GHG Measurement Precision, Reporting Incentives, and
Environmental Liability
Aline Grahn & Jochen Bigus
Freie Universität Berlin, September 2016
Abstract
This paper analyses two questions in a simple principal-agent-framework: (1) How does the
firm’s liability for environmental damages induce the firm to precisely measure Greenhouse
Gas (GHG) emissions? (2) How does environmental liability affect the manger’s incentives to
manipulate the report on GHG emissions given that the report can be used as evidence before
court?
We obtain the following results: (1a) Under strict liability, neither the shareholders of the firm
(as principal) nor the manager are interested in GHG measurement precision. (1b) In contrast,
under a negligence rule, the precision of the GHG indicator becomes important because the
shareholders are only held liable for too high emission levels. Higher precision of the GHG
indicator reduces Type 1-errors and by that, reduces the manager’s compensation. (2a) If the
manager is able to manipulate reported GHG emissions and this manipulation is not observable,
the real emission level will increase. Under strict liability, shareholders suffer from this activity
because of higher damage compensation. (2b) Under a negligence regime, the manager will
manipulate GHG emission reports more the more precise the measurement technique gets.
Shareholders may now benefit from GHG emission report manipulation since it is easier to
escape liability; the more so the higher are expected damages. (3) Possible reputation losses
and punitive damages will induce managers both to reduce manipulation of GHG emission reports and to keep real GHG emission levels low.
Overall, while a negligence regime encourages more precise GHG emission measurement
than strict liability, it also provides stronger incentives for manipulation.
GHG Measurement Precision, Reporting Incentives and
Environmental Liability
1. Introduction
This paper analyses how the environmental liability regime affects the managers’ incentive (1)
to precisely measure and (2) to truthfully report the level of Greenhouse Gas (GHG) emissions,
and by that, (3) to effectively reduce the GHG emission level, given that there is a separation
of ownership and control in the firm.
Increasing GHG emissions are considered to be an important driver of climate change (IPCC,
2014). Thus, the reduction of GHG emissions is a pressing goal on the political agenda. In the
European Union, both countries and firms are required to reduce GHG emissions by 20 % until
2020 in comparison to 1990 (EC, 2015 a). 1 An important GHG emission reduction program is
the European Emission Trading Scheme (EU ETS) which covers about 45% of the GHG emissions in the EU (EC, 2015 b), but currently does not provide proper incentives to reduce emissions due to an excess of emission allowances and too low allowance prices (DEHSt, 2013;
Brockmann et al., 2012).2 Thus, politicians and economists are looking for alternatives.
Environmental liability might be an alternative way to improve incentives. If firms are held
liable for GHG emissions they should be more inclined to reduce them. Liability may require
that courts are able to observe the real GHG emission level. But courts may find it hard to do
so (Goldsmith & Basak, 2001) due to limited precision of technical measurement tools (IPCC,
2006), problems of proper verification and the long-term effects of GHG emissions. Moreover,
1
In some countries, there are even more challenging goals, e.g., Germany aims to reduce GHG emissions by 40 %
until 2020 (BMUB, 2015).
2
The European Commission aims to reform the EU ETS (Commission Regulation No 176/2014, EC, 2015c).
1
the EU Commission Regulations provides a series of options and margins of discretion to the
firms how to measure GHG emission levels (EC, 2012a and 2012b) which mainly stem from
the argument of unreasonable costs: “improvements from greater accuracy shall be balanced
against the additional costs” (EC, 2012b, Art. 8).
For instance, firms may measure emission levels directly by output or activity data, or indirectly
by input data (measurement-based versus calculation-based methodology). With output data
there are problems of precise measurements and proper verification, especially considering the
long-term effects of emissions. The precision on measurement techniques partly depends on the
very polluting facility, that is, on the type, design and age of equipment, chemical handling
practices but also on legislation, e.g. on chemical recovery requirements (IPCC, 2006). If emission levels would be too high based on output data, firms may rather want to estimate emission
levels indirectly based on “activity data” (EC, 2012 b, Art. 27 (2)), e.g. by the quantity of fuel
or material processed during the reporting period, that is, by input information. The GHG emission level is then indirectly derived from the activity data combined with standardized calculation factors. However, there are choice options regarding the calculation factors. On the one
hand default values are given through guideline values partly referring to the IPCC guidelines
(cf. EC, 2012 b, Annex VI), but on the other hand, Annex II grants several choice options for
some tiers in determining the calculation factors permitting uncertainties of partly more than ±
10 % (cf. EC, 2012 b, Annex II). There is also discretion on monitoring plans, firms may ap-
plicate standardized or simplified monitoring plans (EC, 2012b, Art. 13, 26). 3 Usually privately
run technical assurance companies perform the monitoring. They are paid by the polluting firm.
3
Firms have discretion to choose the tier for their monitoring concept while tier 1 describes the minimum standard.
The requirements increase and the permitted uncertainties decrease the higher the applied tier is.
2
To sum up, it is important to note that there are pronounced information asymmetries and that
the firms have considerable discretion on measuring and reporting GHG emission levels.
Thus, it is not surprising that firms tend to voluntarily disclose poor information on the level
and on the measurement of GHG emissions (e.g., Sullivan, 2009, Sullivan & Gouldson, 2012,
Kolk et al., 2008, DEFRA, 2010 Stanny, 2013, Matisoff et al. 2013 and González-González &
Zamora-Ramírez, 2013). Within the ETS trading scheme there is mandatory reporting but yet
there are no studies explicitly analyze the reporting quality of mandatory disclosures.
Our model addresses two components of reporting quality: the precision of measurement which
is likely to influence the precision of reporting given that manipulation is not possible. The
second component refers to the incentives for reporting manipulation. While precision refers to
the incentives to improve the precision of technical measurement devices in order to reduce the
measurement error (reduce the standard deviation), manipulation considers the manager’s incentives to report lower GHG emission levels than they actually are (shifting the mean).
We analyze how environmental liability rules affect the precision and manipulation of GHG
emission reports. There are three scenarios: (1) no liability for GHG emissions, (2) strict liability and (3) a negligence rule. With strict liability, the firm − effectively the firm’s shareholders −
are held liable when GHG emissions cause damages. With a negligence regime, the firm is held
liable if GHG emissions cause damages and, additionally, if the firm acted negligently, i.e. if
the firm failed to meet the standard of due care specified by legal rules. The court decides on
whether due care has been met based on the GHG emission report. In the European Union,
firms generally face strict liability (see Directive 2004/35/EG). Still, as we intend to provide a
normative analysis we will also investigate the cases of no liability and of a negligence rule.
Since it is the firm’s management which decides both on the GHG emission level but also on
the reporting, we explicitly consider the agency problem between shareholders and managers.
3
Only managers are able to observe the real GHG emission level. Shareholders may have to pay
damage payments, though. Thus, shareholders rationally will set up a bonus contract which
provides monetary rewards to the manager not only to increase financial performance but also
environmental performance, that is, to decrease GHG emission levels. The manager can increase both types of performance by higher respective effort levels. Neither effort level is observable and thus, not contractible. The bonus contract is based on a financial and an environmental performance measure instead, which are both biased, though.
We find that the liability regime strongly influences the incentives for GHG emission measurement precision, GHG emission reporting manipulation and the real GHG emission levels. With
no liability, GHG emission levels do not matter neither to the shareholders nor to the manager
such that they are not part of the bonus contract. There is no need for precision nor for manipulation and GHG emission levels are high.
Under strict liability, shareholders want to reduce GHG emissions to the efficient level and –
given that reporting manipulation is not possible – the manager will do that. Neither the shareholders nor the manager are interested in GHG measurement precision. However, if managers
are able to manipulate the GHG emission report, they will do so and the real emission level
will increase. Shareholders suffer from this activity because of higher damage compensation.
Under a negligence rule, the precision of the GHG indicator matters because the shareholders
are only held liable for emission levels exceeding the standard of due care. Higher precision of
the GHG indicator reduces Type 1-errors − reported GHG emissions are high even though real
GHG emissions are low − and by that, increases the manager’s compensation. But if managers
are able to manipulate the GHG emission report, they will be more inclined to do so when
measurement precision is high. The simple reason for this is that manipulation pays off most
when it is the only reason for biased emission reports. In contrast to a strict liability regime,
4
shareholders now may benefit from GHG emission report manipulation since it is easier to meet
the due standard of care and to escape liability; the more so the higher are expected damages.
To sum up, there is a trade-off: While only a negligence regime induces the firm to precisely
measure GHG emission levels, it also provides incentives to both managers and shareholders
to manipulate GHG emission reporting. Only if reputation losses in markets sufficiently punish
the firm for manipulation, managers will not manipulate GHG emission reports. However, reputation losses induce managers to increase real GHG emission levels. Punitive damages will
mitigate the latter incentive.
For both academics and policy-makers it might be an interesting insight that the incentives for
measurement precision and reporting manipulation of GHG emissions crucially depend on the
liability regime. Given the insights of our analysis, policy-makers should install a negligence
regime accompanied by a public register that disclose sever cases of false reporting on GHG
emission levels. If stock or product markets sufficiently penalize firms for misreporting and if
there are punitive damages, a negligence regime will offer both the benefit of improved measurement precision and weak incentives for report manipulation.
To the best of our knowledge, this paper is the first one to analyze the delicate interaction between environmental liability rules and GHG emission reporting incentives given that there is
a separation of ownership and control. It contributes to three different strands of literature.
First and most importantly, it contributes to the emerging literature on GHG emission reporting
which so far rather uses empirical methods but lacks theoretical underpinning (e.g., Sullivan,
2009, Sullivan & Gouldson, 2012, Kolk et al., 2008, DEFRA, 2010 Stanny, 2013, Matisoff et
al. 2013 and González-González & Zamora-Ramírez, 2013). In contrast to this literature, we
especially emphasize the impact of the environmental liability regime on reporting choices and
we distinguish between incentives for measurement precision and for reporting manipulation.
5
We are not aware of financial accounting literature that links the firm’s choices on reporting
quality to liability regimes. Yue, Richardson & Thornton (1997) find a partial disclosure equilibrium implying that firms will withhold environmental liabilities exceeding a threshold level
when outside stakeholders can impose political costs. However, Yue et al. (1997) do not model
the drivers of environmental liability, such as the liability regime, and do not analyze how reporting incentives affect liability. In the auditing literature, however, it is well known that the
very characteristics of auditor liability affect audit quality (e.g. Schwartz 1997, Willekens &
Simunic 2007, Laux & Newman 2010, Bigus 2015). We assume the shareholders to be held
liable while the agent (manager) is not liable whereas the audit literature assumes the agent
(auditor) to be held liable and the shareholders not. A manager can receive performance-based
compensation, while an auditor is not allowed to. Our paper also distinguishes between effects
on measurement precision and reporting manipulation, while the audit literature usually captures report manipulation only, and does not consider reputation losses.
Second, the paper adds to the small literature on principal-agent-models that especially consider
an environmental context (Gabel & Sinclair-Desgagné 1993, Sinclair-Desgagné & Gabel 1996,
Goldsmith & Basak 2001, Lothe & Myrtveit 2003). 4 One common way to implement the environmental performance is the assumption of a multi-task agent (Holmström & Milgrom, 1991,
Feltham & Xie, 1994). A common result of this literature is that bonus contracts require sufficiently unbiased environmental performance indicators (EPI). This is in line with the seminal
finding of Holmström & Milgrom (1991) that linking compensation to performance measures
4
There are a few principal-agent-models examining special environmental problems, e.g. the relationship between
countries (Helm & Wirl 2014). Wood et al. (2012) analyse the conflict of interest between house owners and
tenants concerning the investment in energy-efficient equipment in Australia. The principal-agent-model of
Vernon & Meier (2012) deals with in energy-efficient investments in the trucking industry. In contrast to our
model, the agent prefers energy-efficient investments, but not the principal.
6
works better the less noisy the performance measure is. We add to this literature by endogenizing the bias of the EPI. More specifically, we show how the liability regime affects the manager’s incentive for both measurement precision and reporting manipulation.
