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\begin{document}

\author{Marcelo Caffera\thanks{%
Email: marcaffera@um.edu.uy} \\
%EndAName
Universidad de Montevideo \and \noindent Juan Dubra\thanks{\textit{%
Correspondence address}: Departamento de Econom\'{\i}a, Universidad de
Montevideo, Prudencio de Pena 2440, Montevideo 11600, Uruguay. Email:
dubraj@um.edu.uy} \\
%EndAName
Universidad de Montevideo\\
Universidad Torcuato Di Tella}
\title{Getting Polluters to Tell the Truth\textbf{\thanks{%
Hugo Hopenhayn noted a bug in a previous version of this paper, and we are
grateful for his help. We also thank Atila Abdulkadiroglu, Ezequiel Aguirre,
Anil Arya, Jean-Pierre Beno\^{\i}t, Carlos Chavez, Marcelo Cousillas,
Federico Echenique, Jeff Ely, N\'{e}stor Gandelman, Jonathan Glover, Ana Mar%
\'{\i}a Ib\'{a}\~{n}ez, Matt Jackson, Larry Kotlikoff, Carlos Lacurcia,
Preston McAfee, Stephen Morris and Francesco Squintani.}} \bigskip }
\date{First Draft, April 2005.\\
This Draft, October, 2006}
\maketitle

\begin{abstract}
We study the problem of a regulator who must control the emissions of a
given pollutant from a series of industries when the firms' abatement costs
are unknown. We develop a mechanism in which the regulator asks firms to
report their abatement costs and implements the most stringent emissions
standard consistent with the firms' declarations. He also inspects one of
the firms in each industry which declared the cost structure consistent with
the least stringent emissions standard and with an arbitrarily small
probability, he discovers whether the report was true or not. The firm is
punished with an arbitrarily small fine if and only if its report was false.

This mechanism is simple, is implementable in practice, its unique
equilibrium is truth telling by firms, it implements the first best
pollution standards and shares some features of the regulatory processes
actually observed in reality.

\smallskip\ 

\textbf{Keywords:} Efficient Emissions Standards, Command and Control, Truth
Telling, Full Nash Implementation.

\textsl{Journal of Economic Literature} \textbf{Classification numbers:}
D02, D78, D82, Q20, Q52, Q53.
\end{abstract}

\section{Introduction}

In this paper we study the problem of a regulator who must control the
emissions of a given pollutant from a series of industries. He wants firms
to produce the optimal amount of pollution, when both the firms' abatement
costs and the costs of pollution to society are considered. Such a regulator
faces a fundamental problem faced by every regulator worldwide: that he
rarely knows the exact nature of the pollution abatement technology of
firms, which of course influences the optimal pollution level to be chosen.
The regulator must therefore rely on whatever he can learn about firms'
costs from the information they are willing to provide. Given the importance
of the problem of regulating polluters, the issue of how to truthfully
extract information about their costs has been at the heart of both academic
and policymaking discussions for almost three decades.

We posit a model in which the regulator asks firms to declare what their
cost functions are and uses these announcements to set an emissions standard
for each industry: a maximum allowable level of emissions for every firm in
that industry. After receiving the reports, the regulator implements in each
industry the most stringent standard consistent with the declarations of the
firms in that industry. He also inspects one of the firms in each industry
which declared the cost structure consistent with the least stringent
emissions standard (the firms most likely to be lying). With an arbitrarily
small probability, he discovers whether the report was true or not. A firm
which was sampled is punished with an arbitrarily small fine if and only if
its report was false.

This mechanism has several important features. First, it is very simple, and
therefore applicable in practice. In fact, as we will discuss later in more
depth, it is very similar to the mechanism actually used in several
countries, including the United States' National Pollutant Discharge
Elimination System. Second, it fully implements truth telling by the firms,
and results in the regulator setting the efficient standard in each
industry. That is, since the \textit{unique} equilibrium of this game is for
firms to tell the truth, the informational asymmetry disappears, and the
total welfare of society is maximized. Finally, a third advantage of the
mechanism is that it is budget balanced: it implies no costs for the
regulator.

There are other studies that have proposed mechanisms that both implement
truth telling by the firms and result in an efficient level of pollution.
The two most relevant works in this area are: Kwerel (1977) who obtains
truth telling as one of potentially many equilibria when the regulator sells
pollution licenses (which are assumed to be traded in a perfectly
competitive market) and subsidizes firms which buy them in excess of their
needs;\footnote{%
See Montero (2007) who fixes a problem with Kwerel's mechanism, but still
retains the undesirable feature of unbalanced budget.} Dasgupta, Hammond and
Maskin (1980) who use the Groves-Clarke mechanism to obtain dominant
strategy truth telling with an unbalanced budget. Spulber (1988) presents a
mechanism that, contrary to what happens with ours, does not attain the
first best outcomes. There are a few problems with these prior studies, the
main ones being that one does not observe the proposed mechanisms in
practice and that they posses equilibria other than truth telling (which may
be the cause why one doesn't observe these other mechanisms). We believe
that there are two main reasons why those mechanisms are not observed in
reality. Moreover, our mechanism is free of those problems. The first reason
why we don't observe those mechanisms in reality is that they are
complicated. This has been a standard criticism about the literature of
optimal mechanism design. We believe that another reason why previously
proposed mechanisms are not observed is that they are based on taxes,
subsidies, or tradeable permits and these types of instruments have several
implementation problems as compared to classic \textquotedblleft command and
control\textquotedblright\ instruments. Although these types of instruments
have been used recently, they have applied only in very specific contexts,
and their implementation has been slow for several reasons. For example,
regulators in some countries are not educated in environmental economics and
do not see the advantages of these instruments in terms of
cost-effectiveness and efficiency; they see \textquotedblleft
command-and-control\textquotedblright\ instruments as stronger statements of
support for environmental protection. Moreover, other regulators may think
that it is immoral to let firms pollute just because they paid some taxes,
or because they purchased pollution permits. Policymakers may also be
reluctant to impose further costs on firms because of the impact on
employment. Also, incentive-based instruments shift control decisions from
regulatory staff to polluting firms, possibly affecting the regulator's job
security and prestige.\footnote{%
These and other arguments are well documented in the literature. See for
example Bohm and Russell (1985)%
%TCIMACRO{\U{b8} }%
%BeginExpansion
\c{}
%EndExpansion
Russell and Powell (1996), Lewis (1996), Keohane, Revesz and Stavins (1998).}

Another problem with the existing theorems in the literature, is that they
focus on whether truth telling is a Nash equilibrium of the revelation game,
and not on whether truth telling is the unique equilibrium. If declaring
large abatement costs is an equilibrium that yields higher profits for all
firms, one will not observe firms telling the truth, but rather
overestimating their costs. Our theorem is free from that problem, since its
unique equilibrium is truth telling.\footnote{%
Dasgupta et al. also criticize Kwerel for the assumption that permits are
traded in perfectly competitive markets and because of the weak
\textquotedblleft implementation\textquotedblright\ concept: that truth
telling is a Bayesian Nash equilibrium. An additional problem of Kwerel is
that his regulator has an unbalanced budget. In Dasgupta et al., if one
requires a balanced budget one only obtains that truth telling is a Bayes
Nash equilibrium (and neither uniqueness, nor dominant strategy
implementation).}

Section \ref{novelty} discusses the relationships among our mechanism and
those in the literature on implementation, but it suffices here to stress
three points. First, the implementability of the regulator's rule in our
setting does not follow from any of the existing theorems. Second, we
believe that the least that one must demand from a mechanism is that its
unique equilibrium is truth telling, and not just \textquotedblleft truth
telling is an equilibrium\textquotedblright\ so in that regard, our
mechanism presents an improvement over the relevant literature. Finally, our
focus is not on the novelty of the theoretical arguments in the
implementation of the regulator's rule, but on the possibility of actually
implementing it in real contexts.

