Abstract
Bureaucratic shirking and corruption are prevalent in developing countries. This paper presents a delegation model where a government authorizes an inspector to monitor a polluting firm. The inspector may shirk in monitoring and may accept bribery when discovering noncompliance. We distinguish between two types of environmentally friendly actions, emission abatement and investment to enhance abatement technology, and investigate how bureaucratic shirking and corruption affect firms’ incentives of taking these actions. A corruptible inspector exerts more effort in monitoring the firm (an effortinducing effect) but fails to enforce environmental regulations when discovering noncompliance (a nonenforcement effect), compared to an (incorruptible) bureaucratic inspector. Moreover, the firm strategically makes more investment to reduce the corruptible inspector’s monitoring effort (a strategic effect on monitoring). We find that investment and abatement decrease when corruption becomes more widespread, only if the corruptible inspector has sufficiently small bargaining power. Moreover, the corruptible inspector’s higher bargaining power leads to higher investment and abatement.
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Notes
For example, Kireeva (2016) reports that a former official with the Russian government’s environmental watchdog was found to have taken bribes from one of the most polluting companies, which routinely made emissions of sulfur dioxide exceeding the allowable amount.
Gowen (2014) vividly illustrates the bureaucratic idleness in India.
See Sect. 1.1 for a review of the literature.
We assume that there is a bijection between the inspector’s effort and the probability that the inspector discovers the firm’s emission (called monitoring intensity). With this assumption, we will use monitoring effort and monitoring intensity interchangably.
Relatedly, Egger and Winner (2005) find that corruption is a stimulus for industrial investment, from crosscountry data including both developed and developing countries.
The conclusion provides some discussion on the possibility of empirically testing our theoretical implications.
While we do not explicitly model the environmental damage function, a sufficiently large environmental damage caused by a small amount of emission is sufficient to generate a corner solution of zero emissions in the firstbest. Examples include emission of certain toxic chemicals and untreated nuclear waste.
Given the zero emission level in the first best, we assume that emission is illegal and should be fined under the law. However, one can also interpret the fine as a pollution tax without changing the main analysis.
We assume that investment is materialized as tangible physical equipments, which are easily observable.
Alternatively to the assumption that the inspector will discover nothing with probability \(1p\), we can assume that with probability \(1p\), the inspector will procure no evidence of emission, so the firm will not be fined, as in Mookherjee and Png (1995). Moreover, given risk neutrality assumed below, another interpretation of p is that it represents the proportion of unlawful emissions that can be detected by the inspector. These alternative interpretations do not change the analysis.
See also Hicks (2010) for examples in China and India.
We assume that once emission is reported by the inspector, the firm will be fined. However, in some countries, the legal process to impose the fine is so long and complicated that the expected fine is lower than the fine stipulated under the law. To consider this possibility with our model, one can interpret t as the expected fine rate which is the product of the fine rate stipulated in the law by the probability that the fine will be imposed when emission is reported.
We assume, however, that the inspector cannot falsely report emission levels that are greater than what they are. That is, we assume that extortion is not feasible. We thank an anonymous referee for pointing this out.
There are various reasons to believe that the inspector will not deviate from the illicit agreement made with the firm. As explained in Mookherjee and Png (1995), there may be a continuing relationship between the two parties so that the long term cost of reneging outweighs the oneshot gain. Even if the inspector rotates among firms, concern for his reputation will enter: Oneshot defection could preclude the inspector from the possible chance of collecting bribes from other firms. Finally, the firm may resort to physical or other harassment or threaten to make the bribery publicly known if the inspector reneges. Many other studies, e.g., Fredriksson and Svensson (2003), Damania (2002), and Damania et al. (2004), also adopt the assumption that the bribetaker does not renege after accepting bribes.
This is similar to the capacity commitment discussed in the industrial organization literature (e.g. Dixit 1979).
Note that the ambiguity of the effect when \(k<{{\bar{k}}}\left( I^{*}\right)\) critically depends on the timing that investment is made before a and p. If investment is made at the same time as a and p, then the investment loses its role as a commitment device, so the strategic effect on monitoring does not exist any more. The following equations determine the equilibrium in which a, I and p are chosen simultaneously: \(c_{2}\left( {{\widehat{a}}},{{\widehat{I}}}\right) +r=0;\)\(c_{1}\left( {{\widehat{a}}},{{\widehat{I}}}\right) =t\left( \pi {{\widehat{p}}}k+\left( 1\pi \right) p_{0}\right) ;\)\(e^{\prime }\left( {{\widehat{p}}}p_{0}\right) =kt\left( 1{{\widehat{a}}}\right) .\) Applying the Cramer Rule gives \(sign\left( \frac{d{{\widehat{a}}}}{d\pi }\right) =sign\left( \frac{d{{\widehat{I}}}}{d\pi }\right) =sign\left( {{\widehat{p}}}kp_{0}\right)\).