Third, we add to the law and economics literature which usually do not account for agency
problems inside the polluting firm (e.g., Schäfer & Ott, 2004, Endres, Friehe & Rundshagen,
2015). Shavell (2007: 170-175) addresses a principal-agent context assuming that the agent can
cause harm by her actions and can be held liable but has limited assets (vicarious liability). The
principal is also liable and has sufficient assets to pay damages. While Shavell analyses under
which circumstance the agent still will exert efficient care he does not analyze how the liability
rules affect the agents’ incentives for measurement precision and reporting manipulation. Further, we assume that the firm is held liable but not the agent; the so-called business judgment
rule makes it difficult to make the manager responsible (e.g., Reinhardt et al. 2008, Bricker,
2013, Told, 2015). Another related paper is Polinsky & Shavell (2012) who analyze whether
voluntary or mandatory disclosure on product risks is socially beneficial. They find that mandatory disclosure rules are more valuable to costumers while the firm’s incentives to acquire
information are stronger under voluntary disclosure. Whether mandatory or voluntary disclosure is socially preferable also depends on the product liability regime. Polinsky & Shavell
(2012) ignore an owner-manager conflict and assume that the firm discloses new information
truthfully while we endogenize the manager’s incentives in a principal-agent framework.
Section 2 contains the basic model only addressing the management’s incentives for measurement precision on GHG emissions. Section 3 extends the analysis to GHG emission report manipulation. Section 4 concludes.
7
2. Model analysis: incentives for GHG emission measurement precision
Before we analyze the impact of strict liability and of a negligence regime, we first present the
assumptions, the case of no liability and the social optimum as a benchmark.
2.1 Basic assumptions
We analyze a two-stage model with risk-neutral actors. The shareholders of a firm emitting
greenhouse gases (GHG) delegate tasks to a professional manager. The shareholders represent
the principal, the manager is the agent. The manager has two tasks: exerting high effort in order
to maximize shareholder value and deciding on the level of GHG emissions. The decision on
the level of GHG emissions is a decision on using adequate technologies and procedures which
is a strategic decision and not really time-consuming. Thus, we assume that reducing emissions
does not compete with the “business as usual” task to maximize shareholder value. 5 Still, the
manager suffers a disutility, e.g. resources devoted to use adequate technologies.
The shareholders (and other parties such as courts) are unable to directly observe or to control
the managerial effort to maximize shareholder value and to influence the GHG emission level.
In order to keep the model simple, those are the only information asymmetries, everything else
is assumed to be common knowledge.
Thus, in order to align the interests shareholders need to provide incentives to the manager in a
compensation contract based on verifiable indicators for financial performance and for environmental performance (FPI and EPI) which we call 𝑥𝑥� and 𝑦𝑦�, respectively, both of which are
5
The appendix briefly addresses a binding capacity constraint that connects both tasks via a reaction function. It
turns out that the qualitative results on the effects of the different liability regimes do not change.
8
stochastic variables (Holmström & Milgrom, 1991). Following the literature, we assume linear
compensation contracts (e.g. Holmström & Milgrom,1991, Feltham & Xie, 1994).
The FPI 𝑥𝑥�, such as a stock price, depends on the manager’s effort for “business as usual” 𝑏𝑏 and
is uniformly distributed, 𝑥𝑥� ~ 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢(𝑏𝑏 − 𝜋𝜋, 𝑏𝑏 + 𝜋𝜋), with 𝑏𝑏 ≥ 𝜋𝜋 ≥ 0 and an expected value
𝐸𝐸 [𝑥𝑥� ] = 𝑏𝑏. Analogously, the EPI 𝑦𝑦� depends on the manager’s decision e with regard to GHG
emissions. For simplicity, 𝑦𝑦� is also uniformly distributed: 𝑦𝑦� ~ 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢(𝑒𝑒 − 𝜀𝜀, 𝑒𝑒 + 𝜀𝜀), with
𝑒𝑒 ≥ 𝜀𝜀 ≥ 0 and an expected value 𝐸𝐸 [𝑦𝑦�] = 𝑒𝑒. The distribution parameter 𝜀𝜀 can be interpreted as
the measurement error of the firm’s GHG emissions. 6 Since the firm’s reported emission level
and the optimal emission level defined by law cannot be negative either, we consistently assume
𝑒𝑒 ∗ ≥ 𝜀𝜀 ≥ 0 and 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 ≥ 𝜀𝜀 ≥ 0, respectively. We will derive 𝑒𝑒 ∗ and 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 below.
Table 1: Manager’s Tasks
tasks
𝑏𝑏 − manager’s effort for business as usual
𝑒𝑒 − absolute level of the firm’s GHG emissions, 0 ≤ 𝑒𝑒 ≤ 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
performance indicator
𝑥𝑥� − (traditional) financial performance indicator, 𝑥𝑥� ~ 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢�𝑏𝑏 − 𝜋𝜋, 𝑏𝑏 + 𝜋𝜋�, 𝐸𝐸 (𝑥𝑥� ) = 𝑏𝑏
𝑦𝑦� − environmental performance indicator,
�) = 𝑒𝑒
𝑦𝑦� ~ 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢�𝑒𝑒 − 𝜀𝜀, 𝑒𝑒 + 𝜀𝜀�, 𝐸𝐸 (𝑦𝑦
Both the FPI 𝑥𝑥� and EPI 𝑦𝑦� can be used for contracting because they are assumed to be verifiable.
For simplicity, we assume that the manager is unable to manipulate 𝑥𝑥� – which seems plausible
if it is a stock price. In the basic model, the manager truthfully reports the environmental performance indicator 𝑦𝑦�.
The manager’s decision on GHG emission affects the level of future environmental damages.
If the manager does nothing to reduce the GHG emission level, it reaches its maximum level
6
As explained above, the nature of GHG emissions makes their precise measurement very challenging.
9
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 and future damages are 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 then. If there is some effort to reduce GHG emissions, future
damages amount to 𝐷𝐷(𝑒𝑒) =
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝐷𝐷𝑚𝑚𝑚𝑚𝑥𝑥 . With no GHG emissions there will be no damage. Since
environmental damages usually take effect with a considerable time delay 7, we follow Segerson
& Tietenberg (1992) and discount the future damages by the factor 𝛿𝛿 (0 < 𝛿𝛿 < 1) to determine
their present value. Depending on the liability setting, the firm can be held liable for environmental damages.
Since principals and managers are risk-neutral such as in Segerson & Tietenberg (1992), their
respective utility functions are reflected by:
𝑈𝑈 𝑃𝑃 = 𝑥𝑥 − 𝑠𝑠(𝑥𝑥, 𝑦𝑦)
𝑈𝑈 𝑀𝑀 = 𝑠𝑠(𝑥𝑥, 𝑦𝑦) −
1
1
𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2
2
2
(1a)
(1b)
In the absence of environmental liability, the principal’s utility 𝑈𝑈 𝑃𝑃 is defined by the financial
performance of the firm less the remuneration of the manager 𝑠𝑠 which depends on the performance measures 𝑥𝑥� and 𝑦𝑦�. The manager’s utility 𝑈𝑈 𝑀𝑀 is defined by her salary s based on the
indicators for financial and environmental performance minus the disutilities for her “business
as usual”-effort and the disutility for her effort to reduce GHG emissions under the maximum
level 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 . 8 There is a disutility to the manager, e.g. induced by foregone leisure time or simply
because she does not like the task or because more emission reduction is more costly and reduces the manager’s budget. The disutilities are each assumed to depend quadratically on the
7
Probably the most severe consequence of GHG emissions, the climate change, is predicted to take it’s full effect
“by the late 21st century and beyond”. (IPCC, 2014, p. 8).
8
The maximum level 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 can be interpreted as the level of GHG emissions of the firm which occur when the
manager does not do anything to reduce them.
10
particular effort and are expressed in monetary terms through an appropriate unit of the particular disutility parameter, that is, 𝑐𝑐𝑏𝑏 and 𝑐𝑐𝑒𝑒 , respectively. Without loss of generalization, we as-
sume the manager to generate a zero reservation wage in an alternative employment.
In order to obtain interior solutions, we assume that the manager’s disutility is convex and that
her disutility exceeds the present value of maximum damages, 𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 > 𝛿𝛿𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 . Otherwise it
would be socially desirable to avoid any GHG emission.
Figure 1: Timeline of events
We will analyze how strict liability and a negligence rule will affect the manager’s choice to
precisely measure and to correctly report GHG emission levels. With strict liability, the firm,
that is, the firm’s shareholders, are held liable whenever GHG emissions cause damages. The
firm has sufficient assets to pay damages. The manager is not held liable. 9 With a negligence
regime, the firm is held liable if GHG emissions cause damages and, additionally, if the firm
9
The environmental law in the European Union defines a liability of the firm, not of the management (see
2004/35/EG, Art. 8). In principle, corporation law allows shareholders to ask managers for damage compensation
if they violated their duties towards shareholders. However, the so-called “business-judgment”- rule makes it difficult to make managers liable (see Reinhardt et al., 2008, Bricker 2013, Told 2015). Moreover, even if the manager
would be held liable, she usually has too limited assets to fully cover environmental damages. Thus, with limited
assets, the manager’s incentive to prevent environmental damages would not significantly improve.
11
acted negligently, i.e. if the firm failed to meet the standard of due care specified by legal rules.
We follow the law and economics literature and assume that the standard of due care is defined
by the socially optimal emission level. We will derive the socially optimal emission level in
Section 2.3. The court decides on whether due care has been met based on the GHG emission
report y which we above assumed to be verifiable. 10 We also assume that victims face zero
transaction costs when bringing a lawsuit (Shavell, 2007). If transactions were too high, we
would obtain the simple result that victims will not sue, and consequently, that the shareholders
and the manager will not care about emission levels in the first place.
2.2 No environmental liability
In the absence of environmental liability, the shareholders maximize utility according to (1a)
taking into account the manager’s utility function (1b). The shareholders (principal) design a
linear compensation scheme of the form 𝑠𝑠 (𝑥𝑥, 𝑦𝑦) = 𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 and decide how to choose 𝛼𝛼, 𝛽𝛽
and 𝛾𝛾 before the manager (agent) decides on the action set (𝑏𝑏, 𝑒𝑒) and before the respective
performance indicators (𝑥𝑥, 𝑦𝑦) realize. Thus, the principal aims to maximize expected utility
𝑚𝑚𝑚𝑚𝑚𝑚 𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝑠𝑠(𝑥𝑥, 𝑦𝑦) = 𝑏𝑏 − (𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾)
𝛼𝛼,𝛽𝛽,𝛾𝛾,𝑏𝑏,𝑒𝑒
•
(2)
with respect to the individual rationality constraint (IR), that is the zero reservation wage
the manager could receive in an alternative employment:
𝐸𝐸𝑈𝑈 𝑀𝑀 = 𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 −
10
1
1
𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 ≥ 0
2
2
(3)
This implies a mandatory report on GHG emission levels. If neither the manager’s effort to reduce GHG emis-
sions (e) nor the EPI y were observable, courts would be unable to find negligent behavior. Again, this in turn will
induce shareholders and the manager not to care about emission levels in the first place.
12
•
and with respect to the two incentive compatibility constraints (IC 1 and IC 2) that ensure that the manager will take the actions that maximize her expected utility:
𝜕𝜕𝜕𝜕𝑈𝑈 𝑀𝑀
= 0 = 𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏∗
𝜕𝜕 𝑏𝑏
(4a)
𝜕𝜕𝜕𝜕𝑈𝑈 𝑀𝑀
= 0 = 𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )
𝜕𝜕 𝑒𝑒
(4b)
Introducing Lagrange-Multipliers 𝜆𝜆, µ and 𝜐𝜐 for each constraint, yields the following Lagrange
function:
𝐿𝐿 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 + 𝜆𝜆 �𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 −
1
1
𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 �
2
2
(5)
+ µ(𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏) + 𝜐𝜐�𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)�
Solving this optimization problem yields the results summarized in Table 2 (see also appendix
A1). The asterisk always marks the individual optimum for the manager’s working effort in the
particular scenario.
Table 2: Results with no environmental liability
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
GHG emission level
environmental compensation contract parameter
𝛾𝛾 = 0
𝐸𝐸𝑈𝑈 𝑃𝑃 =
expected utility of the principal
1 1
∙
2 𝑐𝑐𝑏𝑏
The consequence of no environmental liability is that the compensation contract ignores envi1
ronmental performance, 𝑠𝑠(𝑥𝑥, 𝑦𝑦) = − ∙
1
2 𝑐𝑐𝑏𝑏
+ 1 ∙ 𝑥𝑥 = 𝑠𝑠(𝑥𝑥). Consistently, the manager does not
do anything to reduce emission levels, GHG emissions reach the maximum level 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 . Future
environmental damages reach the maximum level 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 .
13
2.3 Socially optimal GHG emission level
The socially optimal emission level is derived through minimizing the expected social cost
function 𝐸𝐸𝐸𝐸(𝑒𝑒) (Shavell, 2007) which consist of the discounted expected damages from GHG
emissions (Segerson & Tietenberg, 1992) and the manager’s disutility to reducing them:
𝐸𝐸𝐸𝐸 (𝑒𝑒) = 𝛿𝛿
𝑒𝑒
1
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 .