We have argued that our mechanism is simple, shares some features of some
regulatory practices around the world, implements truth telling and the
efficient level of pollution, and is budget balanced. Also, we have argued
that one of the reasons why one does not observe in practice alternative
mechanisms that have been proposed in the literature is because they were
complicated and relied on taxes and subsidies, which may be too difficult to
implement for regulators. We now turn to the discussion of our assumptions.

\section{Discussion of Assumptions}

Our model is very similar to that in Kwerel (1977) and Dasgupta et al.
(1980). In some dimensions our model is more general, and the conclusion of
the theorem is stronger, but we make two additional assumptions.\footnote{%
Like both these works, our model can be applied more generally, and not just
to the problem of a regulator trying to fix the right level of pollution. As
will become clear, the main idea is just to induce Bertrand-like competition
among firms.} First, we assume that if the regulator samples one firm, it
can find out, with probability $\varepsilon ,$ for $\varepsilon $
arbitrarily small, whether the report of abatement costs was true or not.
Second, we assume that in each of $m$ industries there are at least two
firms with the same cost functions.

With the first assumption the asymmetry of information between the regulator
and the firms ceases to be absolute. The assumption is quite weak for at
least three reasons. First, we assume that the regulator inspects and
samples just one firm out of a potentially large pool. Second, we assume
that in case the inspection is successful and it provides some information,
the regulator only learns whether the report was true or not, but in case of
a false report, he does not get to know the true cost function. Third, and
most important, the regulator only finds out whether the report is true or
not with an arbitrarily small chance. That is, we fix any $\varepsilon >0,$
and the regulator only learns whether the report is true with probability $%
\varepsilon .$

Our assumption that the asymmetry of information is not absolute is also a
reasonable one in the context we study. First, regulators worldwide engage
in controlling or monitoring the statements of polluters about the abatement
technology to be used, so our assumption reflects a common practice. In the
US for example, before starting their operations firms are required to
present an exhaustive description of their production processes, abatement
technology and costs in order to obtain a pollution discharge permit.%
\footnote{%
Several countries have copied extensively the US National Pollutant
Discharge Elimination System, including our own Uruguay. Most of such
systems share the \textquotedblleft inspection\textquotedblright\ features
of the US system that we are interested in.} Second, this common practice is
well founded, since the regulators can check each piece of information
provided by the firm, and assess its validity, or even in some cases be more
proactive by pointing out to firms how other businesses have coped with the
same abatement problems. Engineers from the Environmental Protection Agency
study the different abatement technologies available to a particular type of
industrial activity and then establish effluent standards for each category
of polluter and place of discharge (see Field, 1997). Since the regulation,
and the standard-setting, occur at a basic \textquotedblleft process
level\textquotedblright\ and not at a more complicated \textquotedblleft
plant-level,\textquotedblright\ the processes involved are standard across
industries, and the regulator has a deep knowledge about costs as
illustrated, for example, in the following quotation from the Environmental
Protection Agency (1992).\bigskip

\begin{quote}
``The document provides a generic process-by-process assessment of pollution
prevention opportunities for the Kraft segment of the pulp and paper
industry. The process areas covered are: wood yard operations, pulping and
chemical recovery, pulp bleaching, pulp drying and papermaking, and
wastewater treatment. These process areas are further broken down by
specific process (e.g., oxygen delignification as one specific process under
the pulping and chemical recovery area). For each specific process there is
a description, a cost estimate, a discussion of applicability, and estimate
of environmental benefits.''\footnote{%
Similar quotations can be found for other industries. See for example EPA
(2002) for the iron and steel industries and their process by process
regulation.}\bigskip
\end{quote}

Both the way the regulatory process takes place, and the depth of the
knowledge of the regulator about each individual process suggest that the
asymmetry of information between firms and the regulators is not absolute,
so that our assumption seems appropriate. In addition the reason why the
assumption that the mechanism designer can inspect declarations has not been
used in general is not present in our model: a sizeable part of the
literature on mechanism design concerns the case in which the private
information is about preferences of the individual, which of course can not
be inspected (or declared false). Since, in our case, the inspections
concern verifiable information, the assumption is justifiable in this case.

Our second assumption is that for any way of generating the pollutant of
interest, there are at least two firms that generate it from the same
source. In the case of CO$_{2},$ for example, we assume that for each way of
generating it, through diesel engines, or through burning of coal, there are
at least two firms that generate it in the same way. We call an
\textquotedblleft industry\textquotedblright\ the collection of all firms
that generate the pollutant in the same way, and we assume that each
industry has at least two firms. Naturally, all firms in the industry have
identical cost functions. The assumption follows from the way the regulatory
process works (i.e. setting emissions standards on a process by process
basis). If a firm buys cows and delivers leather shoes, it won't have the
same abatement costs as a firm that buys cows and delivers leather seats for
cars. But both firms will first produce raw hides and then tan the leather.
Since both firms need to abate its pollution levels at each individual task,
each of which is also undertaken in other firms producing different goods,
our assumption reflects the fact that even very complicated production
processes are based on some elementary processes that are repeated in
several firms even across industries. Another reason why the assumption of
at least two firms per industry is not so restrictive is that our exact same
model would apply if it was common knowledge that costs in the same industry
are just \textquotedblleft vertical\textquotedblright\ translations of each
other. That is, if firm $1$ has a cost function of $c,$ and firm $2$ a cost
function of $c+k$, those cost functions are \textquotedblleft
identical\textquotedblright\ as far as our mechanism is concerned.
Therefore, if a firm in California and a firm in New York buy their
abatement technology from a firm in New York, and the price in California is
just the price in New York plus shipping, those two firms can be modeled as
having identical costs. Finally, as we will argue in Section \ref{dif}, even
if there are some firms that have cost functions that no other firm in the
whole economy share, our mechanism can still be used. Suppose that the
regulator can estimate the cost functions of these firms and produce
estimates which are \textquotedblleft close\textquotedblright\ to the truth.
Then, the unique equilibrium of our mechanism (when it is applied among the
firms in industries with at least two firms) is still truth telling, and the
standards set for each industry are \textquotedblleft
close\textquotedblright\ to the first-best, complete information, ones.