For the case of China, see Beech (2016). For the U.S., see e.g. https://bribery.uslegal.com/federallawsonbribery/.
In this subsection, we slightly abuse notations by continuing using notations \(\gamma _{I}\), \(\gamma _{F}\), \(k_{I}\) and \(k_{F}\) with somewhat different meanings.
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We thank the coeditor Michael Finus and two anonymous referees for constructive comments and suggestions. We also thank participants in the EAERE 2017 Annual Conference (Athens), Singapore Economic Review Conference 2017 and the 5th Annual Xiamen University International Workshop on Economic Analysis of Institutions and seminar participants in Nanyang Technological University for helpful comments.
Appendices
Appendix 1: Proofs
Proof
(Lemma 1) Differentiating the system of equations, (3) and (4), with respect to \(\pi\), gives
Note that these are done for any given investment level I, so the derivatives are in the partial notion. Ditto for the following derivatives. In matrix notation, we have
By the Cramer’s rule,
Simplifying them gives
and
Given the curvature assumptions, \(c_{11}e^{\prime \prime }+\pi t^{2}k^{2}>0\). Therefore, \(sign\left( \frac{\partial a^{*}}{\partial \pi }\right) =sign\left( \frac{\partial p^{*}}{\partial \pi }\right) =sign\left( p^{*}kp_{0}\right)\), which proves Part 1 of the lemma.
To show Part 2, steps similar to the above establish
Then \(\frac{\partial }{\partial k}\left( p^{*}k\right) =\frac{\partial p^{*}}{\partial k}k+p^{*}=\frac{kc_{11}t\left( 1a^{*}\right) +p^{*}c_{11}e^{\prime \prime }}{c_{11}e^{\prime \prime }+\pi t^{2}k^{2}}\ge 0\) given our curvature assumptions. Obviously \(\lim _{k\rightarrow 0}p^{*}k<p_{0}\). Meanwhile from (4) we have \(\lim _{k\rightarrow 1}p^{*}>p_{0}\) and so \(\lim _{k\rightarrow 1}p^{*}k>p_{0}\). By these facts and the continuity of \(p^{*}k\), the threshold \({{\bar{k}}}\), which depends on I, must exist.
Steps similar to the above give us
and
which prove Parts 3–4. \(\square\)
Proof
(Proposition 2) Define \(H\left( I\right) \equiv\)\(\frac{dL\left( I\right) }{dI}\). With the first order condition \(H\left( I\right) _{I=I^{*}}=0\), using the implicit function theorem we have \(\frac{dI^{*}}{d\pi }=\frac{\frac{\partial H\left( I^{*}\right) }{\partial \pi }}{H^{\prime }\left( I^{*}\right) }\). Given convexity of \(L\left( I\right)\), \(H^{\prime }\left( I^{*}\right) \ge 0\), so \(\frac{dI^{*}}{d\pi }\ge 0\) if and only if \(\frac{\partial H\left( I^{*}\right) }{\partial \pi }\le 0\). From the last line of (22) in “Appendix 2”, we have
where the first term on the second line has sign \(sign\left( \frac{\partial a^{*}}{\partial \pi }\right)\) given the curvature assumptions; in the second term,
given \(c_{111}=e^{\prime \prime \prime }=0\). Equation (18) becomes
where the terms in the square brackets and the second term are nonpositive due to our assumptions on c and e.
Therefore, if \(k\ge {{\bar{k}}}\left( I^{*}\right)\), then \(\frac{\partial a^{*}}{\partial \pi }\ge 0\) by Lemma 1, and thus \(\frac{\partial H\left( I\right) }{\partial \pi }\le 0\) so \(\frac{dI^{*} }{d\pi }\ge 0\); moreover, \(\frac{da^{*}}{d\pi }=\frac{da^{*}}{dI} \frac{dI^{*}}{d\pi }+\frac{\partial a^{*}}{\partial \pi }\ge 0\) given \(\frac{da^{*}}{dI}\ge 0\) and \(\frac{\partial a^{*}}{\partial \pi }\ge 0\) because of \(k\ge {{\bar{k}}}\left( I^{*}\right)\) by Lemma 1. Part 1 is thus proved.