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
2
(6)
Optimization yields (see Appendix A2):
𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑑𝑑
𝑐𝑐𝑒𝑒
𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑑𝑑 = 𝛿𝛿
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚
> 0.
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
(7)
The social optimum is positive, 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 > 0, if 𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 > 𝛿𝛿𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 holds which we assumed in
Section 2.1. Comparative statics show that the socially optimal GHG emission level decreases
with higher maximum damages 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 and higher discount factors 𝛿𝛿 as well as with lower costs
to reduce emissions 𝑐𝑐𝑒𝑒 .
2.4 Strict liability
In this scenario, the firm has to pay damages to victims of environmental damages caused by
the GHG emissions. Damages need to be paid regardless of how much the manger reduced the
emission level. The principals now also consider expected damage payments:
max 𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝑏𝑏 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝛼𝛼,𝛽𝛽,𝛾𝛾,𝑏𝑏,𝑒𝑒
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 .
(8)
Recall the assumption that the manager is not held liable, that is, her objective function remains
unchanged. Lagrangian optimization yields the results summarized in Table 3. The principal
fully bears the cost of environmental damages and thus, designs the compensation contract in a
way that the manager fully internalizes the damage payments and thus, chooses the socially
14
optimal emission level 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 . Consequently, the remuneration now also depends on the environ-
mental performance indicator 𝑦𝑦. Any increase in 𝑦𝑦 reduces the manager’s compensation.
Table 3: Results under strict liability (see Appendix A3)
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
GHG emission level
environmental compensation contract parameter
expected utility of the principal
𝐸𝐸𝑈𝑈 𝑃𝑃 =
𝑑𝑑
= 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜
𝑐𝑐𝑒𝑒
𝛾𝛾 = −𝑑𝑑
1 1
1 𝑑𝑑
∙ − 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
�
2 𝑐𝑐𝑏𝑏
2 𝑐𝑐𝑒𝑒
The expected utility of the principal compared to the no liability case decreases by
𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
1 𝑑𝑑
2 𝑐𝑐𝑒𝑒
� > 0. This expression can be interpreted as a kind of agency costs caused by the
introduction of strict liability. The manager is “punished” for increasing GHG emissions. However, the principal needs to compensate the manager’s disutility from reducing GHG emissions
and thus, increases the fixed salary parameter 𝛼𝛼. This in turn reduces the principal’s expected
utility by 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
1 𝑑𝑑
2 𝑐𝑐𝑒𝑒
�.
Another important implication of the results in Table 3 is that the noise of the environmental
performance indicator as measured by 𝜀𝜀 is not of interest for any of the players. Under strict
liability, measurement precision is not important.
2.5 Negligence rule
With a negligence regime, the firm is held liable if GHG emissions cause damages and, additionally, if the firm acted negligently, i.e. if the firm failed to meet the standard of due care
15
specified by legal rules. We follow the law and economics literature and assume that the standard of due care is defined by the social optimum (Shavell, 2007). According to (7), the legislator
should define a threshold level of the EPI which firms should not exceed:
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 .
(9)
Recall that only the indicator 𝑦𝑦 is verifiable but not the real emissions 𝑒𝑒. Since the EPI on
average will reflect the true emission level, 𝐸𝐸�𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � = 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 , 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 might well serve as a benchmark to judge whether emission levels are “too high”.
Thus, if there is a damage, the firm only will be held liable, if the environmental performance
indicator indicates emission levels exceeding 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 . With lower indicated emission levels, there
is no liability. Consequently, the principal’s utility function has two sections.
𝑈𝑈 𝑃𝑃 =
⎧𝑥𝑥 − 𝑠𝑠(𝑥𝑥, 𝑦𝑦)
𝑖𝑖𝑖𝑖 𝑦𝑦 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
⎨ 𝑥𝑥 − 𝑠𝑠(𝑥𝑥, 𝑦𝑦) − 𝛿𝛿 𝑒𝑒 𝐷𝐷
⎩
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚
𝑖𝑖𝑓𝑓 𝑦𝑦 > 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
𝑑𝑑
𝑐𝑐𝑒𝑒
(10)
We first concentrate on the case that the EPI 𝑦𝑦 indicates negligence, that is, 𝑦𝑦 > 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 . Recall
that the EPI is biased and uniformly distributed between 𝑒𝑒 − 𝜀𝜀 and 𝑒𝑒 + 𝜀𝜀. Ex ante, the probability that the EPI exceeds the standard level 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 amounts to:
P�𝑦𝑦 > 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � = 1 − P�𝑦𝑦 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � = 1 − 𝐹𝐹�𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � = 1 −
The expected utility of the principal then reads:
𝐸𝐸𝑈𝑈 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − (𝑒𝑒 − 𝜀𝜀 )
≥0
2𝜀𝜀
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 ∙ P�𝑦𝑦 > 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 �
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 + 𝜀𝜀
= 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1 −
�
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
2𝜀𝜀
𝑒𝑒
(11)
(12)
Again, Lagrangian optimization is applied to derive the results in Table 4.
16
Table 4: Results under a negligence rule given that liability is possible: 𝑃𝑃�𝑦𝑦 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � < 1 (see
Appendix A4)
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
GHG emission level
environmental compensation contract parameter
expected utility of the
principal
𝑑𝑑
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
−
≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
2𝑐𝑐𝑒𝑒 2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑 )
𝛾𝛾 = −𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ ) = −𝑐𝑐𝑒𝑒 �
𝐸𝐸𝑈𝑈 𝑃𝑃 =
𝑑𝑑
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
+
�<0
2𝑐𝑐𝑒𝑒 2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑 )
1 1 𝑐𝑐𝑒𝑒
1 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗
∙ − (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2 − 𝑑𝑑𝑒𝑒 ∗ � −
�
2 𝑐𝑐𝑏𝑏 2
2
2𝜀𝜀
As is shown in the Appendix A4, the emission level 𝑒𝑒 ∗ is lower than the socially optimal level
𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 and lower than the negligence standard 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 . Why should the principal want the manager
to voluntarily reduce the GHG emission level more than necessary even though this is costly
(due to a higher manager’s disutility)? The reason is that the environmental performance indi-
cator 𝑦𝑦 is not perfect and might indicate high emission levels even though real (but unverifiable)
GHG emissions do not exceed the social optimum. This type 1 error makes the firm liable even
though it actually did not do wrong. Since the manager’s compensation is reduced by the principal’s liability payments, the noise in the EPI 𝑦𝑦 matters. In accordance to the basic principal-
agent literature (e.g., Holmström & Milgrom 1991, Lothe & Myrtveit 2003) we find that the
manager’s compensation becomes more sensitive to the EPI the more precise the EPI gets. 11
Lower real emission levels than required by the standard will make it less likely that the EPI
wrongly indicates negligence. The type 1 error decreases as well as the damage payments related to it. As long as the reduced expected damage payments exceed the additional managerial
costs of reducing real emission levels it makes sense to reduce emission levels below the standard level required by law. Thus, in contrast to strict liability, the negligence rule tends to induce
excessive care due to the imprecise measurement of real emission levels.
11 𝜕𝜕𝛾𝛾
𝜕𝜕𝜕𝜕
> 0, i.e. with smaller 𝜀𝜀 (more precise EPI) 𝛾𝛾 decreases, but because 𝛾𝛾 < 0 it’s absolute value increases.
17
Reducing real GHG emissions pays more with a more precise EPI, i.e. lower values of 𝜀𝜀. With
a smaller measurement error, it is more likely that the EPI will correctly reflect lower real emis-
sions and that the principal can escape liability. This is also shown by comparative statics (see
Appendix A5):
𝜕𝜕𝑒𝑒 ∗
>0
𝜕𝜕𝜕𝜕
(13)
if 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑐𝑐𝑒𝑒 > 0.
Thus, with lower measurement error, there will be lower real emission levels in equilibrium.
The principal’s expected utility also increases with lower measurement error:
𝜕𝜕𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
𝑑𝑑 ∙ 𝑒𝑒 ∗
=−
∙ �𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗ � ≤ 0.
𝜕𝜕𝜕𝜕
2 ∙ 𝜀𝜀 2
(14)
The important insight here is that in contrast to a strict liability rule, the firm’s shareholders
have an interest in higher precision of the environmental performance indicator.
Given that the measurement error can be reduced and the real emission level is not too high,
there must be the case that the EPI never indicates negligence. In mathematical terms, we know
from (13) that the real emission level 𝑒𝑒 ∗ decreases if the measurement error 𝜀𝜀 decreases. Thus,
there is an 𝜀𝜀̂ below which the belonging 𝑒𝑒�∗ is so small that 𝑒𝑒�∗ + 𝜀𝜀̂ ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 holds for sure and, as
a consequence, liability can be ruled out. The point when P�𝑦𝑦 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � = 1 holds determines
the threshold 𝜀𝜀̂ below which liability is ruled out (see Appendix A5):
𝜀𝜀̂ = −
3𝑑𝑑
1𝑑𝑑
𝑐𝑐𝑒𝑒
�1 + 8 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
+
4𝑐𝑐𝑒𝑒 4𝑐𝑐𝑒𝑒
𝑑𝑑
(15)
Thus, we complement the principal’s expected utility function from relation (12) by the nonnegligence case and make it depend on the threshold measurement error 𝜀𝜀̂.
18
𝐸𝐸𝑈𝑈 𝑃𝑃 = �
𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾
𝑖𝑖𝑖𝑖 𝜀𝜀 ≤ 𝜀𝜀̂
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 + 𝜀𝜀
𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝑑𝑑 ∙ 𝑒𝑒 ∙ �1 −
� 𝑖𝑖𝑖𝑖 𝜀𝜀 > 𝜀𝜀̂
2𝜀𝜀
(16)
The second part of this utility function has already been analyzed (see (14)). For analyzing the
first part, which implies no negligence and no liability, the principal’s expected utility in equilibrium results to:12
𝐸𝐸𝐸𝐸
𝑃𝑃
𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙.
2
1 1 𝑐𝑐𝑒𝑒
1 1 𝑐𝑐𝑒𝑒 𝑑𝑑
∗
2
= ∙ − (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ) = ∙ − � + 𝜀𝜀�
2 𝑐𝑐𝑏𝑏 2
2 𝑐𝑐𝑏𝑏 2 𝑐𝑐𝑒𝑒
(17)
Again, the principal’s expected utility increases the more precise the EPI gets:
𝑑𝑑
𝜕𝜕𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙.
=
𝜕𝜕𝜕𝜕
−𝑐𝑐𝑒𝑒 � + 𝜀𝜀� < 0. The reasoning is the same as before. The lower is the measurement error of
𝑐𝑐𝑒𝑒
the environmental performance indicator 𝑦𝑦, the less the manager needs to reduce the real emis-
sion level 𝑒𝑒 ∗ in order to avoid negligence. Consequently, the principal saves the managerial
cost of reducing the real emission level.
Following this argumentation, the principal reaches the highest expected utility for the most
precise GHG emission report. Then, 𝜀𝜀 = 0 is valid and the real emission level equals the so-
cially optimal level, i.e. 𝑒𝑒 ∗ = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 . As a consequence, the maximum principal’s expected utility
1
is achieved, when there is no measurement error, 𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙.,𝑚𝑚𝑚𝑚𝑚𝑚 = ∙
1
2 𝑐𝑐𝑏𝑏
−
1 𝑑𝑑2
2 𝑐𝑐𝑒𝑒
.
Analogously, the maximum measurement error, 𝜀𝜀 = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 , minimizes the principal’s expected
1
utility (see Table 4): 𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛,𝑚𝑚𝑚𝑚𝑚𝑚 = ∙
1
2 𝑐𝑐𝑏𝑏
−
𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚
2
. This minimum expected utility under negli1
gence still exceeds the principal’s expected utility under strict liability, 𝐸𝐸𝑈𝑈 𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = ∙
1
2 𝑐𝑐𝑏𝑏
−
Figure 2: Principal’s expected utility under a negligence rule depending on the measurement
error of the environmental performance indicator, EU P (ε)
12
Optimization is restricted by the “no-negligence-constraint” 𝑒𝑒 ∗ + 𝜀𝜀 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 . In equilibrium, this constraint holds
as an equality. See Appendix A5.
19
𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
1 𝑑𝑑
2 𝑐𝑐𝑒𝑒
�. The reason is that under a negligence rule there is a chance to escape liability
when damage occurs while there is none under strict liability. Figure 2 illustrates how the prin-
cipal’s expected utility in equilibrium depends on the measurement error of the EPI and also
shows the lower and constant expected utility with strict liability.