A stronger version of our assumption that there are at least two firms with
identical abatement cost functions has been used in the theory of yardstick
competition. In the seminal Schleifer (1985), for example, the regulation of
a \textit{single} industry is based on the fact that all firms in the
industry have identical cost functions. Similarly, our assumption is related
to the assumption in Varian (1994) that participants in a regulated industry
have common knowledge about each other's costs.

Another, less disputable, assumption that we make is that the regulator can
fine the firms for lying. This is consistent with the practice of pollution
regulators worldwide. In Uruguay, for example, as a consequence of
``forgery'' in the cost declaration, the person in charge of filling the
reports about the abatement technology can be imprisoned. Another potential
punishment is the temporary closing of the plant. Similar practices are
common elsewhere. It is worth emphasizing that for our mechanism to work,
the fine can be arbitrarily small. If fines were large, even a small
probability of a false report being uncovered would suffice to make truth
telling a dominant strategy. In our mechanism the fine is used exclusively
for breaking ties.

We also assume that total damages to society are known or can be estimated.
Although this has been the standard assumption in this branch of the
literature (see Kwerel (1977) and Dasgupta et al., 1980) it is quite strong.
As we will argue later, however, our mechanism is robust to whether the
regulator knows total damages exactly, or approximately, or just wants to
set a total level of emissions for the whole economy. The first extension is
relevant if one is able to estimate total damages to society approximately,
and is concerned that the emissions standards will be approximately correct.
We show that is indeed the case: our mechanism still fully implements truth
telling, and if the regulator's estimate of total damages are close to the
true damages, then the emissions standards that result from our mechanism
are close to the ones that would be implemented if the regulator knew
exactly the damages to society and abatement costs. In a second relaxation
of the assumption that the regulator knows damages, we investigate how our
mechanism fares when the regulator does not know, or is not interested in,
damages to society, but rather on achieving a certain level of emissions for
the whole economy. This extension is important because in practice it is
common to proceed in that way. Moreover, the adoption of the Kyoto Protocol
implies that the regulatory agencies must find the most efficient way to
achieve a certain level of emissions for the economy as a whole. We show
that our mechanism can be used to determine the standards which minimize the
total cost to society of complying with, say, the Kyoto level of aggregate
emissions.

In this note we are only concerned with the problem of setting the right
emissions standards. The enforcement of those standards is a different
issue, and we therefore omit its study. Our mechanism does not assume that
there is perfect enforcement, only that higher emissions standards are
better for firms. If there is perfect enforcement, then our mechanism
maximizes total welfare to society. If there isn't, the emissions standards
are the correct ones, but if firms violate the standards, welfare is not
maximized, and the regulator must try to maximize compliance subject to its
enforcement budget (see footnote \ref{cost} for more on this issue).
Therefore, our mechanism separates the problem of setting the right standard
from the problem of enforcing it.

\section{The Model\label{model}}

There are $m$ industries and $n^{i}$, for $i=1,...,m$, firms in industry $i.$
Firms in $I^{1}=\left\{ 1,...,n^{1}\right\} $ are those in industry $1,$
firms in $I^{2}=\left\{ n^{1}+1,...,n^{1}+n^{2}\right\} $ are those in
industry $2$ and $I^{i}$ is the set of firms in industry $i.$ Each industry
has at least $2$ firms.

The total damages to society coming from pollution are a convex and twice
differentiable function $D:\mathbf{R}_{+}\rightarrow \mathbf{R}_{+}$, with $%
D^{\prime }>0,$ $D^{\prime \prime }\geq 0,$ where total damages are given by 
$D\left( X\right) $ and $X$ is the total pollution from every firm in every
industry: 
\begin{equation*}
X^{i}=\sum_{j\in I^{i}}x_{j},i=1,...,m\quad \text{and\quad }%
X=\sum_{1}^{m}X^{i}
\end{equation*}%
As is standard in this branch of the literature, we make the strong
assumption that the regulator knows or is able to estimate $D(X),$ but we
relax this assumption in Section \ref{unknown}. This simplification is aimed
at focusing on the problems that arise due to the asymmetric information
between the regulator and the firms. Also, this definition of damages also
assumes that what matters is the total level of pollution, and not its
geographic distribution. Although this assumption is not essential for our
mechanism to work, it can be justified on the grounds that the pollutant to
be regulated is \textquotedblleft uniformly mixed\textquotedblright\ in the
sense that only the amounts emitted are relevant, and not their place of
generation.\footnote{%
Less importantly, it is the standard assumption in this strand of the
literature.}

Let $\mathcal{C}$ be the set of all functions $c$ such that $c^{\prime
}\left( x\right) $ is negative, strictly increasing and for all $x$%
\begin{equation}
D^{\prime }\left( x\right) +c^{\prime }\left( 0\right) <0.\footnote{%
This assumption rules out the possibility that firms declare a cost function
that would make the optimal standard for that industry equal to $0.$ Since $%
D^{\prime }$ can be bounded, it does not require that $c^{\prime }\left(
0\right) =\infty $ for all admissible $c^{\prime }.$ This assumption is
reasonable for regulation of industries or processes that are already
functioning, since it just reflects the fact that regulators have chosen not
to prohibit those industries or processes.}  \label{interior}
\end{equation}%
Each firm in industry $i$ can abate its pollution level using an abatement
technology which has a cost of $c^{i}\left( \cdot \right) \in \mathcal{C}.$
That is, $c^{i}\left( x_{j}\right) $ for firm $j$ polluting a level $x_{j}$
in industry $i$ is the difference in profits from (a) not engaging in
abatement, and (b) abating its potential pollution to level $x_{j}.\footnote{%
\label{cost}If $c^{i}\left( x_{j}\right) $ is interpreted as the cost of
abating pollution to $x_{j},$ one is implicitly assuming that there is
perfect enforcement, and therefore our mechanism will maximize total
welfare. If $c^{i}\left( x_{j}\right) $ is interpreted as the cost of having
a standard of $x_{j},$ one is not assuming perfect enforcement, only that
higher standards are better. In that case, our mechanism sets the right
standard, but eschews the issue of whether they will be enforced.}$ Note
that all firms in each industry have the same cost function and we will
assume that this is common knowledge.

Before continuing with the presentation of the model, we remark that the
assumptions made so far about $D$ and the set of possible cost functions of
the firms are the same as the ones that have been used in the papers most
related to this. In particular, Kwerel (1977) and Dasgupta et al (1980) both
assume known damages. Kwerel also assumes convex differentiable $D$ and $c$%
's, and an analogue of (\ref{interior}). Dasgupta et al. do not assume
convexity, but do assume that there exists a unique minimum for the problem
of the regulator, which is all we use of the convexity conditions and
equation (\ref{interior}). Therefore, our assumptions so far are equivalent
to the ones in the relevant literature.

The cost function $c_{j}^{i}$ of firm $j$ in industry $i$ is unknown to the
regulator and to firms in industries other than $i.$ They only know that $%
c^{i}\in \mathcal{C}$ for $i=1,...,m$ and that the profile $c=\left(
c^{1},c^{2},...,c^{m}\right) $ is drawn from $\mathcal{C}^{m}$ using some
probability distribution $P$ which is common knowledge. In the mechanism of
this paper, the regulator asks firms to report their cost functions. In
spite of the informational asymmetry, the regulator can inspect one firm.
With probability $\varepsilon >0$ he finds out whether the report was
truthful or not, with probability $1-\varepsilon $ the inspection is
inconclusive. In case the regulator discovers that the report was not true,
he does not find out the true $c^{i},$ but only that the report was false.

