \bar{\phi}^{2}$,
will still make cost-efective to induce compliance for every firm and will
not have an effect on the minimum expected costs of the program, namely $%
\tsum \limits_{i=1}^{n}c_{i}(s_{{\large i}}^{{\large \ast }})+\mu \tsum
\limits_{i=1}^{n}\pi _{{\large i}}^{\ast }.$

Therefore, if the optimal policy induces compliance for all $i$, the
cost-minimizing shape of the fine must be such that the linear component $%
\phi $ is set as high as possible. The value of the progressive component $%
\gamma $ is irrelevant in equilibrium as long as $0\leq \gamma \leq \min %
\left[ \frac{\beta _{i}}{\mu _{i}}\right] \times \bar{\phi}^{2},$ where $%
\bar{\phi}$ is the chosen level of $\phi $.

\textit{If the regulator is going to induce non-compliance, }how does it have%
\textit{\ }to\textit{\ }choose $\phi $ and $\gamma $ in order to minimize
the costs of a program that produces $E?$ In other words, can the regulator
decrease the expected costs of the program by altering the fine structure
(the value of $\phi $ and $\gamma ),$ 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 $\phi $ and $\gamma $ keeping violations, and fines, constant. If $%
f(e-s)=\phi (e-s)+\frac{\gamma }{2}(e-s)^{2},$ changing $\phi $ and $\gamma $
so as to keep $f$ constant requires $\frac{e-s}{2}=-\frac{d\phi }{d\gamma }.$
But with $n$ firms, it is impossible to move $\phi $ and $\gamma $ such that 
$\frac{e_{i}-s_{i}}{2}=-\frac{d\phi }{d\gamma }$ for all $i.$ Keeping $f$
contant for all $i$ requires a firm-specific fine parameters. We assume that
this is the case and we then show that the optimal design of the program
calls for a uniform fine structure.