To show Part 2, notice that when \(k\rightarrow 0\), Eq. (19) becomes
where the inequality is derived from \(c_{12}<0\) and \(\lim _{k\rightarrow 0} \frac{\partial a^{*}}{\partial \pi }=\frac{p_{0}t}{c_{11}}\le 0\) using Eq. (15). Consequently \(\frac{dI^{*}}{d\pi }\le 0\) and thus \(\frac{da^{*}}{d\pi }=\frac{da^{*}}{dI}\frac{dI^{*}}{d\pi } +\frac{\partial a^{*}}{\partial \pi }\le 0\) given \(\frac{da^{*}}{dI}\ge 0\) and \(\frac{\partial a^{*}}{\partial \pi }\le 0\) (for small enough k) by Lemma 1, which proves Part 2. \(\square\)
Proof
(Remark 1) Define
where the second equality uses (17) and \(b=kt\left( 1a\right)\). We have \(\frac{d\phi _{FB}}{dI}=c_{22}\ge 0\). Note that \(\phi _{FB}\left( I^{FB}\right) =0\) and \(\phi \left( I^{*}\right) =0\) pin down the firstbest investment and the equilibrium investment respectively. Given our assumptions \(c_{12}<0\) and \(c_{112}\le 0\) (that is, \(c_{2}\left( a,I\right)\) is a concave and decreasing function in a), we have
For any I, we have
where the first inequality uses (20) and the second inequality is due to \(\frac{\pi k^{2}t^{2}}{c_{11}e^{\prime \prime }+\pi k^{2}t^{2}} <1\) (since \(c_{11}>0\) and \(e^{\prime \prime }>0\)) and \(c_{2}\left( 1,I\right) c_{2}\left( a^{*},I\right) \le 0\) (since \(c_{2}\) is decreasing in a by \(c_{21}<0\)). Since \(\phi _{FB}\left( I\right) \le \phi \left( I\right)\) for all I, \(\phi \left( I^{FB}\right) \ge \phi _{FB}\left( I^{FB}\right) =0\); moreover, \(\phi \left( I^{*}\right) =0\). Therefore, we have \(I^{*}\le I^{FB}\) given that \(\phi \left( I\right)\) is an increasing function. \(\square\)
Proof
(Proposition 3) The implicit function theorem gives \(\frac{dI^{*}}{dk}=\left. \frac{\frac{\partial H\left( I\right) }{\partial k}}{H^{\prime }\left( I\right) }\right _{I=I^{*}}\). Again by the convexity of L, \(sign\left( \frac{dI^{*}}{dk}\right) =\)\(sign\left( \frac{\partial H\left( I^{*}\right) }{\partial k}\right)\). By the last line of (22) in “Appendix 2”,
where \(\frac{\partial a^{*}}{\partial k}\ge 0\) by Lemma 1 and
where the second equality makes use of \(c_{111}=e^{\prime \prime \prime }=0\), and the inequality makes use of \(c_{12}<0\), \(c_{121}\le 0\) and \(\frac{\partial a^{*}}{\partial k}\ge 0\) by Lemma 1. So \(\frac{\partial H\left( I\right) }{\partial k}<0\) and consequently we have \(\frac{dI^{*} }{dk}>0\) and \(\frac{da^{*}}{dk}=\frac{\partial a^{*}}{\partial k} +\frac{da^{*}}{dI}\frac{dI^{*}}{dk}\ge 0\) given \(\frac{da^{*}}{dI}\ge 0\) and \(\frac{\partial a^{*}}{\partial k}\ge 0\) by Lemma 1. This proves the first part of the statement. Exactly the same steps apply for the comparative statics with respect to t. \(\square\)
Proof
(Remark 2) First, Eq. (6) and the convexity of \(L\left( I\right)\) imply that \(\frac{dI^{*}}{dr}\le 0\). Recall from Lemma 1 that \(\frac{da^{*}}{dI}\ge 0\) and \(\frac{dp^{*}}{dI}\le 0\). Because \(\frac{dI^{*}}{dr}\le 0\), by the chain rule we get \(\frac{da^{*}}{dr}=\frac{da^{*}}{dI}\frac{dI^{*}}{dr} \le 0\) and \(\frac{dp^{*}}{dr}=\frac{dp^{*}}{dI}\frac{dI^{*}}{dr} \ge 0\). \(\square\)