In this context, a social choice function is a function $f:\mathcal{C}%
^{m}\rightarrow \mathbf{R}_{+}^{m}$ that specifies for each possible profile
of cost functions (one for each industry) the pollution level that each firm
must produce. The regulator wishes to implement the social choice function
that minimizes the total cost of pollution. Technically, given our convexity
assumptions, $f$ is the function $f:\mathcal{C}^{m}\rightarrow \mathbf{R}%
_{+}^{m}$ defined by 
\begin{equation}
f\left( c\right) =\arg \min_{\left( x^{1},...,x^{m}\right) }\left[ D\left(
\sum n^{i}x^{i}\right) +\sum n^{i}c^{i}\left( x^{i}\right) \right] ,
\label{f}
\end{equation}%
for all $c=\left( c^{1},...,c^{m}\right) \in \mathcal{C}^{m},$ where $x^{i}$
is the standard set for industry $i,$ with which all firms in the industry
must comply. As was argued earlier, the only role of our convexity
assumptions is to make the arg min in equation (\ref{f}) unique.

When $c$ in equation (\ref{f}) is the true profile of cost functions, the
function $f$ yields the first best emission levels: the emission levels that
the regulator would choose if he knew the true cost functions. In this paper
we will show that our mechanism allows the regulator to find out the true
profile of cost functions $c,$ and therefore find the first best emission
levels. We will \textit{not}, however, deal with the problem of finding the
best allocations for the whole economy, when firms pay to consumers the
damage caused. In the problem of finding this optimal allocation when firms
have to pay the damage caused, some polluting firms could be forced to close
down due to losses. This difference is relevant because, among other things,
regulatory agencies in some countries care about the impact of their
regulation on the probability of inducing firms to close down. Nevertheless,
our take on this problem is the standard one in the literature on
Environmental Economics (including the papers most related to ours).

It is also worth noting that since our model is static, and we do not
include a player that can enforce collusive agreements (as is sometimes done
in static collusion games), we are eschewing the problem of collusion among
firms. In our static model, the unique equilibrium is truth telling, but if
the game of \textquotedblleft standard setting\textquotedblright\ were
repeated an infinite number of times, other equilibria (including a
collusive outcome in which firms claim high abatement costs) could arise.
Since collusion is a widespread problem, it is a drawback of our model. But
because we lack a decent theory of equilibrium selection for infinitely
repeated games, the same can be said of any static mechanism. Therefore, if
collusion is strongly suspected in the regulation of some pollutant (if
there are few firms, for example) the best alternative may be the method
that has been used the most in the past: estimation of cost functions by the
regulator.

\section{The Mechanism and the Theorem\label{mechanism}}

We now present our mechanism, and then show that it fully implements $f.$
That is, we will show that in the unique equilibrium of the game designed by
the regulator, firms truthfully disclose their cost functions.

For our direct revelation mechanism, firms must announce their cost
functions, and thereby, the cost function of the industry (since the
announcement can in principle depend on the true cost function, a strategy
is a mapping from true cost functions to announcements). For each profile of
announcements $C=\left( C^{1},...,C^{m}\right) ,$ $C^{i}$ will represent the
profile of announcements of firms in industry $i,$ so that 
\begin{equation}
C=\left( C^{1},...,C^{m}\right) =\left( \underset{\text{industry 1}}{%
\underbrace{c^{1},...,c^{n_{1}}}},\underset{\text{industry 2}}{\underbrace{%
c^{n_{1}},...,c^{n_{1}+n_{2}}}}%
,c^{n_{1}+n_{2}+1},...,c^{n_{1}+n_{2}+...+n_{m}}\right) .  \label{C}
\end{equation}%
For each profile $C$ let 
\begin{equation}
x_{j}^{1}=\min \left\{ f_{1}\left( c^{j},c^{p_{2}},...,c^{p_{m}}\right)
:p_{i}\in I^{i},i=2,...,m\right\} .  \label{x}
\end{equation}%
\qquad \qquad \qquad

The number $x_{j}^{1}$ is the emissions standard that would result for
industry $1$ if the regulator believed the announcement of firm $j$ in this
industry$,$ and chose the announcement of a firm $p_{i}$ in each remaining
industry $i\neq j$ which would result in the most stringent standard for
industry $1.$ A firm with a low $x_{j}^{1}$ is most likely telling the
truth, since it is announcing a cost function that could result in a harsh
environmental policy. Similarly, define $x_{j}^{i}$ for $i=2,...,m$ and $%
j\in I^{i}$ to be the standard that would be implemented for industry $i$ if
the regulator believed the announcement of firm $j$ in that industry. Also,
define 
\begin{equation}
\underline{x}^{i}=\min_{j\in I^{i}}x_{j}^{i}\quad \quad \text{and\quad \quad 
}\overline{x}^{i}=\max_{j\in I^{i}}x_{j}^{i}  \label{xbar}
\end{equation}%
to be, respectively, the most (least) stringent standard consistent with the
announcements of firms in industry $i.$

Our mechanism is as follows:

\begin{enumerate}
\item Firms announce their types

\item If in industry $i$ announcements coincide, the regulator samples
randomly one of the firms and inspects it. If the announcements do not all
coincide, the regulator: identifies the firms, or firm, which announced the
cost functions which are consistent with $\overline{x}^{i};$ randomly
selects one of them and inspects this firm with probability $\pi >\left(
n^{i}-1\right) /n^{i},$ and some other firm with probability $1-\pi .$ The
idea is to monitor firms which are most likely lying with a larger
probability. A firm is fined if and only if: its report is false; it is
inspected and the inspection discovers (with probability $\varepsilon $)
that the report was false. The size of the fine does not matter, it can be
as small as one wants.