If the fine structure is firm-specific, we have $f_{i}(\bar{e}%
_{i}-s_{i})=\phi _{i}(\bar{e}_{i}-s_{i})+\frac{\gamma _{i}}{2}(\bar{e}%
_{i}-s_{i})^{2},$ and $f_{i}^{\prime }(\bar{e}_{i}-s_{i})=\phi _{i}+\gamma
_{_{i}}(\bar{e}_{i}-s_{i})$ for each $i.$ Now we ask how to choose $\phi
_{i} $ and $\gamma _{i}$ 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 $\phi _{i}$ and $\gamma
_{i}$) while choosing $\pi _{i}=\pi _{i}^{\ast }=\frac{E\left[
-c_{i}^{\prime }(\bar{e}_{{\large i}})\right] }{f^{\prime }(\bar{e}%
_{i}-s_{i})}.$ In order to answer this question, we evaluate the Lagrangean
of the regulator's problem at $\pi _{i}^{\ast }$ when $\bar{e}_{i}>s_{i}$
and $\tsum \limits_{i}\bar{e}_{i}=E$ and change $\phi _{i}$ and $\gamma _{i}$
such that $df_{i}=0,$ that is $-\frac{d\phi _{i}}{d\gamma _{i}}=\frac{\bar{e}%
_{i}-s_{i}}{2}.$%
\begin{subequations}
\label{ProofProp2}
\begin{equation*}
L=E\left[ \tsum \limits_{i=1}^{n}c_{i}(\bar{e}_{i})\right] +\tsum
\limits_{i=1}^{n}\mu _{i}\pi _{{\large i}}^{\ast }+\tsum
\limits_{i=1}^{n}\beta _{i}\pi _{{\large i}}^{\ast }f_{i}(\bar{e}_{i}-s_{i})
\end{equation*}%
\begin{equation*}
dL=\frac{\partial L}{\partial \phi _{i}}d\phi _{i}+\frac{\partial L}{%
\partial \gamma _{i}}d\gamma _{i}
\end{equation*}%
\begin{eqnarray*}
dL &=&\left[ \mu _{i}\frac{\partial \pi _{{\large i}}^{\ast }}{\partial \phi
_{i}}+\beta _{i}\left[ \frac{\partial \pi _{{\large i}}^{\ast }}{\partial
\phi _{i}}f_{i}(\bar{e}_{i}-s_{i})+\pi _{{\large i}}^{\ast }(\bar{e}%
_{i}-s_{i})\right] \right] d\phi _{i} \\
&&+\left[ \mu _{i}\frac{\partial \pi _{{\large i}}^{\ast }}{\partial \gamma
_{i}}+\beta _{i}\left[ \frac{\partial \pi _{{\large i}}^{\ast }}{\partial
\gamma _{i}}f_{i}(\bar{e}_{i}-s_{i})+\pi _{{\large i}}^{\ast }\frac{(\bar{e}%
_{i}-s_{i})^{2}}{2}\right] \right] d\gamma _{i}
\end{eqnarray*}%
Dividing both sides by $d\phi _{i}\ $and substituting $\frac{d\gamma _{i}}{%
d\phi _{i}}$ for $-\frac{2}{\bar{e}_{i}-s_{i}}$ we obtain 
\begin{eqnarray*}
\frac{dL}{d\phi _{i}} &=&\mu _{i}\frac{\partial \pi _{{\large i}}^{\ast }}{%
\partial \phi _{i}}+\beta _{i}\left[ \frac{\partial \pi _{{\large i}}^{\ast }%
}{\partial \phi _{i}}\left( \phi _{i}(\bar{e}_{i}-s_{i})+\frac{\gamma _{i}}{2%
}(\bar{e}_{i}-s_{i})^{2}\right) \right] \\
&&-\frac{2\mu _{i}}{\bar{e}_{i}-s_{i}}\frac{\partial \pi _{{\large i}}^{\ast
}}{\partial \gamma _{i}}-\beta \left[ \frac{\partial \pi _{{\large i}}^{\ast
}}{\partial \gamma _{i}}\left( 2\phi _{i}+\gamma _{i}(\bar{e}%
_{i}-s_{i})\right) \right]
\end{eqnarray*}%
We know that $\frac{\partial \pi _{i}^{\ast }}{\partial \phi _{i}}=\frac{-E%
\left[ -c_{i}^{\prime }(\bar{e}_{{\large i}})\right] }{\left[ \phi
_{i}+\gamma _{_{i}}(\bar{e}_{i}-s_{i})\right] ^{2}}$ and $\frac{\partial \pi
_{i}^{\ast }}{\partial \gamma _{i}}=\frac{-E\left[ -c_{i}^{\prime }(\bar{e}_{%
{\large i}})\right] }{\left[ \phi _{i}+\gamma _{_{i}}(\bar{e}_{i}-s_{i})%
\right] ^{2}}\times (\bar{e}_{i}-s_{i}).$ Substituting, 
\end{subequations}
\begin{eqnarray}
\frac{dL}{d\phi _{i}} &=&-\frac{E\left[ -c_{i}^{\prime }(\bar{e}_{{\large i}%
})\right] }{\left[ \phi _{i}+\gamma _{_{i}}(\bar{e}_{i}-s_{i})\right] ^{2}}%
\left[ \mu _{i}+\beta _{i}\left( \phi _{i}(\bar{e}_{i}-s_{i})+\frac{\gamma
_{i}}{2}(\bar{e}_{i}-s_{i})^{2}\right) \right]  \label{dL/dfi} \\
&&+\frac{E\left[ -c_{i}^{\prime }(\bar{e}_{{\large i}})\right] }{\left[ \phi
_{i}+\gamma _{_{i}}(\bar{e}_{i}-s_{i})\right] ^{2}}\times (\bar{e}_{i}-s_{i})%
\left[ \frac{2\mu _{i}}{\bar{e}_{i}-s_{i}}+\beta _{i}\left( 2\phi
_{i}+\gamma _{i}(\bar{e}_{i}-s_{i})\right) \right]  \notag
\end{eqnarray}%
And after some operations we obtain 
\begin{subequations}
\label{ProofProp2prima}
\begin{equation*}
\frac{dL}{d\phi _{i}}=\frac{E\left[ -c_{i}^{\prime }(\bar{e}_{{\large i}})%
\right] }{\left[ \phi _{i}+\gamma _{_{i}}(\bar{e}_{i}-s_{i})\right] ^{2}}%
\left[ \mu _{i}+\beta _{i}\left( \phi _{i}(\bar{e}_{i}-s_{i})+\frac{\gamma
_{i}}{2}(\bar{e}_{i}-s_{i})^{2}\right) \right] >0
\end{equation*}%
This means that the regulator can decrease the costs of a program that
induces a violation $(\bar{e}_{i}-s_{i})$ for each firm by decreasing $\phi
_{i}$ (and increasing $\gamma _{_{i}}$ accordingly so as to keep the
equilibrium fine constant).