Figure 3 illustrates the relation between measurement error of the EPI and the real emission
level 𝑒𝑒 ∗ in equilibrium. This relation is non-monotonic. As long as 𝜀𝜀 does not exceed the thresh-
old level 𝜀𝜀̂, the manager chooses the real emission level 𝑒𝑒 ∗ in a way that liability is ruled such
that 𝑒𝑒 ∗ + 𝜀𝜀 = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 is fulfilled. Thus, with higher error 𝜀𝜀, the real emission level 𝑒𝑒 ∗ decreases in
equilibrium. However, if the measurement error exceeds the threshold level 𝜀𝜀̂, the manager
cannot avoid the firm’s liability. This changes the incentives for the real emission level. Since
higher measurement errors reduce the marginal benefit for further real GHG emission reduction, there is less of an incentive to reduce real emission levels. Note, however, that due to the
measurement error of the EPI, a negligence regime generally induces lower real GHG emission
levels than a strict liability rule and lower than would be socially optimal.
20
Figure 3: Real GHG emission levels in equilibrium depending on the measurement error of
the environmental performance indicator, 𝑒𝑒 ∗ (𝜀𝜀)
Result 1: Strict liability provides efficient incentives to reduce GHG emissions but does
not induce the manager to precisely measure GHG emissions. With a negligence regime,
precision of GHG emission measurement becomes important. With more precise measurement it is less likely that the measurement device reports non-tolerable GHG emissions even
though real emission are lower and not indicating negligence. Due to this type-1-error, real
GHG emission levels are generally lower than the socially optimal emission level.
3. Model extension: incentives for GHG reporting manipulation
3.1 Change in assumptions
As pointed out above, both the nature of GHG emissions and the discretion provided by existing
reporting guidelines makes the verification of the GHG emission reports very difficult. Consequently, managers and/or shareholders might take advantage of those verification problems.
We therefore now delete the assumption of the basic model that the manager truthfully reports
21
the environmental performance indicator 𝑦𝑦. Instead, we introduce a third activity of the manager
in 𝑡𝑡 = 1, namely emission reporting manipulation or reporting manipulation 𝑖𝑖. Higher levels of
emission reporting manipulation tend to lower reported GHG emissions:
𝐸𝐸 [𝑦𝑦] = 𝑒𝑒 − 𝑖𝑖.
(19)
We assume 0 ≤ 𝑖𝑖 ≤ 𝑒𝑒. 13 Reporting manipulation induces a disutility for the manager which
needs to be added to her utility function in (1b) and which implies a forth restriction for the
optimization problem:
𝜕𝜕𝜕𝜕𝑈𝑈 𝑀𝑀
= 0 = −𝛾𝛾 − 𝑐𝑐𝑖𝑖 𝑖𝑖 ∗
𝜕𝜕 𝑖𝑖
(20)
Similar to the other two manager’s actions, we assume 𝑖𝑖 not to be verifiable. Contracts still can
only be made based on the EPI 𝑦𝑦 or FPI x. Otherwise, there is symmetric information.
There is a downside of reporting manipulation which is expected reputation losses in markets.
There is evidence that firms suffer significant reputation losses in capital markets for financial
misrepresentation. 14 We might also expect negative reactions in product markets because costumers are not willing to buy the firm’s products anymore. 15 We assume that expected reputation losses increase with the extent of emission reporting manipulation:
13
14
Consequently, we also adjust the assumption on the measurement error: 𝑒𝑒 − 𝑖𝑖 ≥ 𝜀𝜀 ≥ 0.
For instance, Karpoff et al. (2008) found that 585 firms targeted by SEC enforcement actions for financial
misrepresentation from 1978 to 2002 lost 38% of their market value on average; about two thirds of the loss can
be attributed to the loss of reputation. Interestingly, 61% of the firms were not sued, but still suffered a significant
reputation loss. There is less quantitative evidence on the reputation losses of managers. In order to keep the model
simple, we assume only firms to suffer from reputation losses.
15
We can imagine other negative consequences, such as fines or other sanctions imposed by public authorities.
22
𝑅𝑅 ∙ 𝛿𝛿 ∙
𝑖𝑖
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
= 𝑟𝑟 ∙ 𝑖𝑖
(21)
where 𝑅𝑅 is the maximal future reputation loss; 𝛿𝛿 is a discount factor. The following sections
analyze the incentives for GHG emission reporting manipulation under strict liability as well
as under negligence. We will distinguish between the subcases where there is a reputation loss
and where there is none. We do not analyze the case of no liability since obviously there is no
incentive to manipulate GHG emission reports when there is no liability.
3.2 Emission reporting manipulation under strict liability
We adjust the manager’s utility function for the additional disutility related to the reporting
manipulation activity. 16 The principal needs to consider a possible reputation loss.
In the absence of possible reputation losses (𝑅𝑅 = 0), the resulting manipulation level in equi-
librium is positive, 𝑖𝑖 ∗ = 𝑐𝑐 𝑑𝑑+𝑐𝑐 . As one can expect, manipulation pays more with higher damage
𝑒𝑒
𝑖𝑖
payments and decreases with the cost parameters 𝑐𝑐𝑒𝑒 and 𝑐𝑐𝑖𝑖 . Since the manager manipulates the
reported emission level 𝑦𝑦, she is able to raise the real GHG emission level accordingly. Thus,
real GHG emissions are higher than in the initial strict liability model.
16
1
1
1
𝐸𝐸𝑈𝑈 𝑀𝑀 = 𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾(𝑒𝑒 − 𝑖𝑖) − 2 𝑐𝑐𝑏𝑏 𝑏𝑏 2 − 2 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 − 2 𝑐𝑐𝑖𝑖 𝑖𝑖 2
23
Table 5: Results with emission reporting manipulation under strict liability
No emission reporting manipulation
principal’s objective function
individual optima:
GHG emission
level and manipulation level
environmental compensation contract
parameter
expected utility of
the principal in
equilibrium
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 −
𝑑𝑑𝑑𝑑
𝑑𝑑
∗
𝑒𝑒 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑐𝑐 = 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜
𝑒𝑒
2
Emission reporting manipulation and reputation loss
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 −
𝛾𝛾(𝑒𝑒 − 𝑖𝑖) − 𝑑𝑑𝑑𝑑
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 −
𝛾𝛾(𝑒𝑒 − 𝑖𝑖) − 𝑑𝑑𝑑𝑑 − 𝑟𝑟𝑟𝑟
𝑑𝑑
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑐𝑐 + 𝑐𝑐
𝑖𝑖 ∗ = 𝑐𝑐
𝑑𝑑
𝑒𝑒
𝑑𝑑
𝑒𝑒+𝑐𝑐𝑖𝑖
𝑒𝑒 +𝑐𝑐𝑖𝑖
𝛾𝛾 = − 𝑐𝑐
𝛾𝛾 = −𝑑𝑑
1
Emission reporting manipulation, but no reputation loss
1
1 𝑑𝑑
∙ 𝑐𝑐 − 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 2 𝑐𝑐 �
𝑒𝑒
𝑏𝑏
1
2
1
𝑑𝑑
− 2 𝑖𝑖 ∗
�1 −
𝑖𝑖 ∗ = 𝑐𝑐
𝑐𝑐𝑒𝑒
𝑐𝑐𝑒𝑒 +𝑐𝑐𝑖𝑖
𝑑𝑑
𝑒𝑒 +𝑐𝑐𝑖𝑖
𝑐𝑐𝑒𝑒
𝑐𝑐𝑖𝑖𝑑𝑑
𝑑𝑑
1
𝑒𝑒 +𝑐𝑐𝑖𝑖
1 𝑑𝑑
𝑒𝑒
1
2
�
−
𝑟𝑟
1
𝑐𝑐
𝑑𝑑
𝑒𝑒+𝑐𝑐𝑖𝑖
𝑒𝑒
𝑟𝑟𝑐𝑐𝑒𝑒
𝑐𝑐𝑖𝑖 (𝑐𝑐𝑒𝑒+𝑐𝑐𝑖𝑖 )
� − �
𝑐𝑐
𝛾𝛾 = 𝑐𝑐
∙ 𝑐𝑐 − 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 2 𝑐𝑐 �
𝑏𝑏
𝑟𝑟
𝑐𝑐𝑖𝑖
𝑐𝑐𝑒𝑒 +𝑐𝑐𝑖𝑖 𝑐𝑐𝑒𝑒
𝑒𝑒 +𝑐𝑐𝑖𝑖
𝑑𝑑
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑐𝑐 + 𝑐𝑐
𝑖𝑖
+
=
(𝑟𝑟𝑐𝑐𝑒𝑒 − 𝑑𝑑𝑐𝑐𝑖𝑖 )
𝑑𝑑
𝑟𝑟
∙ 𝑐𝑐 − 2𝑖𝑖 𝑖𝑖 ∗ �𝑐𝑐 − 𝑐𝑐 � −
𝑏𝑏
𝑑𝑑𝑒𝑒 ∗ − 𝑟𝑟𝑖𝑖 ∗
𝑒𝑒
𝑖𝑖
The principal will anticipate reporting manipulation and will reduce the responsiveness of managerial compensation to the EPI. Thus, the absolute value of the compensation parameter 𝛾𝛾 is
now lower than before. 17 Compared to the model without manipulation, the expected utility of
the principal is reduced by
𝑑𝑑 ∗
𝑖𝑖 .
2
The principal anticipates that the firm’s real emissions 𝑒𝑒 ∗ are
higher than without manipulation which results in higher future damage compensation.
If the principal suffers from reputation losses, the manager will be less inclined to manipulate
the EPI. There will be no report manipulation with sufficiently large reputation losses, i.e.: 18
17 𝑐𝑐𝑖𝑖𝑑𝑑
𝑐𝑐𝑒𝑒+𝑐𝑐𝑖𝑖
18
< 𝑑𝑑 holds because of 𝑐𝑐
𝑐𝑐𝑖𝑖
𝑒𝑒 +𝑐𝑐𝑖𝑖
< 1.
An interesting implication of this corner solution is that the GHG emission level 𝑒𝑒 ∗ will be expanded to its
maximum, 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 . That’s the only way for the firm to truthfully commit that they did not manipulate and therefore
should not be blamed for emission reporting manipulation through a reputation loss.
24
𝑑𝑑
𝑟𝑟
≤
𝑐𝑐𝑒𝑒 𝑐𝑐𝑖𝑖
(22)
Given that (22) does not hold, the manipulation incentive is mitigated but does not disappear.
Under this assumption, 𝛾𝛾 remains negative but its absolute value decreases compared to the
situation without reputation loss. The principal anticipates that the manager manipulates the
GHG emission report. To achieve lower manipulation levels, the principal allows some more
units of GHG emissions because otherwise the manager’s incentive compatibility constraints
would not be met and the manager would not sign the compensation contract. As a consequence,
the principal weakens the relation between the manager’s remuneration and the EPI by reducing
the absolute value of 𝛾𝛾.
Not surprisingly, reputation losses reduce the principal’s expected utility even more. 19 Thus,
under strict liability, the principal should be interested in using manipulation-proof environmental performance indicators. Consistently with prior analysis under strict liability, the measurement error, 𝜀𝜀, is not relevant.
3.3 Emission reporting manipulation under a negligence regime
Under a negligence regime, only the principal’s expected utility differs to the case of strict
liability. The manager’s expected utility does not change nor do the constraints of the optimization problem.