\item The emissions standards $\left( \underline{x}^{1},...,\underline{x}%
^{m}\right) $ are implemented.\bigskip
\end{enumerate}

A strategy for a firm in the game that this mechanism defines is a
continuous function $s:\mathcal{C}\rightarrow \mathcal{C}$ that announces a
cost function for each possible type (real cost function) of the firm. We
now present our main result.\bigskip

\noindent \textbf{\textit{Theorem 1. }}\textit{The efficient (first best,
full information) social choice function }$f$ \textit{defined by equation (%
\ref{f}) is fully implementable. That is, the unique equilibrium of the
direct revelation mechanism, is truth telling.}\bigskip

The proof is in the appendix.\bigskip

\noindent \textbf{\textit{Remark 1. }}\textit{It is worth emphasizing that
our mechanism has a unique equilibrium. Using assumptions similar to ours
(except for auditing and two firms per industry) Kwerel (1977) and Dasgupta
et al. (1980) study the issue of whether truth telling is an equilibrium.%
\footnote{%
In Dasgupta et al., when the budget is not required to be balanced, their
version of the Groves-Clarke mechanism is implementable in dominant
strategies. If one requires balanced budget, as we do, they only obtain that
truth telling is an equilibrium. In Kwerel the budget is not balanced 
\textit{and} uniqueness does not obtain.} The issue of uniqueness and
whether truth telling is the only equilibrium is not analyzed. As has been
argued in the literature on mechanism design the issue of multiplicity is
very relevant, especially if the equilibria arising from the game can be
Pareto ranked, and it becomes focal to lie. See Moore (1992) p. 186, fn. 5
and the references therein.}\bigskip

\noindent \textbf{\textit{Remark 2. }}\textit{The central idea of the proof
is very simple (which may also help in the actual implementation of the
mechanism). First, if one firm is telling the truth, it is optimal for all
to tell the truth, since they will relax the standard, and reduce the
probability of a fine (to 0). That shows that truth telling is an
equilibrium. Second, if all firms are lying, one will have an expected
probability of inspection which is larger than the rest. That firm has an
incentive to slightly undercut any other fixed firm in its announcement,
since it will strictly reduce the chance of an inspection and change the
standard only slightly.}\bigskip

From Remark 2 one can see that if one is willing to use a stronger
equilibrium concept, in particular, trembling hand perfection, then one can
simplify the mechanism even further by eliminating the probability of the
inspection of the firms which are not declaring $\overline{x}^{i}$. That
inspection is used to get rid of equilibria in which in some industry one or
more firms declare the truth, and two or more firms declare cost functions
which yield standards that are more stringent than the ones corresponding to
the truth. Without these inspections, those firms have no incentive to
deviate, since the standard will be very low (stringent) even if they
declare the truth, and they are not fined if they lie. If one took a finite
type space, and all firms mixed on all of their types, there would always be
a chance that a firm would be the only one declaring the cost function
consistent with the low standard, and it would then be a best response to
play the truth.\footnote{%
The size of the type space would have to be fairly large for the small $%
\varepsilon $ and small fine to be enough incentive for firms to undercut
each other.}

From a practical point of view, the application of the mechanism as it may
present two difficulties. First, the type space may be too large, and it may
be too hard for firms to estimate exactly which is its cost function.
Second, and related to the previous point, two firms \textquotedblleft
trying\textquotedblright\ to declare the truth may not declare the exact
same cost function, and it would not make much sense to punish them in that
case. A solution to both of these problems is to present the firms with a
fairly large (but finite) menu of cost functions that can be declared, and
the authority deems the statement to be true if it is close enough to the
truth (the inspection, instead of declaring truth or not, would declare
whether the statement is close to the truth or not, which is even easier for
the regulator).\footnote{%
One choice of a finite type space for which the unique equilibrium is
telling the closest \textquotedblleft declarable type\textquotedblright\ to
the truth is the following. Partition the interval $\left[ 0,M\right] ,$ for
large $M$ into intervals of length $1.$ The menu of cost functions that can
be declared is that of costs which have constant derivative in those
intervals, and the derivative is a multiple of $1/K$ (for large $K$). The
mechanism is the same, only that the regulator declares a firm to be lying
if its declaration is not close enough to the truth (with the metric of the
supremum).}

It is worth noting that our mechanism can also be used to elicit the optimal
Pigouvian taxes. We do not pursue this route here, but only sketch how the
mechanism would work. In that case, the regulator still wishes to implement $%
f$ from equation (\ref{f}). If he knew the true cost functions $c,$ he would
calculate $x=\left( x^{1},...,x^{m}\right) ,$ then he would set $t=D^{\prime
}\left( \sum n^{i}x^{i}\right) .$ Then, the firms' problem in industry $i$
would be to choose $x$ to minimize $c\left( x\right) +tx,$ so that the
optimal emission would be characterized by 
\begin{equation*}
-c^{i\prime }\left( x\right) =t=D^{\prime }\left( \sum n^{i}x^{i}\right) .
\end{equation*}%
Since this is the first order condition of the regulator's problem when
choosing the optimal $x,$ we see that the firm's problem yields the first
best levels of pollution.

In order to use our mechanism to implement $f$ via taxes, the regulator
would calculate (as before) for each firm $j$ in industry $i,$ the highest
tax rate consistent with the firm's declaration. Then, he would calculate
for each industry $i,$ $\overline{t}^{i}=\max t_{j}^{i}$ and $\underline{t}%
^{i}=\min t_{j}^{i}.$ Then, in the mechanism, $\overline{t}^{i}$ would play
the role of $\underline{x}^{i}$ and $\underline{t}^{i}$ the role of $%
\overline{x}^{i}.$

\section{Different Assumptions\label{dif}}

In this Section we briefly discuss three variants of our assumptions and of
the mechanism that still fully implement truth telling. This is relevant
since the institutional settings may vary from country to country, making
some versions impossible to implement, while rendering others feasible.
These extensions are simple applications of the main idea behind our
Theorem, and this simplicity just illustrates how powerful our basic
mechanism is.

Before turning to the variations of the model, we note that our two main
assumptions are necessary for the mechanism to fully implement truthtelling.
If the regulator had no way of finding out whether the firms are lying, the
following would be an equilibrium. Suppose there is a maximum potential
pollution level in each industry $\overline{X}_{i}$ (when firms do not
engage in abatement), and that firms in industry $i$ report a cost function $%
c_{i}$ such that 
\begin{equation*}
D^{\prime }\left( \sum n_{i}\overline{X}_{i}\right) =-c_{i}^{\prime }\left( 
\overline{X}_{i}\right)
\end{equation*}%
and the regulator then sets the non binding standard $\overline{X}_{i}$ in
industry $i.$ Also, note that even if the regulator could find out whether a
report was true with probability $\varepsilon ,$ as in our model, if a firm
were alone in the industry, it would maximize profits by declaring a cost
function that yields $\overline{X}_{i}$ as its standard, provided $%
\varepsilon $ and the fine are sufficiently small.

There are several variants of the mechanism that also yield truthtelling as
the unique equilibrium. Here we analyze two. The first variant is concerned
with our main assumption: that there are at least two firms in each
industry. The second analyzes the case where damages to society are unknown,
or there is no interest in determining them.

\subsection{Industries with one firm.}

Suppose industries $1$ through $k$ have just one firm, $%
n^{1}=n^{2}=...=n^{k}=1,$ and that industries $k+1$ through $k+m$ have at
least two firms, as has been our assumption so far. As before, we let $I^{i}$
be the set of indexes of firms in industry $i,$ even for industries with $1$
firm. Again, the regulator wishes to implement the social choice function
that minimizes the net cost of pollution. Technically, he wishes to
implement the function $f:\mathcal{C}^{k+m}\rightarrow \mathbf{R}_{+}^{k+m}$
defined by 
\begin{equation}
f\left( c\right) =\arg \min_{\left( x^{1},...,x^{k+m}\right) }\left[ D\left(
\sum n^{i}x^{i}\right) +\sum n^{i}c^{i}\left( x^{i}\right) \right] ,
\label{monopoly}
\end{equation}%
for all $c=\left( c^{1},...,c^{k+m}\right) \in \mathcal{C}^{k+m}$. We endow $%
\mathcal{C}$ with the sup norm.