Now, decreasing $\phi _{i}$ has a limit and this limit is $\phi _{i}=0.$
Under a negative value of $\phi _{i}$ it will always exist a (sufficiently
small) level of violation that makes the fine negative. But a negative fine
violates our assumption that $f\geq 0$ for all levels of violations. On the
other hand, there is no theoretical maximum value for $\gamma _{i}.$ 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: $f(\bar{e}%
_{i}-s_{i})=\frac{\gamma }{2}(\bar{e}_{i}-s_{i})^{2}$ for all $i.$ The
regulator always decreases monitoring costs by increasing $\gamma ,$ for the
same level of violation. This is true for all firms and therefore it must
set $\gamma $ as high as possible for all firms. Because we are in the case
where the regulator induces non-compliance, condition $\mu _{i}\gamma >\beta
_{i}\phi ^{2}$ for all $i=1,...,n$ must hold. And because we have just said
that the cost minimizing shape of the penalty function requires $\phi _{i}=0$
for all $i=1,...,n,$ the above condition only requires $\gamma >0.$ In
conclusion, if the optimal policy induces expected non-compliance, the best
shape of the penalty function is one in which the linear component $\phi =0$
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 $%
P^{n}=(s_{1}^{n},s_{2}^{n},...,s_{n}^{n},\pi _{1}^{n},\pi _{2}^{n},...\pi
_{n}^{n},f^{n}),$ with $f^{n}=\frac{\gamma }{2}(e_{i}-s_{i})^{2}$ for all $i$
(with $\gamma $ as high as possible following the results above), $\pi
_{i}^{n}=\frac{E\left[ -c_{i}^{\prime }(\bar{e}_{{\large i}}^{n})\right] }{%
\gamma (\bar{e}_{i}^{n}-s_{i}^{n})}$ and $\tsum \limits_{i=1}^{n}\bar{e}%
_{i}^{n}=E.$ Now consider an alternative policy $%
P^{c}=(s_{1}^{c},s_{2}^{c},...,s_{n}^{c},\pi _{1}^{c},\pi _{2}^{c},...\pi
_{n}^{c},f^{c})$ such that $s_{i}^{c}=\bar{e}_{i}^{n}$ and $\pi _{i}^{c}=\pi
_{i}^{n}$ for all $i,$ and $f^{c}=\phi (e_{i}-s_{i})$ for all $i$ with $\phi
=\gamma \times \max_{i}\left[ \bar{e}_{i}^{n}-s_{i}^{n}\right] .$ By
construction, this policy induces expected compliance because $\pi
_{i}^{c}f^{c\prime }=\pi _{i}^{c}\phi =\pi _{i}^{c}\gamma \times \max_{i}%
\left[ \bar{e}_{i}^{n}-s_{i}^{n}\right] \geq E\left[ -c_{i}^{\prime }(\bar{e}%
_{{\large i}}^{n})\right] =E\left[ -c_{i}^{\prime }(s_{{\large i}}^{c})%
\right] $ for all $i.$ Moreover, $P^{c}$ is cheaper than $P^{n}$ in expected
terms because expected abatement costs are the same under both programs ($%
s_{i}^{c}=\bar{e}_{i}^{n}$ for all $i),$ expected monitoring costs are the
same under both programs ($\pi _{i}^{c}=\pi _{i}^{n}$ for all $i),$ but
under policy $P^{c}$ there are no expected sanctioning costs because there
are no expected violations, \textbf{Q.E.D}.

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\end{subequations}

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