Proof
(Remark 3) Using (21) in “Appendix 2”, under the optimal \(I^{*}\), we have \(\frac{dL\left( I^{*}\right) }{d\pi }=\left( 1a^{*}\right) c_{11}\frac{\partial a^{*}}{\partial \pi }\) by the envelope theorem, where \(\frac{\partial a^{*}}{\partial \pi }\ge 0\) if and only if \(k\ge {{\bar{k}}}\left( I^{*}\right)\) by Lemma 1. \(\square\)
Proof
(Proposition 4) Proposition 6 in “Appendix 4” shows \(\frac{dI^{*}}{dk_{F}}\ge 0\) and \(\frac{dI^{*}}{dk_{I}}\ge 0\) for all \(k_{F}\) and \(k_{I}\). We also have \(\frac{da^{*}}{dk_{F}}=\frac{\partial a^{*}}{\partial k_{F}} +\frac{da^{*}}{dI}\frac{dI^{*}}{dk_{F}}>0\) and \(\frac{da^{*}}{dk_{I} }=\frac{\partial a^{*}}{\partial k_{I}}+\frac{da^{*}}{dI}\frac{dI^{*}}{dk_{I}}>0\), using (27), (28), and (29) in the proof of Lemma 2 in “Appendix 4”. Note that \(k_{I}\equiv \frac{1\lambda \gamma _{I}}{1+\lambda \gamma _{F}}k\) is decreasing in \(\lambda\), \(\gamma _{I}\), and \(\gamma _{F}\), while \(k_{F}=k~\) is independent of \(\lambda\), \(\gamma _{I}\), and \(\gamma _{F}\). Therefore by the chain rule, \(I^{*}\) and \(a^{*}\) are decreasing in \(\lambda\), \(\gamma _{I}\), and \(\gamma _{F}\). \(\square\)
Proof
(Remark 4) First, derivation analogous to the proof of Proposition 3 shows \(\frac{dI^{*}}{dk_{F}}>0\) and \(\frac{dI^{*}}{dk_{I}}>0\) for all \(k_{F}\) and \(k_{I}\). We also have \(\frac{da^{*}}{dk_{F}}=\frac{\partial a^{*}}{\partial k_{F}} +\frac{da^{*}}{dI}\frac{dI^{*}}{dk_{F}}>0\) and \(\frac{da^{*}}{dk_{I} }=\frac{\partial a^{*}}{\partial k_{I}}+\frac{da^{*}}{dI}\frac{dI^{*}}{dk_{I}}>0\) by (27), (28), and (29) in the proof of Lemma 2 in “Appendix 4”. It remains to show how the punishment parameters \(\lambda\), \(\gamma _{I}\), and \(\gamma _{F}\) affect \(k_{I}\) and \(k_{F}\) under each of the specifications.
Part 1. By the chain rule,
When \(k\rightarrow 1\), \(k_{F}\rightarrow 1\), a constant, independent of \(\lambda\), and therefore \(\frac{dk_{F}}{d\lambda }\rightarrow 0\). Since \(\frac{da^{*}}{dk_{F}}\frac{dk_{F}}{d\lambda }\rightarrow 0\) and \(\frac{da^{*}}{dk_{I}}\frac{dk_{I}}{d\lambda }<0\) (by \(\frac{da^{*} }{dk_{I}}>0\) and \(\frac{dk_{I}}{d\lambda }<0\)), we have \(\frac{da^{*} }{d\lambda }<0\). Similarly, we can prove \(\frac{dI^{*}}{d\lambda }<0\). The same argument applies for the comparative statics for \(\gamma _{I}\), and \(\gamma _{F}\).
When \(k\rightarrow 0\), we have \(k_{I}\rightarrow 0\), a constant, while \(k_{F}\rightarrow\)\(\lambda \left( \gamma _{I}+\gamma _{F}\right)\), which is increasing in \(\lambda\), \(\gamma _{F}\) and \(\gamma _{I}\). Following an argument similar to the above, by the chain rule, \(I^{*}\) and \(a^{*}\) are increasing in \(\lambda\), \(\gamma _{I}\), and \(\gamma _{F}\).