Suppose that the regulator can estimate, not necessarily exactly, the cost
functions of industries $1$ through $k$ and call $\widehat{c}^{i}$ those
estimates. As before, the regulator will ask firms in industries $k+1$
through $k+m$ to report their cost structures. For each profile of
announcements, 
\begin{equation*}
C=\left( C^{k+1},...,C^{k+m}\right) =\left( \underset{\text{industry }k+1}{%
\underbrace{c^{k+1},...,c^{k+n^{k+1}}}}%
,...,c^{k+n^{k+1}+n^{k+2}+...+n^{k+m}}\right)
\end{equation*}%
let 
\begin{equation*}
\widehat{x}^{1}=\min \left\{ f_{1}\left( \widehat{c}^{1},...,\widehat{c}%
^{k},c^{p_{1}},...,c^{p_{m}}\right) :p_{i}\in I^{k+i},i=1,...,m\right\}
\end{equation*}%
and similarly for industries $2,...,k.$ As before, define 
\begin{equation*}
x_{j}^{k+1}=\min \left\{ f_{k+1}\left( \widehat{c}_{1},...,\widehat{c}%
_{k},c^{j},...,c^{p_{m}}\right) :p_{i}\in I_{k+i},i=2,...,m\right\}
\end{equation*}%
and similarly for industries $k+2$ through $k+m.$ The definitions of $%
\overline{x}^{i}$ and $\underline{x}^{i}$ are as before, from equation (\ref%
{xbar}).

Consider the following mechanism:

\begin{enumerate}
\item The regulator estimates a cost function $\widehat{c}^{i}$ for firms in
industries $i=1,...,k.$

\item Firms in industries $k+1$ through $k+m$ announce their types

\item If in industry $i=k+1,...,k+m$ announcements coincide, the regulator
samples randomly one of the firms and inspects it. If the announcements do
not all coincide, the regulator: identifies the firms, or firm, which
announced the cost functions which are consistent with $\overline{x}^{i};$
randomly selects one of them and inspects this firm with probability $\pi
>\left( n^{i}-1\right) /n^{i},$ and some other firm with probability $1-\pi
. $ A firm is fined if and only if: it is sampled; its report is false; the
inspection discovers (with probability $\varepsilon $) that the report was
false..

\item The emissions standards $\left( \widehat{x}^{1},...,\widehat{x}^{k},%
\underline{x}^{k+1},...,\underline{x}^{k+m}\right) $ are implemented.\bigskip
\end{enumerate}

\noindent \textbf{\textit{Theorem 2. }}\textit{For any estimates }$\left( 
\widehat{c}^{1},...,\widehat{c}^{k}\right) $ \textit{the unique equilibrium
of the direct revelation mechanism, is truth telling. Moreover, the
standards }$\left( \widehat{x}^{1},...,\widehat{x}^{k},\underline{x}%
^{k+1},...,\underline{x}^{k+m}\right) $ \textit{are continuous in }$\left( 
\widehat{c}^{1},...,\widehat{c}^{k}\right) $\textit{\ so that if the
estimated }$\left( \widehat{c}^{1},...,\widehat{c}^{k}\right) $\textit{\ are
close to the truth, the standards in all industries will be close to the
first best standards.}\bigskip

\noindent \textbf{Proof}. The proof that the unique equilibrium is truth
telling mirrors exactly the proof of Theorem 1, and is therefore omitted.

Continuity of the standards follows from applying Berge's Maximum Theorem
(see Aliprantis and Border (1999), p. 539) to $F\left( c\right) $ in
equation (\ref{monopoly}): when $\left( c^{k+1},...,c^{k+m}\right) $ are
fixed in their true levels, 
\begin{equation}
D\left( \sum n^{i}x^{i}\right) +\sum_{1}^{k}\widehat{c}^{i}\left(
x^{i}\right) +\sum_{k+1}^{k+m}n^{i}c^{i}\left( x^{i}\right)  \label{costs}
\end{equation}%
is a function of $\left( \widehat{c}^{1},...,\widehat{c}^{k}\right) $ and $%
x=\left( x^{1},...,x^{k+m}\right) .$ Then, the set $x\left( \widehat{c}%
\right) $ of minimizers of (\ref{costs}) is upper hemicontinuous, and
therefore continuous, as was to be shown.\qed

\subsection{Unknown Damages\label{unknown}}

In this section we consider two extensions to our basic model that address
the question of whether our mechanism works when either $D$ is unknown, or
irrelevant.

Suppose first that the regulator is able to estimate $D.$ Then, as in the
previous section, we have that the mechanism works, and that if the estimate
of $D$ is accurate, the emissions standards will be close to the complete
information ones.\bigskip

\noindent \textbf{\textit{Theorem 3. }}\textit{For any estimate }$\widehat{D}
$ \textit{the unique equilibrium of the direct revelation mechanism of
Section \ref{mechanism}, is truth telling. Moreover, the standards are
continuous in }$\widehat{D}$\textit{\ so that if the estimated }$\widehat{D}$%
\textit{\ is close to the truth, the standards in all industries will be
close to the first best standards.}\bigskip

The proof of Theorem 3 is similar to that of Theorem 2, and is therefore
omitted.

Another extension of the model that is relevant is one in which total
damages to society are irrelevant. Consider the case of a country that wants
to achieve a certain level of pollution $\overline{X}$ in the most efficient
way. This could be the case, for example, of countries that adopted the
Kyoto Protocol: they have committed to achieving by 2012 a certain level of
emissions. Europe, for instance, must abate its 1990 levels of green house
gases by 8\%. The problem of the regulator is therefore to find the
standards for each industry that minimize the total costs of abatement, and
that achieve the desired level of emissions. Formally, suppose that the
Kyoto standard is $\overline{X},$ and let 
\begin{equation*}
\Gamma \left( \overline{X}\right) =\left\{ \left( x^{1},...,x^{m}\right)
:\sum n^{i}x^{i}\leq \overline{X}\right\} .
\end{equation*}%
Then, the regulator wants to implement $f$ from 
\begin{equation*}
f\left( c\right) =\arg \min_{\left( x^{1},...,x^{m}\right) \in \Gamma \left( 
\overline{X}\right) }\sum n^{i}c^{i}\left( x^{i}\right) .
\end{equation*}%
We have that our mechanism still implements truth telling, and this results
in the complete information standards for this problem.\bigskip

\noindent \textbf{\textit{Theorem 4. }}\textit{For any }$\overline{X}$ 
\textit{the unique equilibrium of the direct revelation mechanism of Section %
\ref{mechanism}, is truth telling.}\bigskip

The proof is identical to that of Theorem 1, and is therefore omitted.