19
𝐸𝐸𝑈𝑈 𝑃𝑃 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑅𝑅.𝐿𝐿. < 𝐸𝐸𝑈𝑈 𝑃𝑃 𝑤𝑤𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑅𝑅.𝐿𝐿. < 𝐸𝐸𝑈𝑈 𝑃𝑃 𝑤𝑤𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑖𝑖
25
Table 6: Results with emission reporting manipulation under a negligence regime
No emission reporting
manipulation
principal’s objective function
individual optima:
GHG emission
level and manipulation level
environmental compensation contract
parameter
expected utility of
the principal in
equilibrium
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽
−𝛾𝛾𝛾𝛾 − 𝑑𝑑𝑑𝑑 �1 −
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 −𝑒𝑒+𝜀𝜀
2𝜀𝜀
Emission reporting manipulation, but no reputation loss
�
𝑑𝑑
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 2𝑐𝑐
− 2(𝜀𝜀𝑐𝑐
𝑒𝑒+𝑑𝑑)
𝑑𝑑
𝑒𝑒 +𝑑𝑑)
1
1
∙
2 𝑐𝑐
−
(𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2
1
−𝑑𝑑𝑒𝑒 ∗ �2 −
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 −𝑒𝑒 ∗
2𝜀𝜀
�
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
1
1
∙ 𝑐𝑐
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑒𝑒
2𝑐𝑐𝑖𝑖 (𝑑𝑑+𝜀𝜀𝑐𝑐𝑒𝑒
1
+
)
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑒𝑒
2(𝑑𝑑+𝜀𝜀𝑐𝑐𝑒𝑒)
𝑏𝑏
2
−
1
𝑑𝑑𝑒𝑒
𝑑𝑑𝑐𝑐𝑖𝑖
𝑒𝑒 (𝑐𝑐𝑒𝑒
∗
𝑖𝑖
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑒𝑒
𝑒𝑒
𝜀𝜀𝜀𝜀𝑐𝑐𝑒𝑒
+ (𝑑𝑑+𝜀𝜀𝑐𝑐
+𝑐𝑐 )
𝑒𝑒 )(𝑐𝑐𝑒𝑒+𝑐𝑐𝑖𝑖)
𝑖𝑖 = (𝑑𝑑+𝜀𝜀𝑐𝑐 ) +
2(𝑐𝑐
2𝑐𝑐
𝑖𝑖
𝜀𝜀𝜀𝜀𝑐𝑐𝑒𝑒2
𝑒𝑒
𝑐𝑐𝑖𝑖(𝑑𝑑+𝜀𝜀𝑐𝑐𝑒𝑒)(𝑐𝑐𝑒𝑒+𝑐𝑐𝑖𝑖 )
𝑑𝑑𝑒𝑒
𝑐𝑐
𝑑𝑑
𝑒𝑒 +𝑐𝑐𝑖𝑖 )
𝑐𝑐 𝑑𝑑
−
𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒
𝛾𝛾 = − 2(𝑑𝑑+𝜀𝜀𝑐𝑐
− 2(𝑐𝑐 𝑖𝑖+𝑐𝑐 )
)
𝑐𝑐𝑖𝑖 𝑑𝑑
𝜀𝜀𝜀𝜀𝑐𝑐𝑒𝑒 2
2(𝑐𝑐𝑒𝑒 +𝑐𝑐𝑖𝑖)
+ (𝑑𝑑+𝜀𝜀𝑐𝑐
𝑐𝑐𝑒𝑒
+ �
1
𝑐𝑐𝑖𝑖
𝑐𝑐
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 −𝑒𝑒 ∗ + 𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −𝑒𝑒 ∗ )
𝑐𝑐𝑖𝑖
2𝜀𝜀
2𝜀𝜀
− 2𝑐𝑐
𝑑𝑑
∗ 2 � �1
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 + 𝛾𝛾𝛾𝛾 −
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 −𝑒𝑒+𝑖𝑖+𝜀𝜀
� −𝑟𝑟𝑟𝑟
𝑑𝑑𝑑𝑑 �1 −
𝑚𝑚𝑚𝑚𝑚𝑚
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 2(𝑑𝑑+𝜀𝜀𝑐𝑐
)
2(𝑐𝑐𝑒𝑒 +𝑐𝑐𝑖𝑖 )
− 2 𝑐𝑐𝑒𝑒 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒
−𝑑𝑑𝑒𝑒 ∗ �2 −
�
𝑒𝑒)
𝑑𝑑𝑐𝑐𝑖𝑖
𝛾𝛾 = −
2
𝑏𝑏
𝑐𝑐𝑒𝑒
2
�
2𝜀𝜀
𝑒𝑒 (𝑐𝑐𝑒𝑒+𝑐𝑐𝑖𝑖 )
𝑖𝑖 =
𝛾𝛾 = −𝑐𝑐𝑒𝑒 �2𝑐𝑐 + 2(𝜀𝜀𝑐𝑐
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 −𝑒𝑒+𝑖𝑖+𝜀𝜀
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 2(𝑑𝑑+𝜀𝜀𝑐𝑐
∗
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
𝑒𝑒
−𝑑𝑑𝑑𝑑 �1 −
− 2𝑐𝑐
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 −𝛾𝛾(𝑒𝑒 − 𝑖𝑖)
Emission reporting manipulation
and reputation loss
2
1
𝑒𝑒
𝑒𝑒)(𝑐𝑐𝑒𝑒 +𝑐𝑐𝑖𝑖)
1
𝑒𝑒
𝑖𝑖
1
1
∙ 𝑐𝑐 − 2 𝑐𝑐𝑒𝑒 2 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2 �𝑐𝑐 + 𝑐𝑐 �
𝑏𝑏
1
−𝑑𝑑𝑒𝑒 ∗ �2 −
𝑐𝑐
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 −𝑒𝑒 ∗ + 𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
� − 𝑟𝑟 𝑐𝑐 (𝑒𝑒
− 𝑒𝑒 ∗ )
𝑐𝑐 𝑒𝑒 𝑚𝑚𝑚𝑚𝑚𝑚
𝑐𝑐𝑖𝑖
2𝜀𝜀
𝑒𝑒
−𝑒𝑒 ∗ )
𝑖𝑖
Table 6 summarizes the results under a negligence regime for three different scenarios: (a) when
there is no reporting manipulation, (b) when there is manipulation, and no reputation loss of the
principal due to manipulation, (c) when there is manipulation and a reputation loss of the principal. First, we analyze the results without reputation loss. In order to keep the interpretation of
results simple, we will assume in the following the cost parameters to be equal: 𝑐𝑐𝑒𝑒 = 𝑐𝑐𝑖𝑖 = 𝑐𝑐.
This simplification does not affect qualitative results but allows us to highlight the impact of
the liability regime and of reputation losses.
In the absence of reputation losses, in equilibrium there is reporting manipulation (𝑖𝑖 ∗ is greater
than 0) and consequently, real GHG emissions increase. Those findings are qualitatively the
same as under strict liability.
26
�
𝑖𝑖
What is different to strict liability is that the principal may benefit from GHG emission reporting
manipulation. Recall, that with a negligence regime, the principal only pays damage compensation when she is found negligent which in turn depends on what the EPI reports. Since the
EPI is the only verifiable measure of real GHG emissions, manipulation of the EPI reduces the
probability of being held liable. Thus, the principal benefits from emission reporting manipulation given that 2𝑑𝑑 > 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 holds, i.e. if the incremental future damage is sufficiently large.
If 2𝑑𝑑 ≤ 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 holds, manipulation still can make sense as long as the GHG emission report is
sufficiently precise, i.e. 𝜀𝜀 ≤ 𝜀𝜀̅. 20 Here, we need to talk about the link between EPI precision and
incentives to manipulate. Less precise indicators, that is higher values of 𝜀𝜀, lead to less emission
reporting manipulation activity,
𝜕𝜕𝑖𝑖 ∗
𝜕𝜕𝜕𝜕
< 0. The reason is that less precise measurement techniques
reduce the marginal benefit of manipulation activities. Recall that under strict liability, manipulation incentives were not tied to 𝜀𝜀.
Thus, under a negligence rule, more precise measurement techniques imply higher marginal
benefits for the manipulation activity. As a consequence, for less precise measurement techniques, 𝜀𝜀 > 𝜀𝜀̃, manipulation does not pay off for the principal.
This is an interesting result: If (a) the firm’s shareholders suffer from sufficiently high damage
compensation and/or (b) the measurement technique is sufficiently precise, the shareholders
benefit from the manager’s manipulation activities under a negligence regime.
20
𝜀𝜀̅ is defined as the threshold error for which the principal’s expected utility with GHG emission reporting ma-
nipulation is the same as without reporting manipulation: 𝜀𝜀̃ = −𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 + �𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 2 −
𝑑𝑑 2
𝑐𝑐 2
+ 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2
27
Result 2: When we allow for GHG emission reporting manipulation, the manager is
inclined to manipulate reports which in turn impairs the need to reduce real GHG emissions. Thus, emissions increase and so do future damages. Under strict liability, the
firm’s shareholders are fully responsible for damages and thus, they are worse-off when
GHG reporting manipulation is possible. In contrast, under a negligence regime, the
firm’s shareholders may benefit from GHG reporting manipulation since this reduces
the likelihood of being held negligent and of damage compensation payments. This
“benefit” is higher, the more precise the GHG measurement is.
When we introduce a reputation loss, we obtain analogous results to the findings under strict
liability: The level of reporting manipulation 𝑖𝑖 ∗ decreases while real GHG emissions 𝑒𝑒 ∗ in-
crease. Because the principal anticipates the possible future reputation loss through the emission
reporting manipulation, the contract design will induce the manager to reduce manipulation.
However, this comes up with the cost of higher real GHG emission levels in equilibrium in
order to meet the manager’s incentive compatibility constraints. How to induce the manager
not to manipulate the emission report and to reduce real GHG emissions?
3.4 Punitive damages
One way out might be punitive damages, that is, when the firm’s shareholders are held liable in
excess of the environmental damage. Punitive damages are expected to provide stronger incentives to reduce GHG emissions. We therefore now assume that the present value of damage
compensation equals 𝜔𝜔𝜔𝜔
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 where 𝜔𝜔 > 1 reflects the punitive character.
As a consequence, the firm’s emissions 𝑒𝑒 ∗ which are specified in the right column of Table 6
(negligence, reporting manipulation, and reputation losses) depend on 𝜔𝜔𝜔𝜔 instead of 𝑑𝑑. It can
28
be shown that with higher punitive damages, GHG emissions in equilibrium decrease, that is,
𝛿𝛿𝑒𝑒 ∗
𝛿𝛿𝛿𝛿
< 0. 21 Given that in the absence of punitive damages, reputation losses induce the manager
to choose GHG emission levels beyond the socially desirable level, there must be one 𝜔𝜔 which
ensures the efficient GHG emissions.
Since
𝜕𝜕𝑒𝑒 ∗
𝜕𝜕𝜔𝜔
< 0 and
𝜕𝜕𝜕𝜕
𝜕𝜕𝑒𝑒 ∗
> 0, we get
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
< 0, such that more punitive damages makes the man-
ager’s compensation more sensitive to 𝜔𝜔 (recall that 𝛾𝛾 < 0). Consistently, the fixed salary 𝛼𝛼
increases, since
𝜕𝜕𝜕𝜕
𝜕𝜕𝑒𝑒 ∗
= 𝑐𝑐 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 2𝑒𝑒 ∗ ) ≤ 022 and consequently,
𝜕𝜕𝜕𝜕
𝜕𝜕𝜔𝜔
≥ 0.
There is also an indirect effect of punitive damages on real emission levels. Other things being
equal, punitive damages induce the manager to manipulate emission reporting more which in
turn tends to increase real emission levels. The direct effect on reducing emission levels is
stronger, though, such that
𝜕𝜕𝑒𝑒 ∗
𝜕𝜕𝜔𝜔
< 0.
We know from the model without reporting manipulation (Section 2.5) that a negligence rule
may provide excessive incentives to reduce GHG emissions due to the type-1-error of the measurement device. This is quite the opposite problem to the one above. Consequently, in such as
setting, the regulator may want to appropriately limit liability in order to increase GHG emissions to the efficient level, thus, 0 < 𝜔𝜔 < 1.
∗
21 𝜕𝜕𝑒𝑒
𝜕𝜕𝜔𝜔
=
−𝜀𝜀𝜀𝜀(𝑐𝑐(2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 +𝜀𝜀)+𝑑𝑑+2𝑟𝑟)
4(𝜀𝜀𝜀𝜀+𝑑𝑑𝑑𝑑)2
< 0 because 𝑑𝑑, 𝑐𝑐, 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑟𝑟, 𝜔𝜔 > 0 and also 𝜀𝜀 ≥ 0. Even if we relax the assumption
𝑐𝑐𝑒𝑒 = 𝑐𝑐𝑖𝑖 = 𝑐𝑐 and allow for 𝑐𝑐𝑒𝑒 ≠ 𝑐𝑐𝑖𝑖 , we obtain
22
𝛿𝛿𝑒𝑒 ∗
𝛿𝛿𝛿𝛿
< 0.
In the model with reporting manipulation 2𝑒𝑒 ∗ ≥ 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 must hold, otherwise the expected EPI would be negative
which does not make sense: 𝐸𝐸[𝑦𝑦] = 𝐸𝐸[𝑒𝑒 − 𝑖𝑖] = 𝑒𝑒 ∗ − (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ ) = 2𝑒𝑒 ∗ − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ≥ 0.