Part 2. Given that \(k_{I}\equiv k\left( 1\lambda \gamma _{I}\right) \left( 1\lambda \gamma _{F}\right)\) is decreasing in \(\gamma _{I}\) while \(k_{F}\equiv k+\left( 1k\right) \lambda \gamma _{F}\) is independent of \(\gamma _{I}\), by the chain rule, \(I^{*}\) and \(a^{*}\) are decreasing in \(\gamma _{I}\). When \(k\rightarrow 1\), we have \(k_{I}\rightarrow \left( 1\lambda \gamma _{I}\right) \left( 1\lambda \gamma _{F}\right)\), which is decreasing in \(\lambda\) and \(\gamma _{F}\), while \(k_{F}\rightarrow 1\), a constant. Following an argument similar to the proof of Part 1, by the chain rule, \(I^{*}\) and \(a^{*}\) are decreasing in \(\lambda\) and \(\gamma _{F}\). When \(k\rightarrow 0\), we have \(k_{I}\rightarrow 0\), a constant, while \(k_{F}\rightarrow\)\(\lambda \gamma _{F}\), which is increasing in \(\lambda\) and \(\gamma _{F}\). Therefore by the chain rule, \(I^{*}\) and \(a^{*}\) are increasing in \(\lambda\) and \(\gamma _{F}\). \(\square\)
Appendix 2: Derivation of \(\frac{dL\left( I\right) }{dI}\) and \(\frac{d^{2}L\left( I\right) }{dI^{2}}\) in Section 2.3
Using the firm’s Stage 2 first order condition (3), (5) can be written as
We have
where the third equality makes use of (16). By (17) in the proof of Lemma 1, (22) can be written as (6).
The second order derivative is
where the last term equals
by the assumption of \(c_{111}=e^{\prime \prime \prime }=0\).
Substituting this back to (23) yields
where the third equality comes from (16) and the inequality comes from the assumptions \(e^{\prime \prime }\ge 0\), \(c_{22}\ge 0\), \(c_{122}\ge 0\), and \(c_{11}c_{122}\ge 2c_{12}c_{121}\).
Appendix 3: Sign Ambiguity of \(\frac{da^{*}}{d\pi }\): Numerical Examples
The following numerical examples illustrate the ambiguity of the sign of \(\frac{da^{*}}{d\pi }\) in Case 2 of Table 1 (where \(\frac{da^{*}}{dI} \ge 0\), \(\frac{dI^{*}}{d\pi }\ge 0\) and \(\frac{\partial a^{*}}{\partial \pi }\le 0\)), assuming \(e=\frac{e_{0}}{2}\left( pp_{0}\right) ^{2}\) and \(c=\frac{c_{0}}{2I}a^{2}\). Part 3 of Lemma 1 shows that \(\frac{da^{*}}{dI}\ge 0\) is always satisfied. In Fig. 2, we let \(\pi =0.1\), \(t=0.2\), \(p_{0}=0.05\), \(e_{0}=0.3\), \(c_{0}=0.05\), and \(r=0.001\). When \(k\in \left( 0.32,0.36\right)\), we have \(\frac{dI^{*} }{d\pi }\ge 0\) and \(\frac{\partial a^{*}}{\partial \pi }\le 0\), while \(\frac{da^{*}}{d\pi }\ge 0\). In Fig. 3, we let \(\pi =0.7\), \(t=0.05\), \(p_{0}=0.15\), \(e_{0}=0.05\), \(c_{0}=0.05\), and \(r=0.001\). When \(k>0.22\), we have \(\frac{dI^{*}}{d\pi }\ge 0\) and \(\frac{\partial a^{*} }{\partial \pi }\le 0\), while \(\frac{da^{*}}{d\pi }<0\).
Appendix 4: Derivation of the Model in Section 3
Applying the Cramer Rule to (11) and (12) we obtain the following lemma which is analogous to Lemma 1 in Sect. 2.
Lemma 2
For any given I , in Stage 2, we have the following.

1.
More corruption (higher \(\pi\)) increases abatement,\(a^{*}\), and decreases the corruptible inspector’s monitoring intensity \(p^{*}\), if and only if \(p^{*}k_{F}>p_{0}\).

2.
There exists threshold \({{\bar{k}}}_{F}\left( I\right) \in \left( 0,1\right)\) such that \(p^{*}k_{F}>p_{0}\) if and only if \(k_{F}>{{\bar{k}}}_{F}\left( I\right)\).

3.
Other things being equal, a higher \(k_{F}\), \(k_{I}\), ortinduces more abatement.

4.
Higher investment (I)increases abatement and decreases monitoring.