\section{On the Novelty of Our Theorems\label{novelty}}

We believe that the main merit of our results is their applicability given
the simplicity of the mechanism and of the proof, which makes it
\textquotedblleft likely\textquotedblright\ that players will understand
their incentives.\footnote{%
We thank Matt Jackson for many of the references in this Section, and for
his comments regarding the importance of the simplicity of the mechanism and
the proof.} In particular, we do not use some of the standard techniques,
like cross reporting, used in the literature on implementation with complete
information. Nevertheless, in this section we argue that our results are
new, and discuss the relationship with the literature on mechanism design.

First, our results do not follow from any of the existing theorems in the
literature. That is, there is no theorem that ensures that the social choice
correspondence defined by equation \ref{f}, or any selection from it, is
fully implementable in Nash equilibrium. The results in Jackson, Palfrey and
Srivastava (1994) do not apply either to our mechanism, or to the simpler
version in which there is only one industry and two firms. Most importantly,
their theorems are for implementation in undominated Nash, and our results
are full Nash implementation (we get uniqueness without requiring that the
strategies be undominated). Moreover, their Theorem 1 is for three or more
firms, and their Theorem 3 requires the existence of a \textquotedblleft
worse outcome\textquotedblright\ that is not present in our setup.\footnote{%
A worse outcome in that setting would be a standard of $0$ and for each firm
a lottery which yields the fine with probabilty $\varepsilon .$ We do not
need to include such an outcome in our space of allocations for our
mechanism to work. Our mechanism inspects only one firm.}

Second, although inspections and fines have been used in the past and it is
\textquotedblleft known\textquotedblright\ that they help in the
implementation problem, our assumptions are weaker and different than the
ones that have been used before. For example, the important works of
Mookherjee and P'ng (1989) and Ortu\~{n}o-Ortin and Roemer (1993) used
costly but perfectly informative inspections and sizeable fines. Our
inspections can be as uninformative as one wants, and the fines can be
arbitrarily small. Arya and Glover (2005) use a public signal that may be
only slightly correlated with the player's reports to implement truth
telling (to the owner of a firm) by a manager and his auditor. In their
model, however, fines for lying can be large.

Finally, our results are not subject to the criticisms to full
implementation in complete information that have been raised by Chung and
Ely (2003), since our setup is, in their terminology, one of
\textquotedblleft private values\textquotedblright .

\section{Summary}

We have presented a mechanism that may help in solving the important problem
of how to get polluters to tell the truth about their abatement costs. Our
solution is simple, shares some features of how the actual regulatory
process works in the US and other places, it implements truth telling by
firms and the efficient level of pollution. Also, we have argued that one of
the reasons why one does not observe in practice alternative mechanisms that
have been proposed in the literature is because they were complicated and
relied on taxes and subsidies, which may be too difficult to implement for
regulators.

Our main assumption is that there are at least two firms in each industry.
We have argued that this is a reasonable assumption, and we have shown how
our mechanism can still be used even when that assumption is not satisfied.

\section{Appendix}

\noindent \textbf{Proof of Theorem 1. Truth Telling is an Equilibrium}. We
first show that truth telling is an equilibrium. Without loss of generality,
consider the situation of firm $1$ when all other firms in all industries
are reporting the true costs $\left( c^{1},c^{2},...,c^{m}\right) $. Notice
that declaring the true $c^{1}$ leads to the implementation of $%
x_{2}^{1}=...=x_{n^{1}}^{1}$ consistent with all the declarations of firms $%
2 $ through $n^{1}.$ If firm $1$ reports $\widehat{c}^{1}\neq c^{1},$ two
things could happen, depending on the profile of types announced by
industries $2,...,m$:

\begin{itemize}
\item $x_{1}^{1}\geq x_{j}^{1}$ for all $j=2,...,n^{1}.$ In this case the
same standard is implemented in industry $1,$ and the firm could be fined.

\item $x_{1}^{1}<x_{j}^{1}$ for all $j=2,...,n^{1}.$ In this case, a harsher
standard is implemented for industry $1$.
\end{itemize}

Since, no matter what is the profile of types announced in the other
industries, firm $1$ is worse off deviating, and hence, declaring the truth
is better than declaring anything else, proving that truth telling is an
equilibrium.\bigskip

\noindent \textbf{There is no other equilibrium. }Suppose there is a profile
of strategies $s=\left(
s_{1}^{1},...,s_{n^{1}}^{1},...,s_{n^{1}+...+n^{m}}^{m}\right) $ such that
for some industry $i$ and firm $j$, $s_{j}^{i}\left( c_{lie}^{i}\right) \neq
c_{lie}^{i}$ for some $c_{lie}^{i}\in \mathcal{C}$ in the support of $P^{i}$
(the probability distribution over industry $i$'s types induced by $P$) and
suppose it is an equilibrium. That is, suppose there is an equilibrium
without truth telling. Without loss of generality, suppose $i=1.$ Then, for
any state $c=\left( c_{lie}^{1},...,c^{m}\right) $ the profile of
announcements $C=\left( C^{1},C^{2},...,C^{m}\right) $ (see equation (\ref{C}%
)) is such that not all firms in industry $1$ are telling the truth. Notice
that \textit{all} announcement in $C^{1}$ are lies, since if one firm were
telling the truth all firms would be strictly better off telling the truth,
since (relative to lying) they would weakly increase the standard for the
industry, and strictly reduce the chance of being fined (to $0$). Since one
firm is inspected the average chance of a firm being inspected is $1/n^{1}.$
Take any firm that in state $c_{lie}^{1}$ has a probability $p$ (depending
on other firms' declarations) of being inspected that is (weakly) larger
than $1/n^{_{1}}.$ Suppose it is firm $1.$ We will show that firm $1$ is
strictly better off slightly undercutting the announcement of firm $2$ (or
any other firm in industry $1$): firm $1$ will never be one of the firms
announcing a cost function consistent with $\overline{x}^{1}$ (the ones
inspected with higher probability) and therefore will strictly reduce its
chance of being inspected and fined.

The standards claimed by firms $2,...,n^{1}$ in industry $1$, $\left\{
x_{2}^{1},x_{3}^{1},...,x_{n^{1}}^{1}\right\} $ depend on: the strategies $%
s_{j}^{i}\left( \cdot \right) $ of the firms $j$ in other industries $i$ and
on their types $c^{i}$ (true cost functions)$.$ If, given $s_{j}^{i}$ for
firms $j$ in other industries $i$, firm $1$ in industry $1$ can ensure that
for all\ of the types of the other industries $x_{1}^{1}$ will be strictly
(but slightly) smaller than $x_{2}^{1}$ then the probability of inspection
when $c_{lie}^{1}$ happens will be $\left( 1-\pi \right) /\left(
n^{1}-1\right) $. That way, it reduces the standard only slightly, but is
sampled with a probability of%
\begin{equation*}
\frac{1-\pi }{n^{1}-1}<\frac{1}{n^{1}}\leq p
\end{equation*}%
which corresponds to the probability of being inspected for a firm that
claimed a standard different from the maximum. Since firm $1$ was lying, it
will be strictly better off with that deviation (by strictly reducing the
chance of being inspected and fined). To establish the existence of such a
deviation, notice that for the declaration $\widehat{c}_{2}\equiv
s_{2}^{1}\left( c_{lie}^{1}\right) $ of firm $2,$ firm $1$ can always
declare a $\widetilde{c}$ defined by 
\begin{equation}
\widetilde{c}^{\prime }\left( x\right) =\delta \widehat{c}_{2}^{\prime
}\left( x\right)  \label{deviation}
\end{equation}%
for $\delta <1,$ close enough to $1.$ Such a $\widetilde{c}$ is in $\mathcal{%
C}$ (it has a negative and strictly increasing first derivative) so we now
show that it yields a strictly higher utility for firm $1$ by showing that
any profile of declarations of cost functions of firms in the other
industries $C^{-1}=\left( C^{2},...,C^{m}\right) $ induces a standard $%
x_{1}^{1}$ smaller than $x_{2}^{1}$ for industry $1.$ Hence for this type of
the other industries the probability of inspection for firm $1$ is $\left(
1-\pi \right) /\left( n^{1}-1\right) $. Since the type was arbitrary the
overall probability of inspection will be $\left( 1-\pi \right) /\left(
n^{1}-1\right) .$