29
Result 3: If the firm suffers from sufficiently big reputation losses when manipulating GHG
emission reports, managers will not engage in reporting manipulation under either liability
regime. However, this comes up with the cost of higher real GHG emissions. Appropriately
defined punitive (environmental) damages will mitigate the problem of excessive GHG
emissions. Thus, in the presence of reputation losses, the regulator should introduce punitive
damages.
4. Conclusion
How does the liability regime for environmental damages induce the manager to precisely
measure and to truthfully report GHG emissions? These questions are important for properly
governing GHG emissions, especially since there is considerable technical and legal discretion
on the measurement and reporting of GHG emissions. This is the first paper highlighting the
interaction between the environmental liability regime and GHG emission reporting incentives.
This paper adopts a principal-agent-model where a manager decides on real GHG emission
levels and on the extent of manipulating reported GHG emission levels. Real GHG emission
levels are assumed not to be verifiable, but reported GHG emission levels are.
We allow for three liability regimes: no liability, strict liability and a negligence rule. With strict
liability, the firm is held liable whenever GHG emissions cause damages. With a negligence
regime, the firm is held liable if GHG emissions cause damages and, additionally, if the firm
acted negligently, that is, failed to meet the standard of due care specified by legal rules.
If there is no reporting manipulation, we find that a strict liability rule tends to provide efficient
incentives to reduce GHG emissions. However, a strict liability rule does not induce the manager to measure GHG emissions as precisely as possible. With a negligence regime, however,
30
it is important to know whether the firm’s GHG emissions exceed the standard level allowed
by law or not. With more precision on GHG emission measurement it is less likely that the
measurement device reports non-tolerable GHG emission levels even though real emission are
lower and not indicating negligence. Due to this type-1-error, real GHG emission levels are
generally lower than the socially optimal emission level.
When we allow for GHG emission reporting manipulation, the manager is inclined to manipulate reports which in turn impairs the need to reduce real GHG emissions. Thus, emissions
increase and so do future damages. Under strict liability, the firm’s shareholders are fully responsible for damages and thus, they are worse-off when GHG reporting manipulation is possible. In contrast, under a negligence regime, the firm’s shareholders may benefit from GHG
reporting manipulation since this reduces the likelihood of being held negligent and of damage
compensation payments. This “benefit” is higher, the more precise the GHG measurement is.
Thus, overall, we find that while a negligence regime better encourages more precise GHG
emission measurement than strict liability, it also provides stronger incentives for manipulation.
Regulatory bodies may still be able to capture the benefit of a negligence rule (incentives for
more precise GHG emission measurement) while mitigating its negative effect on reporting
manipulation – given that the financial and/or product markets sufficiently penalize the firm by
reputation losses and that there are punitive damages. Thus, regulatory bodies might want to set
up a public register for firms that have proven to report manipulated GHG emission levels.
There are also limitations to mention. We do not thoroughly analyze the impact of manager’s
personal liability and do not explicitly allow for manager’s capacity constraints. Basic analyses
suggest that qualitative results may not change too much, though. Moreover, we assume that
victims do not bear transaction costs to bring a lawsuit while in fact these costs are positive.
This would add a third player and require a game-theoretical extension.
31
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35
Appendix
A1: Results under No Liability
Table A 1: Results without Liability
Lagrange-Multipliers
manager’s actions
compensation contract parameter
Expected Utility
𝜆𝜆 = 1
µ=0
𝜐𝜐 = 0
𝑏𝑏 ∗ =
1
𝑐𝑐𝑏𝑏
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
1 1
𝛼𝛼 = − ∙
2 𝑐𝑐𝑏𝑏
𝛽𝛽 = 1
𝛾𝛾 = 0
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 ∗ − 𝛼𝛼 =
𝐸𝐸𝑈𝑈 𝑀𝑀 = 0
1 1
∙
2 𝑐𝑐𝑏𝑏
First, it can be noticed that IC 1 and IC 2 are not binding because the Lagrange-Multipliers
become zero. This is not surprising because, without risk aversion of one player, the first-best
contract can be achieved 23. The manager’s actions cannot be observed directly by the principal,
but without risk aversion the principal can implement a forcing contract. The manager is always
pushed to her zero reservation utility by the fixed salary part 𝛼𝛼 - such that a deviation from her
optimal actions 𝑏𝑏 ∗ and 𝑒𝑒 ∗ , expressed by IC 1 and IC 2, does not change her expected utility and
the Lagrange-Multipliers become zero.
Second, since the principal is assumed to bear no liability costs, emissions will reach the max1
imum level 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 . The compensation contract has the form of 𝑠𝑠(𝑥𝑥, 𝑦𝑦) = − ∙
1
2 𝑐𝑐𝑏𝑏
23
+ 1 ∙ 𝑥𝑥 = 𝑠𝑠(𝑥𝑥).
See e.g. Holmström (1979), Feltham & Xie (1994) or Segerson & Tietenberg (1992).
36
A2: Socially optimal emission levels
The expected social cost of GHG emissions reads
1
𝑒𝑒
𝐸𝐸𝐸𝐸(𝑒𝑒) = 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 + 𝛿𝛿
𝐷𝐷
2
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚
(A 1)
First partial derivative:
𝜕𝜕𝜕𝜕𝜕𝜕
1
= 0 = −𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒) + 𝛿𝛿
𝐷𝐷
𝜕𝜕𝜕𝜕
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚
(A 2)
Proof of the sufficient condition:
𝜕𝜕𝜕𝜕𝜕𝜕 2
= 𝑐𝑐𝑒𝑒 > 0 𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝜕𝜕 2 𝑒𝑒
(A 3)
Rearranging the first partial derivative:
(A 4)
𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝛿𝛿
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 1
𝑑𝑑
= 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑒𝑒
𝑐𝑐𝑒𝑒
A3: Results under strict liability
The principal’s objective function under strict liability is:
max 𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝛼𝛼,𝛽𝛽,𝛾𝛾,𝑏𝑏,𝑒𝑒
The resulting Lagrange-Function is reflected by:
𝐿𝐿 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚
1
1
𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 − 𝑈𝑈�
2
2
+ µ(𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏) + 𝜐𝜐�𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)�
+ 𝜆𝜆 �𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 −
(A 5a)
(A 5b)
37
Optimization yields the results summarized in Table A 2. IC 1 and IC 2 are again not binding
because the particular Lagrange-Multipliers become zero. Also the manager’s effort for “business as usual” does not change because the liability payments only depend on the emission
level. The manager chooses the socially optimal emission level 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 derived in 2.3.
Table A 2: Results under Strict Liability
Lagrange-Multipliers
manager’s actions
compensation contract
parameter
Expected Utility
𝜆𝜆 = 1
µ=0
𝜐𝜐 = 0
𝑏𝑏 ∗ =
1
𝑐𝑐𝑏𝑏
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑑𝑑
= 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜
𝑐𝑐𝑒𝑒
1 1
1 𝑑𝑑
𝛼𝛼 = − ∙ + 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
�
2 𝑐𝑐𝑏𝑏
2 𝑐𝑐𝑒𝑒
𝛽𝛽 = 1
𝛾𝛾 = −𝑑𝑑
𝐸𝐸𝑈𝑈 𝑃𝑃 = 𝑏𝑏 ∗ − 𝛼𝛼 =
𝐸𝐸𝑈𝑈 𝑀𝑀 = 0
1
The compensation contract is 𝑠𝑠(𝑥𝑥, 𝑦𝑦) = − ∙
1 1
1 𝑑𝑑
∙ − 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
�
2 𝑐𝑐𝑏𝑏
2 𝑐𝑐𝑒𝑒
1
2 𝑐𝑐𝑏𝑏
+ 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
1 𝑑𝑑
2 𝑐𝑐𝑒𝑒
� + 1 ∙ 𝑥𝑥 − 𝑑𝑑 ∙ 𝑦𝑦. With higher
absolute values of 𝑦𝑦, the manager gets less compensation. Because 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
1 𝑑𝑑
2 𝑐𝑐𝑒𝑒
> 0 holds, the
fixed part of the salary, 𝛼𝛼, increases in comparison to the no liability case. Fixed salary increases
in order to compensate for the manager’s disutility from decreasing emission levels. The increase in fixed salary decreases the principal’s expected utility implying higher agency cost.
A4: Results under negligence given that liability is possible, 𝑷𝑷�𝒚𝒚 ≤ 𝒚𝒚𝒐𝒐𝒐𝒐𝒐𝒐 � < 𝟏𝟏
The expected utility of the principal reads:
38
𝐸𝐸𝑈𝑈 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
= 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 ∙ P�𝑦𝑦 > 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 �
𝑒𝑒
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1 −
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 + 𝜀𝜀
�
2𝜀𝜀
(A 6a)
In analogy to the case of strict liability and no liability the Lagrange function follows as:
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 + 𝜀𝜀
�
𝑒𝑒𝑚𝑚𝑚𝑚𝑥𝑥
2𝜀𝜀
1
1
+ 𝜆𝜆 �𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 − 𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 − 𝑈𝑈�
2
2
+ µ(𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏) + 𝜐𝜐�𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)�
𝑒𝑒
𝐿𝐿 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 − 𝛿𝛿
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1 −
(A 6b)
The results of the Lagrange optimization are summarized in Table A 3.
Table A 3: Results under Negligence (when liability is possible: 𝑃𝑃�𝑦𝑦 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � < 1)
Lagrange-Multipliers
manager’s actions
compensation contract
parameter
Expected Utility
𝜆𝜆 = 1
µ=0
𝜐𝜐 = 0
𝑏𝑏 ∗ =
1
𝑐𝑐𝑏𝑏
𝑑𝑑
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
−
2𝑐𝑐𝑒𝑒 2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑)
1 1 𝑐𝑐𝑒𝑒
𝛼𝛼 = − ∙ + �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 − 𝑒𝑒 ∗ 2 �
2 𝑐𝑐𝑏𝑏 2
𝛽𝛽 = 1
𝑑𝑑
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
𝛾𝛾 = 𝑐𝑐𝑒𝑒 (𝑒𝑒 ∗ − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ) = −𝑐𝑐𝑒𝑒 �
+
�
2𝑐𝑐𝑒𝑒 2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑)
1 1
𝑐𝑐𝑒𝑒
1 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗
𝐸𝐸𝑈𝑈 𝑃𝑃 = ∙ − − (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2 − 𝑑𝑑𝑒𝑒 ∗ � −
�
2 𝑐𝑐𝑏𝑏
2
2
2𝜀𝜀
𝐸𝐸𝑈𝑈 𝑀𝑀 = 0
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
Again IC 1 and IC 2 are not binding, just as in the case of strict liability and no liability. The
1
compensation contract has the form of 𝑠𝑠(𝑥𝑥, 𝑦𝑦) = − ∙
1
2 𝑐𝑐𝑏𝑏
+
𝑐𝑐𝑒𝑒
2
�𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 − 𝑒𝑒 ∗ 2 � + 1 ∙ 𝑥𝑥 +
𝑐𝑐𝑒𝑒 (𝑒𝑒 ∗ − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ) ∙ 𝑦𝑦. The optimal contract depends on the firm’s emission level 𝑒𝑒 ∗ in equilib-
rium. The parameter 𝛾𝛾 is again negative because 𝑐𝑐𝑒𝑒 (𝑒𝑒 ∗ − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ) < 0. The link between the
39
manager’s compensation and the EPI – as reflected by 𝛾𝛾 – becomes stronger the more precise
the EPI gets. Moreover, the following relations hold:
𝑒𝑒 ∗ ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 .
𝑒𝑒 ∗ ≥ 0.
Proof of (A 6c): 𝑒𝑒 ∗ ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
𝜕𝜕𝑒𝑒 ∗
> 0 𝑖𝑖𝑖𝑖 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑐𝑐𝑒𝑒 > 0
𝜕𝜕𝜕𝜕
(A 6c)
(A 6d)
(A 6e)
The firm’s emission level in equilibrium is defined by:
𝑒𝑒 ∗ =
𝑑𝑑
𝜀𝜀𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + 2 �𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝜀𝜀�
𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑
If 𝑒𝑒 ∗ ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 would be valid:
𝑑𝑑
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
�= 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
−
�
2𝑐𝑐𝑒𝑒 2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑)
𝑒𝑒 ∗ ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 ↔ 𝑒𝑒 ∗ − 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 ≤ 0
Inserting 𝑒𝑒 ∗ and multiply with 𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑:
Rearranging:
𝑑𝑑
𝜀𝜀𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + �𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝜀𝜀� − 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 𝜀𝜀𝑐𝑐𝑒𝑒 − 𝑑𝑑𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 ≤ 0
2
𝑑𝑑
𝜀𝜀𝑐𝑐𝑒𝑒 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 � − �𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 + 𝜀𝜀� ≤ 0
2
(A 7a)
(A 7b)
(A 7c)
(A 7d)
𝑑𝑑
Inserting 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑐𝑐 :
Rearranging:
𝜀𝜀𝑐𝑐𝑒𝑒 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 +
Multiplying with 2/𝑑𝑑:
𝑑𝑑
𝑑𝑑
𝑑𝑑
� − �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − + 𝜀𝜀� ≤ 0
𝑐𝑐𝑒𝑒
2
𝑐𝑐𝑒𝑒
𝑑𝑑
𝑑𝑑 2 𝜀𝜀𝜀𝜀
𝜀𝜀𝜀𝜀 − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 +
−
≤0
2
2𝑐𝑐𝑒𝑒 2
2𝜀𝜀 − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 +
𝑑𝑑
− 𝜀𝜀 ≤ 0
𝑐𝑐𝑒𝑒
(A 7e)
(A 7f)
(A 7g)
40
Rearranging:
𝑑𝑑
𝜀𝜀 ≤ 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑐𝑐 = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 .