Proof
(Lemma 2) Using the Cramer Rule, we obtain
and
where \(sign\left( \frac{\partial a^{*}}{\partial \pi }\right) =sign\left( \frac{\partial p^{*}}{\partial \pi }\right) =sign\left( p^{*}k_{F} p_{0}\right)\) which proves Part 1.
To show Part 2, we first establish that
and
given \(\frac{\partial p^{*}}{\partial k_{I}}\ge 0\) and \(\frac{\partial k_{F}}{\partial k_{I}}=\frac{1+\gamma _{F}}{1\gamma _{I}}>0\). Obviously \(\lim _{k_{F}\rightarrow 0}p^{*}k_{F}<p_{0}\). Meanwhile from the inspector’s firstorder condition (12) we get \(p^{*}>p_{0}\) given \(k_{I}>0\), so \(\lim _{k_{F}\rightarrow 1}p^{*}k_{F}>p_{0}\). Combining these facts and \(\frac{\partial \left( p^{*}k_{F}\right) }{\partial k_{F}}\ge 0\) as well as the continuity of \(p^{*}k_{F}\), we conclude that there exists a threshold \({{\bar{k}}}_{F}\left( I\right) \in \left( 0,1\right)\) such that \(p^{*}k_{F}>p_{0}\) if and only if \(k_{F}>{{\bar{k}}}_{F}\left( I\right)\).
To show Part 3, using the Cramer Rule we have
and
Finally
and
which proves Part 4. \(\square\)
Given \(k_{I}=\frac{1\lambda \gamma _{I}}{1+\lambda \gamma _{F}}k\) and \(k_{F}=k\), Parts 2 and 3 in Lemma 2 lead to the following.
Lemma 3
There exists a threshold \({{\widetilde{k}}}\left( I\right) \in \left( 0,1\right)\)such that\(p^{*}k_{F}>p_{0}\)if and only if\(k>{{\widetilde{k}}}\left( I\right)\). A higher bargaining power (k) induces more abatement.
Proof
(Lemma 3) To show the first statement, by the chain rule,
by (25) and (26). Clearly \(\lim _{k\rightarrow 1}k_{I} =\frac{1\lambda \gamma _{I}}{1+\lambda \gamma _{F}}>0\), which implies \(\lim _{k\rightarrow 1}p^{*}>p_{0}\) by (12). Meanwhile \(\lim _{k\rightarrow 1}k_{F}=1\), which then implies \(\lim _{k\rightarrow 1}p^{*}k_{F}>p_{0}\). Similarly \(\lim _{k\rightarrow 0}k_{I}=\lim _{k\rightarrow 0} k_{F}=0\). From (12), this means \(\lim _{k\rightarrow 0}p^{*}=p_{0}\) and so \(\lim _{k\rightarrow 0}p^{*}k_{F}=p_{0}k_{F}<p_{0}\). Combining these and using \(\frac{\partial \left( p^{*}k_{F}\right) }{\partial k}\ge 0\) as well as the continuity of \(p^{*}k_{F}\) establish the existence of the said threshold. For the second statement of the lemma, by the chain rule \(\frac{\partial a^{*}}{\partial k}=\frac{\partial a^{*}}{\partial k_{I} }\left( \frac{1\lambda \gamma _{I}}{1+\lambda \gamma _{F}}\right) +\frac{\partial a^{*}}{\partial k_{F}}>0\) using (27) and (28). \(\square\)
After characterizing the subgame equilibrium in Stage 2, at Stage 1 the firm’s loss function is
We have the following equation, which is the same as (22) except that \(k^{2}\) is replaced by \(k_{F}k_{I}\):
Steps similar to those in Sect. 2 establishes the existence of equilibrium \(I^{*}\). The following result is analogous to Proposition 2 in Sect. 2, given \({{\bar{k}}}_{F}\left( I^{*}\right)\) as defined in Lemma 2.
Proposition 5
At the equilibrium investment \(I^{*}\) :

1.
If \(k_{F}>{{\bar{k}}}_{F}\left( I^{*}\right)\) , a higher proportion of corruptible inspectors \((\pi )\) increases investment and abatement.

2.
If \(k_{F}\rightarrow 0\) or \(k_{I}\rightarrow 0\) , a higher proportion of corruptible inspectors \((\pi )\) decreases investment and abatement.