We will now show that for $\widehat{c}_{2}\equiv s_{2}^{1}\left(
c_{lie}^{1}\right) ,$ the deviation defined by $\widetilde{c}^{\prime
}\left( x\right) =\delta \widehat{c}_{2}^{\prime }\left( x\right) $ (as in
equation \ref{deviation}) induces a standard $x_{1}^{1}$ for industry $1$
that is smaller than $x_{2}^{1}$ $.$ Fix the profile of types $c=\left(
c_{lie}^{1},...,c^{m}\right) $ and given the strategies of firms in other
industries, fix the selection $\left( c_{\ast }^{2},...,c_{\ast }^{m}\right) 
$ from $\left( C^{2},...,C^{m}\right) $ that achieves $x_{1}^{1}$. That is: $%
C^{i}$ contains all the announcements of firms in industry $i,$ and, in
order to calculate $x_{1}^{1}$ the regulator chooses the profile of
announcements (one cost function per industry $i\neq 1$) that makes the
standard $f_{1}\left( \widetilde{c},c_{\ast }^{2},...,c_{\ast }^{m}\right) $
as small as possible. Since $D$ and all $c$s are differentiable, and $%
D^{\prime }\left( x\right) +c^{\prime }\left( 0\right) <0$ for all $c,$ the
solution of the regulator's problem is interior, and hence the first order
condition of the problem of the regulator is satisfied (when calculating $%
x_{1}^{1}$):%
\begin{equation}
D^{\prime }\left( \sum_{i>1}n^{i}x^{i}+n^{1}x_{1}^{1}\right) =-\widetilde{c}%
^{\prime }\left( x_{1}^{1}\right) =-c_{\ast }^{2\prime }\left( x^{2}\right)
=...=-c_{\ast }^{m\prime }\left( x^{m}\right) .  \label{foc}
\end{equation}%
If we had that $f_{1}\left( \widehat{c}_{2},c_{\ast }^{2},...,c_{\ast
}^{m}\right) ,$ the standard claimed by firm $2$ in industry $1,$ when the
regulator selects $\left( c_{\ast }^{2},...,c_{\ast }^{m}\right) ,$ was such
that $x_{1}^{1}\geq f_{1}\left( \widehat{c}_{2},c_{\ast }^{2},...,c_{\ast
}^{m}\right) =\widehat{x}^{1},$ we would have that $f\left( \widehat{c}%
_{2},c_{\ast }^{2},...,c_{\ast }^{m}\right) =\widehat{x}$ would satisfy%
\begin{equation}
D^{\prime }\left( \sum_{i>1}n^{i}\widehat{x}^{i}+n^{1}\widehat{x}^{1}\right)
=-\widehat{c}_{2}^{\prime }\left( \widehat{x}^{1}\right) =-c_{\ast
}^{2\prime }\left( \widehat{x}^{2}\right) =...=-c_{\ast }^{m\prime }\left( 
\widehat{x}^{m}\right) .  \label{foc2}
\end{equation}%
Then, $-\widehat{c}_{2}^{\prime }\left( \widehat{x}^{1}\right) >-\widetilde{c%
}^{\prime }\left( \widehat{x}^{1}\right) \geq -\widetilde{c}^{\prime }\left(
x_{1}^{1}\right) $ and equations (\ref{foc}) and (\ref{foc2}) imply $%
-c_{\ast }^{i\prime }\left( x^{i}\right) <-c_{\ast }^{i\prime }\left( 
\widehat{x}^{i}\right) $ for all $i\neq 1,$ or equivalently, $x^{i}>\widehat{%
x}^{i}.$ Since $D^{\prime }$ is weakly increasing, this in turn means that 
\begin{equation*}
-\widetilde{c}^{\prime }\left( x_{1}^{1}\right) =D^{\prime }\left(
\sum_{i>1}n^{i}x^{i}+n^{1}x_{1}^{1}\right) \geq D^{\prime }\left(
\sum_{i>1}n^{i}\widehat{x}^{i}+n^{1}\widehat{x}^{1}\right) =-\widehat{c}%
_{2}^{\prime }\left( \widehat{x}^{1}\right)
\end{equation*}%
which is a contradiction. We conclude that $\widehat{x}^{1}>x_{1}^{1}.$
Then, $x_{2}^{1}\geq \widehat{x}^{1}>x_{1}^{1}$ means that firm $1$ will be
inspected with probability $\left( 1-\pi \right) /\left( n^{1}-1\right) <p.$
Moreover, by the Theorem of the Maximum, and uniqueness of the optimal $x$
(a consequence of the strict convexity of the $c$s) the new standard $%
x_{1}^{1}$ claimed by firm $1$ when declaring $\widetilde{c}$ will be close
to the standard claimed by firm $2$, and this is true for any arbitrary
profile of types $c=\left( c_{lie}^{1},...,c^{m}\right) .$ As we vary $%
\left( c^{2},...,c^{m}\right) ,$ and compare the overall standards $%
\underline{x}^{1}=\min_{j\in I^{1}}x_{j}^{1}$ resulting from the proposed
equilibrium announcement $s_{1}^{1}\left( c_{lie}^{1}\right) $ of firm $1$
and from the deviation $\widetilde{c}$, we see that three things can happen:
the standard is unchanged, it increases (when $\underline{x}^{1}=x_{1}^{1}$
for the proposed declaration $s_{1}^{1}\left( c_{lie}^{1}\right) $ of firm $%
1 $), or it slightly decreases. The utility of this proposed deviation can
be made arbitrarily close to $0,$ or even positive, as one makes $\delta $
close to $1.$ Since the change in the probability of inspection is discrete,
this proves that for $c_{lie}^{1}$ firm $1$ is strictly better off deviating.

Finally, since strategies are continuous, the same analysis can be conducted
for types $c^{1}$ close to $c_{lie}^{1}.$ Since $c_{lie}^{1}$ is in the
support of the distribution of types for industry $1,$ there is a positive
probability of types for which firm $1$ is strictly better deviating from
the proposed equilibrium, and that is a contradiction.\qed

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\end{document}