(A 7h)
𝑒𝑒
Since we assume 𝜀𝜀 ≤ 𝑒𝑒𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 , also 𝑒𝑒 ∗ ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 holds.
q.e.d.
Proof of (A 6d), 𝑒𝑒 ∗ ≥ 0
𝑒𝑒 ∗ =
𝑑𝑑
𝜀𝜀𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + 2 �𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝜀𝜀�
𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑
(A 8a)
If 𝜀𝜀 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 is valid, all mathematical terms in (A 8a) will be positive per definitionem. As a
consequence, 𝑒𝑒 ∗ has to be positive, too.
Proof of (A 6e),
𝜕𝜕𝑒𝑒 ∗
𝜕𝜕𝜕𝜕
>0
First partial derivative:
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑑𝑑
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
−
2𝑐𝑐𝑒𝑒 2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑)
𝜕𝜕𝑒𝑒 ∗
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑 ∙ 𝑐𝑐𝑒𝑒
=
> 0 𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝜕𝜕𝜕𝜕
2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑 )2
q.e.d.
(A 9a)
(A 9b)
q.e.d.
Derivate of the principal’s expected utility under negligence:
𝐸𝐸𝑈𝑈 𝑃𝑃 =
1 1
𝑐𝑐𝑒𝑒
1 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗
∙ − 𝑈𝑈 − (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2 − 𝑑𝑑𝑒𝑒 ∗ � −
�
2 𝑐𝑐𝑏𝑏
2
2
2𝜀𝜀
First partial derivative with respect to 𝜀𝜀:
(A 10a)
41
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗ 𝜕𝜕𝑒𝑒 ∗
𝜕𝜕𝐸𝐸𝐸𝐸 𝑃𝑃
𝜕𝜕𝑒𝑒 ∗ 𝑑𝑑 𝜕𝜕𝑒𝑒 ∗
= 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ ) ∙
− ∙
+ 𝑑𝑑
∙
+ 𝑑𝑑𝑒𝑒 ∗
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜀𝜀
2𝜀𝜀
𝜕𝜕𝜕𝜕
𝜕𝜕𝑒𝑒 ∗
− 𝜕𝜕𝜕𝜕 ∙ 2𝜀𝜀 − �𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗ � ∙ 2
�
∙�
4𝜀𝜀 2
(A 10b)
With:
𝜕𝜕𝑒𝑒 ∗
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑 ∙ 𝑐𝑐𝑒𝑒
=
𝜕𝜕𝜕𝜕
2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑 )2
(A 10c)
Inserting (A 10c) and simplifying:
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗
𝜕𝜕𝐸𝐸𝐸𝐸 𝑃𝑃
= −𝑑𝑑𝑒𝑒 ∗ ∙
≤0
𝜕𝜕𝜕𝜕
2 ∙ 𝜀𝜀 2
(A 10d)
because 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 ≥ 𝑒𝑒 ∗ is valid (see Proof of (A 6c)) and 𝑑𝑑, 𝑒𝑒 ∗ , 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 , 𝜀𝜀 ≥ 0.
A5: Results under negligence given that there is no liability, 𝑷𝑷�𝒚𝒚 ≤ 𝒚𝒚𝒐𝒐𝒐𝒐𝒐𝒐 � = 𝟏𝟏
Determination of the threshold measurement error 𝜺𝜺� that ensures no liability. With no liability, it must hold:
Inserting 𝑒𝑒�∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑑𝑑
2𝑐𝑐𝑒𝑒
−
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒�∗ + 𝜀𝜀̂
=1
2𝜀𝜀̂
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
2(𝜀𝜀�𝑐𝑐𝑒𝑒 +𝑑𝑑)
𝜀𝜀̂ 2 +
(A 11a)
and rearranging:
3 𝑑𝑑
1 𝑑𝑑
𝜀𝜀̂ −
𝑦𝑦 = 0
2 𝑐𝑐𝑒𝑒
2 𝑐𝑐𝑒𝑒 𝑜𝑜𝑜𝑜𝑜𝑜
(A 11b)
Solution approach for quadratic equations and rearranging:
𝜀𝜀̂1,2 = −
3 𝑑𝑑 1 𝑑𝑑
𝑐𝑐𝑒𝑒
�1 + 8 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
±
4 𝑐𝑐𝑒𝑒 4 𝑐𝑐𝑒𝑒
𝑑𝑑
(A 11c)
42
−
1 𝑑𝑑
�1 + 8 𝑑𝑑𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 drops because 𝜀𝜀̂ ≥ 0.
Check whether 𝜀𝜀̂1 = −
3 𝑑𝑑
+
Whereas 𝜀𝜀̂2 = −
3 𝑑𝑑
4 𝑐𝑐𝑒𝑒
4 𝑐𝑐𝑒𝑒
4 𝑐𝑐𝑒𝑒
𝑐𝑐
1 𝑑𝑑
4 𝑐𝑐𝑒𝑒
−
𝑐𝑐
�1 + 8 𝑑𝑑𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ≥ 0 is valid:
3 𝑑𝑑 1 𝑑𝑑
𝑐𝑐𝑒𝑒
�1 + 8 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ≥ 0
+
4 𝑐𝑐𝑒𝑒 4 𝑐𝑐𝑒𝑒
𝑑𝑑
1 𝑑𝑑
𝑐𝑐𝑒𝑒
�−3 + �1 + 8 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 � ≥ 0
4 𝑐𝑐𝑒𝑒
𝑑𝑑
�1 + 8
𝑐𝑐𝑒𝑒
𝑒𝑒
≥3
𝑑𝑑 𝑚𝑚𝑚𝑚𝑚𝑚
𝑐𝑐𝑒𝑒
𝑒𝑒
≥1
𝑑𝑑 𝑚𝑚𝑚𝑚𝑚𝑚
This is valid because of the assumption 𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 > 𝑑𝑑.
q.e.d.
Principal’s expected utility 𝐄𝐄𝐄𝐄𝐏𝐏 under the “no-liability-constraint” 𝑒𝑒 ∗ + 𝜀𝜀 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
Since the manager receives the reservation utility of zero, and there are no damage payments,
the principal receives the financial income minus the manager’s disutility for her working effort
regarding the business as usual task and the emission reducing task. This leads to an expected
utility of the principal of:
𝐸𝐸𝐸𝐸
𝑃𝑃
𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙.
2
1 1 𝑐𝑐𝑒𝑒
1 1 𝑐𝑐𝑒𝑒 𝑑𝑑
∗ )2
(
= ∙ −
𝑒𝑒
− 𝑒𝑒
= ∙ − � + 𝜀𝜀�
2 𝑐𝑐𝑏𝑏 2 𝑚𝑚𝑚𝑚𝑚𝑚
2 𝑐𝑐𝑏𝑏 2 𝑐𝑐𝑒𝑒
(A 12)
Table A 4 summarizes the results with regard to the equilibrium emission level 𝑒𝑒 ∗ and with
regard to the principal’s expected utility for the case where the principal will not be found negligent and the case where liability is possible.
43
Table A 4: Equilibrium emission level 𝑒𝑒 ∗ and principal’s expected utility with no liability under negligence and
with liability under negligence
𝟎𝟎 ≤ 𝜺𝜺 ≤ 𝜺𝜺�: no liability under negligence
𝑒𝑒 ∗ = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝜀𝜀,
1
𝜕𝜕𝑒𝑒 ∗
1
𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙. = 2 ∙ 𝑐𝑐 −
𝜕𝜕𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙.
𝜕𝜕𝜕𝜕
𝑏𝑏
𝑐𝑐𝑒𝑒
2
𝑑𝑑
�𝑐𝑐 + 𝜀𝜀�
𝑒𝑒
2
𝐸𝐸𝑈𝑈 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 =
𝜕𝜕𝐸𝐸𝑈𝑈 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
𝑑𝑑
𝜕𝜕𝜕𝜕
𝑒𝑒
1
1 𝑑𝑑2
𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙.,𝑚𝑚𝑚𝑚𝑚𝑚 = 2 ∙ 𝑐𝑐 − 𝑈𝑈 − 2 𝑐𝑐 −
𝜀𝜀 = 𝜀𝜀̂ and 𝑒𝑒 ∗ = 𝑒𝑒�∗
1
𝑏𝑏
1
𝑒𝑒
1 𝑑𝑑2
𝑐𝑐𝑒𝑒
2
𝜀𝜀̂ 2 for
𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙.,𝑚𝑚𝑚𝑚𝑚𝑚 = 2 ∙ 𝑐𝑐 − 𝑈𝑈 − 2 𝑐𝑐 for 𝜀𝜀 = 0
and 𝑒𝑒 ∗ = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
𝑏𝑏
𝑒𝑒
𝑒𝑒
= −𝑐𝑐𝑒𝑒 �𝑐𝑐 + 𝜀𝜀� < 0
1
𝑑𝑑
𝑑𝑑
𝑚𝑚𝑚𝑚𝑚𝑚
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 2𝑐𝑐 − 2(𝜀𝜀𝑐𝑐
,
+𝑑𝑑)
= −1 < 0
𝜕𝜕𝜕𝜕
𝜺𝜺 > 𝜺𝜺�: liability possible
𝑒𝑒
𝑒𝑒
𝜕𝜕𝑒𝑒 ∗
𝜕𝜕𝜕𝜕
𝑒𝑒
𝑑𝑑∙𝑐𝑐
𝑚𝑚𝑚𝑚𝑚𝑚
𝑒𝑒
= 2(𝜀𝜀𝑐𝑐
>0
+𝑑𝑑)2
𝑒𝑒
2
1 1 𝑐𝑐𝑒𝑒 𝑑𝑑
1 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗
�
∙ − � + 𝜀𝜀� − 𝑑𝑑𝑒𝑒 ∗ � −
2 𝑐𝑐𝑏𝑏 2 𝑐𝑐𝑒𝑒
2
2𝜀𝜀
𝑑𝑑∙𝑒𝑒 ∗
= − 2∙𝜀𝜀2 ∙ �𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗ � < 0
1
1
𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚
for 𝜀𝜀 = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
1
1
1 𝑑𝑑2
𝑐𝑐𝑒𝑒
𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛,𝑚𝑚𝑚𝑚𝑚𝑚 = 2 ∙ 𝑐𝑐 − 𝑈𝑈 −
and 𝑒𝑒 ∗ = 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
𝑏𝑏
2
𝐸𝐸𝐸𝐸 𝑃𝑃 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛,𝑚𝑚𝑚𝑚𝑚𝑚 = 2 ∙ 𝑐𝑐 − 𝑈𝑈 − 2 𝑐𝑐 −
𝜀𝜀̂ and 𝑒𝑒 ∗ = 𝑒𝑒�∗
𝑏𝑏
𝑒𝑒
2
𝜀𝜀̂ 2 for 𝜀𝜀 =
A6: Emission Reporting manipulation under Strict Liability
The manager’s emission reporting manipulation influences the utility function of the manager
through the additional disutility of the corresponding working effort:
𝐸𝐸𝑈𝑈 𝑀𝑀 = 𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾 (𝑒𝑒 − 𝑖𝑖 ) −
1
1
1
𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2 − 𝑐𝑐𝑖𝑖 𝑖𝑖 2 ≥ 𝑈𝑈 = 0
2
2
2
(A 13a)
The individual rationality constraint (IR) follows directly from equation (A 14a). The incentive
compatibility constraints (IC 1 and IC 2) have to be complemented by one more equation for
the action 𝑖𝑖 (IC 3):
𝜕𝜕𝜕𝜕𝑈𝑈 𝑀𝑀
= 0 = 𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏∗
𝜕𝜕 𝑏𝑏
(A 3b)
44
𝜕𝜕𝜕𝜕𝑈𝑈 𝑀𝑀
(A 13c)
= 0 = 𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )
𝜕𝜕 𝑒𝑒
𝜕𝜕𝜕𝜕𝑈𝑈 𝑀𝑀
(A 13d)
= 0 = −𝛾𝛾 − 𝑐𝑐𝑖𝑖 𝑖𝑖 ∗
𝜕𝜕 𝑖𝑖
Summarizing this relations into a Lagrange function via introducing the additional Lagrange
multiplier 𝜑𝜑 for IC 3 yields:
𝐿𝐿 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 + 𝛾𝛾𝛾𝛾 − 𝑑𝑑𝑑𝑑
+ 𝜆𝜆 �𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 − 𝛾𝛾𝛾𝛾 −
1
1
𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2
2
2
(A 13e)
1
− 𝑐𝑐𝑖𝑖 𝑖𝑖 2 − 𝑈𝑈� + µ(𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏) + 𝜐𝜐�𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)�
2
+ 𝜑𝜑(−𝛾𝛾 − 𝑐𝑐𝑖𝑖 𝑖𝑖 )
Optimization leads to the results in Table A 5.