Proof
(Proposition 5) We apply the implicit function theorem to the firstorder condition \(\frac{dL\left( I^{*}\right) }{dI}=0\) where \(\frac{dL\left( I\right) }{dI}\) is given by Eq. (30). This yields \(\frac{dI^{*}}{d\pi }\ge 0\) if and only if \(\frac{\partial }{\partial \pi }\left( \frac{dL\left( I\right) }{dI}\right) \le 0\) given convexity of \(L\left( I\right)\). We have
With \(c_{111}=e^{\prime \prime \prime }=0\), we have
Thus \(\frac{\partial }{\partial \pi }\left( \frac{dL\left( I\right) }{dI}\right)\) becomes
This equation is similar to (19) in the proof of Proposition 2. If \(k_{F}>{{\bar{k}}}_{F}\left( I^{*}\right)\), by Lemma 2 we have \(\frac{\partial a^{*}}{\partial \pi }\ge 0\), implying \(\frac{\partial }{\partial \pi }\left( \frac{dL\left( I\right) }{dI}\right) \le 0\), and therefore \(\frac{dI^{*}}{d\pi }\ge 0\). Using exactly the same argument in the proof of Proposition 2, we also have \(\frac{da^{*}}{d\pi }\ge 0\).
To show Part 2, note that \(\lim _{k_{F}\rightarrow 0}\frac{\partial }{\partial \pi }\left( \frac{dL\left( I\right) }{dI}\right) =c_{12}\lim _{k_{F} \rightarrow 0}\frac{\partial a^{*}}{\partial \pi }\), where \(\frac{\partial a^{*}}{\partial \pi }\) is given by (24) in the proof of Lemma 2. When \(k_{F}\rightarrow 0\), then clearly \(\frac{\partial a^{*}}{\partial \pi }\le 0\) by Lemma 2. When \(k_{I}\rightarrow 0\), then \(p^{*}\rightarrow p_{0}\) by (12) and \(k_{F}\rightarrow \lambda \left( \gamma _{F}+\gamma _{I}\right) <1\); therefore, \(p^{*}k_{F}<p_{0}\) and thus \(\frac{\partial a^{*}}{\partial \pi }\le 0\) by Lemma 2. Since \(c_{12}<0\), when either \(k_{F}\rightarrow 0\) or \(k_{I}\rightarrow 0\), we have \(\frac{\partial }{\partial \pi }\left( \frac{dL\left( I\right) }{dI}\right) \ge 0\) and thus \(\frac{dI^{*}}{d\pi }\le 0\) and \(\frac{da^{*}}{d\pi }\le 0\). \(\square\)
When \(k\rightarrow 0\), then \(k_{I}\rightarrow 0\). By Lemmas 2 and 3and Proposition 5, the following corollary is straightforward:
Corollary 1
At the equilibrium investment \(I^{*}\):

If \(k>{{\widetilde{k}}}\left( I^{*}\right)\) , a higher proportion of corruptible inspectors \((\pi )\) increases investment and abatement.

If \(k\rightarrow 0\) , a higher proportion of corruptible inspectors \((\pi )\) decreases investment and abatement.
The following proposition corresponds to Proposition 3 in Sect. 2.
Proposition 6
With endogenous investment, a higher \(k_{I}, k_{F}\) , or t increases investment and abatement.
Proof
(Proposition 6) We apply the implicit function theorem to the firstorder condition \(\frac{dL\left( I^{*}\right) }{dI}=0\) where \(\frac{dL\left( I\right) }{dI}\) is given by Eq. (30). This yields \(\frac{dI^{*}}{dk_{I}}\ge 0\) if and only if \(\frac{\partial }{\partial k_{I}}\left( \frac{dL\left( I^{*}\right) }{dI}\right) \le 0\). We have
With \(c_{111}=e^{\prime \prime \prime }=0\), we have
Thus \(\frac{\partial \left( \frac{dL\left( I\right) }{dI}\right) }{\partial k_{I}}\) becomes
where the inequalities makes use of \(c_{12}<0\) and \(c_{121}\le 0\). Given \(\frac{\partial a^{*}}{\partial k_{I}}\ge 0\) by (28), we conclude that \(\frac{\partial \left( \frac{dL\left( I\right) }{dI}\right) }{\partial k_{I}}\le 0\) and consequently we have \(\frac{dI^{*}}{dk_{I}}\ge 0\) and \(\frac{da^{*}}{dk_{I}}=\frac{\partial a^{*}}{\partial k_{I}} +\frac{da^{*}}{dI}\frac{dI^{*}}{dk_{I}}\ge 0\) given \(\frac{da^{*} }{dI}\ge 0\) by (29) and \(\frac{\partial a^{*}}{\partial k_{I}} \ge 0\) by (28). Similar steps apply for the comparative statics with respect to \(k_{F}\) and t. \(\square\)
Since \(k_{F}\) and \(k_{I}\) increase in k, Proposition 6 implies that an increase in k increases investment and abatement. Finally, it is easy to confirm that all the other comparative statics results in Section 2 hold.
4.1 Appendix 4.1: Derivation of the Model in Section 3.1
Under the two alternative specifications, the proofs of Lemma 2, Propositions 5 and 6 do not change as they only involve \(k_{F}\), \(k_{I}\) and t, but not k, \(\gamma _{I}\) or \(\gamma _{F}\).
When the punishment is proportional to the fine evaded for both the inspector and the firm, Lemma 3 still holds. The proof is as follows.
Proof
(Lemma 3) To show the first statement, by the chain rule,
by (25) and (26). Clearly \(\lim _{k\rightarrow 1}k_{I} =1\lambda \left( \gamma _{F}+\gamma _{I}\right) >0\), which implies \(\lim _{k\rightarrow 1}p^{*}>p_{0}\) by (12). Meanwhile \(\lim _{k\rightarrow 1}k_{F}=1\), which then implies \(\lim _{k\rightarrow 1}p^{*}k_{F}>p_{0}\). Similarly \(\lim _{k\rightarrow 0}k_{I}=0\) while \(\lim _{k\rightarrow 0}k_{F}\rightarrow\)\(\lambda \left( \gamma _{F}+\gamma _{I}\right) <1\). From (12), this means \(\lim _{k\rightarrow 0}p^{*}=p_{0}\) and so \(\lim _{k\rightarrow 0}p^{*}k_{F}=p_{0}k_{F}<p_{0}\). Combining these and the continuity of \(p^{*}k_{F}\) establishes the existence of the said threshold. For the second statement of the lemma, by the chain rule \(\frac{\partial a^{*}}{\partial k}=\left( \frac{\partial a^{*}}{\partial k_{I}}+\frac{\partial a^{*}}{\partial k_{F}}\right) \left( 1\lambda \left( \gamma _{F}+\gamma _{I}\right) \right) >0\) by (27) and (28). \(\square\)
When \(k\rightarrow 0\), then \(k_{I}\rightarrow 0\), and therefore, Corollary 1 also holds.
When the punishment of the inspector is proportional to the bribe while the punishment of the firm is proportional to the evaded fine, the proof of Lemma 3 is as follows.
Proof
(Lemma 3) To show the first statement, by the chain rule,
by (25) and (26). Clearly \(\lim _{k\rightarrow 1}k_{I}=\left( 1\lambda \gamma _{I}\right) \left( 1\lambda \gamma _{F}\right) >0\), which implies \(\lim _{k\rightarrow 1}p^{*}>p_{0}\) by (12). Meanwhile \(\lim _{k\rightarrow 1}k_{F}=1\), which then implies \(\lim _{k\rightarrow 1} p^{*}k_{F}>p_{0}\). Similarly \(\lim _{k\rightarrow 0}k_{I}=0\) while \(\lim _{k\rightarrow 0}k_{F}=\)\(\lambda \gamma _{F}<1\). From (12), this means \(\lim _{k\rightarrow 0}p^{*}=p_{0}\) so \(\lim _{k\rightarrow 0}p^{*}k_{F}=p_{0}k_{F}<p_{0}\). Combining these and the continuity of \(p^{*} k_{F}\) establishes the existence of the said threshold. For the second statement, by the chain rule \(\frac{\partial a^{*}}{\partial k} =\frac{\partial a^{*}}{\partial k_{I}}\left( 1\lambda \gamma _{I}\right) \left( 1\lambda \gamma _{F}\right) +\frac{\partial a^{*}}{\partial k_{F} }\left( 1\lambda \gamma _{F}\right) >0\) by (27) and (28). \(\square\)
When \(k\rightarrow 0\), then \(k_{I}\rightarrow 0\), and therefore, Corollary 1 also holds.
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Hong, F., Teh, TH. Bureaucratic Shirking, Corruption, and Firms’ Environmental Investment and Abatement. Environ Resource Econ 74, 505–538 (2019). https://doi.org/10.1007/s1064001900327w
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DOI: https://doi.org/10.1007/s1064001900327w