Table A 5: Results under Strict Liability and Emission Reporting manipulation
LagrangeMultipliers
manager’s
actions
No Reputation Loss
Reputation Loss
𝜆𝜆 = 1
µ=0
𝜐𝜐 = 𝜑𝜑 = −𝑖𝑖 ∗
1
𝑏𝑏 ∗ =
𝑐𝑐𝑏𝑏
𝜆𝜆 = 1
µ=0
𝑟𝑟
𝜐𝜐 = 𝜑𝜑 = −𝑖𝑖 ∗ − 𝑐𝑐
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑑𝑑
𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖
1 1 1 𝑑𝑑2
𝛼𝛼 = 𝑈𝑈 − ∙ + ∙
2 𝑐𝑐𝑏𝑏 2 𝑐𝑐𝑒𝑒
𝑑𝑑
𝑑𝑑
�𝑐𝑐𝑖𝑖 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − �
+
𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖
2
𝑖𝑖 ∗ =
compensation contract parameter
Expected
Utility
𝑑𝑑
𝑑𝑑
+
𝑐𝑐𝑒𝑒 𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖
𝛽𝛽 = 1
𝑐𝑐𝑖𝑖 𝑑𝑑
𝛾𝛾 = −
𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖
1 1
1 𝑑𝑑
�
𝐸𝐸𝑈𝑈 𝑃𝑃 = ∙ − 𝑈𝑈 − 𝑑𝑑 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
2 𝑐𝑐𝑏𝑏
2 𝑐𝑐𝑒𝑒
𝑑𝑑
− 𝑖𝑖 ∗
2
𝐸𝐸𝑈𝑈 𝑀𝑀 = 𝑈𝑈
𝑖𝑖
1
𝑐𝑐𝑏𝑏
𝑏𝑏 ∗ =
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + �
𝑖𝑖 ∗ = 𝑐𝑐
𝑑𝑑
𝑒𝑒 +𝑐𝑐𝑖𝑖
𝑑𝑑 𝑟𝑟
𝑐𝑐𝑒𝑒
− �∙�
− 1�
𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖
𝑐𝑐𝑒𝑒 𝑐𝑐𝑖𝑖
𝑟𝑟𝑐𝑐𝑒𝑒
− 𝑐𝑐 (𝑐𝑐
1
𝑖𝑖 𝑒𝑒+𝑐𝑐𝑖𝑖 )
1
𝑐𝑐 𝑐𝑐
= 𝑐𝑐
𝑐𝑐𝑒𝑒
𝑑𝑑
𝑑𝑑
𝑟𝑟
� − �
𝑐𝑐
𝑒𝑒 +𝑐𝑐𝑖𝑖 𝑐𝑐𝑒𝑒
𝑖𝑖
𝑟𝑟
𝛼𝛼 = 𝑈𝑈 − 2 ∙ 𝑐𝑐 + 𝑐𝑐 𝑒𝑒+𝑐𝑐𝑖𝑖 �𝑐𝑐 − 𝑐𝑐 � �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
1
𝑑𝑑
𝑟𝑟
� − ��
𝑐𝑐
2 𝑐𝑐𝑒𝑒
𝛽𝛽 = 1
𝛾𝛾 = 𝑐𝑐
𝑏𝑏
𝑒𝑒
𝑖𝑖
𝑒𝑒
𝑖𝑖
𝑖𝑖
1
𝑒𝑒 +𝑐𝑐𝑖𝑖
(𝑟𝑟𝑐𝑐𝑒𝑒 − 𝑑𝑑𝑐𝑐𝑖𝑖 )
1 1
𝑐𝑐𝑖𝑖
𝑑𝑑 𝑟𝑟
∙ − 𝑈𝑈 − 𝑖𝑖 ∗ � − � − 𝑑𝑑𝑒𝑒 ∗
2 𝑐𝑐𝑏𝑏
2
𝑐𝑐𝑒𝑒 𝑐𝑐𝑖𝑖
− 𝑟𝑟𝑖𝑖 ∗
𝐸𝐸𝑈𝑈 𝑀𝑀 = 𝑈𝑈
𝐸𝐸𝑈𝑈 𝑃𝑃 =
IC 2 and IC 3 now become binding because of the exchange relationship between 𝑒𝑒 and 𝑖𝑖24.
The business as usual task is not influenced so that 𝑏𝑏∗ and 𝛽𝛽 remain. In comparison to the initial
24
Deviation from the optimal solutions in Table A 5 influence the expected utilities because i is not contractible.
45
strict liability model, the resulting emission level raises by exactly this manipulation level.
Thus, the expected value of the EPI 𝑦𝑦 remains at the optimal level 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 . The principal antici-
pates that the manager manipulated and so she reduces the proportion of the manipulated salary
part 𝛾𝛾 25. As a consequence, the fixed salary parameter 𝛼𝛼 can be reduced too: because the pun-
ishment for high values of 𝑦𝑦 is now lower than before, the principal can reduce 𝛼𝛼 and still push
the manager’s expected utility to the reservation wage. The range of the EPI, 𝜀𝜀, is again neither
important for the principal nor the manager, just as in the previous analysis of strict liability.
If the detection of the manipulation is possible, the reputation loss will affect the expected utility
function of the principal and the Lagrange function will be:
𝐿𝐿 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 + 𝛾𝛾𝛾𝛾 − 𝑑𝑑𝑑𝑑 − 𝑟𝑟𝑟𝑟
+ 𝜆𝜆 �𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 − 𝛾𝛾𝛾𝛾 −
1
1
𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2
2
2
1
𝑐𝑐 𝑖𝑖 2 − 𝑈𝑈� + µ(𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏) + 𝜐𝜐�𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)�
2 𝑖𝑖
+ 𝜑𝜑(−𝛾𝛾 − 𝑐𝑐𝑖𝑖 𝑖𝑖 )
−
(A 13f)
The results of this optimization problem are already included in Table A 5. The introduction of
a reputation loss reduces the level of emission reporting manipulation in equilibrium 𝑖𝑖 ∗ because
the cost parameters and 𝑟𝑟 are positive. Looking closely at 𝑖𝑖 ∗ and 𝑒𝑒 ∗ reveals the additional necessary assumption
𝑑𝑑
𝑐𝑐𝑒𝑒
𝑟𝑟
> . Then, 𝛾𝛾 remains negative but its absolute value decreases compared
𝑐𝑐𝑖𝑖
to the situation without reputation loss. To reduce the level of emission reporting manipulation,
the principal accepts higher levels of GHG emissions. As a consequence, she weakens the relation between the manager’s compensation and the emission level by reducing the absolute
value of 𝛾𝛾. Therefore, she can also reduce the value of the fixed parameter 𝛼𝛼. The expected
25
Again, this fits with general results from agency theory which are presented in the beginning: Linking compen-
sation to performance measures works better the “better” (seemingly also “non-manipulable”) the performance
measures are (e.g. Holmström & Milgrom (1991), Gabel & Sinclair-Desgagné (1993), Lothe & Myrtveit (2003)).
46
utility of the manager is still pushed to her reservation wage. As one could expect, the introduction of the reputation loss into our model lowers the expected utility of the principal even
more 26.
A7: Emission Reporting manipulation under Negligence
Under negligence, the definition of emission reporting manipulation 𝑖𝑖 as well as the expected
utility of the manager do not change so that the constraints of the optimization problem are the
same as before. Only the expected utility of the principal is affected so that the Lagrange function follows as:
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 + 𝑖𝑖 + 𝜀𝜀
�
2𝜀𝜀
1
1
+ 𝜆𝜆 �𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝑒𝑒 − 𝛾𝛾𝛾𝛾 − 𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2
2
2
1
− 𝑐𝑐𝑖𝑖 𝑖𝑖 2 − 𝑈𝑈� + µ(𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏) + 𝜐𝜐�𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)�
2
+ 𝜑𝜑(−𝛾𝛾 − 𝑐𝑐𝑖𝑖 𝑖𝑖 )
𝐿𝐿 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 + 𝛾𝛾𝛾𝛾 − 𝑑𝑑𝑑𝑑 �1 −
(A 14a)
Table A 6 summarizes the results of this optimization problem. The Lagrange multipliers reveal
no surprising results. Only the absolute value of 𝜐𝜐 and 𝜑𝜑 changed. Also 𝑏𝑏 ∗ and 𝛽𝛽 do not change,
which is comprehensible. In accordance with the results under strict liability, the GHG emission
level in equilibrium 𝑒𝑒 ∗ increases and the emission reporting manipulation in equilibrium 𝑖𝑖 ∗ is
greater than 0. Both activity levels in equilibrium depend on the range of the EPI, 𝜀𝜀. Compared
to the situation without emission reporting manipulation, 𝛾𝛾 decreases 27 so that again the linkage
between the EPI 𝑦𝑦 and the manager’s compensation is weakened, just as it was observed under
strict liability. Here again the principal lowers the fixed salary part 𝛼𝛼 because she does not need
26
27
𝐸𝐸𝑈𝑈 𝑃𝑃 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑅𝑅.𝐿𝐿. < 𝐸𝐸𝑈𝑈 𝑃𝑃 𝑤𝑤𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑅𝑅.𝐿𝐿. < 𝐸𝐸𝑈𝑈 𝑃𝑃 𝑤𝑤𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑖𝑖
because
𝑐𝑐𝑖𝑖
𝑐𝑐𝑒𝑒+𝑐𝑐𝑖𝑖
<1
47
to compensate the manager’s punishment for GHG emissions that much. As always in this setting, the manager is pushed to her reservation wage.
Table A 6: Results under Negligence and Emission Reporting manipulation
LagrangeMultipliers
manager’s
actions
compensation contract parameter
No Reputation Loss
Reputation Loss
𝜆𝜆 = 1
µ=0
𝜆𝜆 = 1
µ=0
𝑟𝑟
𝑑𝑑𝑒𝑒 ∗
𝜐𝜐 = 𝜑𝜑 = −𝑖𝑖 ∗ − 𝑐𝑐 + 2𝜀𝜀𝑐𝑐
𝜐𝜐 = 𝜑𝜑 =
𝑏𝑏 ∗ =
1
𝑐𝑐𝑏𝑏
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝑑𝑑𝑐𝑐𝑖𝑖
𝑒𝑒 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
−
2(𝑑𝑑 + 𝜀𝜀𝑐𝑐𝑒𝑒 ) 2𝑐𝑐𝑒𝑒 (𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖 )
𝑑𝑑𝑒𝑒𝑚𝑚𝑎𝑎𝑎𝑎 𝑐𝑐𝑒𝑒
𝑑𝑑
𝑖𝑖 ∗ =
+
2𝑐𝑐𝑖𝑖 (𝑑𝑑 + 𝜀𝜀𝑐𝑐𝑒𝑒 ) 2(𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖 )
∗
1 1 1
𝛼𝛼 = 𝑈𝑈 − ∙ + 𝑐𝑐𝑒𝑒 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 − 𝑒𝑒 ∗ 2 �
2 𝑐𝑐𝑏𝑏 2
1 𝑐𝑐𝑒𝑒 2 ∗ 2
�𝑒𝑒 − 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 �
−
2 𝑐𝑐𝑖𝑖
𝛽𝛽 = 1
𝛾𝛾 = −
Expected
Utility
𝑑𝑑𝑒𝑒 ∗
− 𝑖𝑖 ∗
2𝜀𝜀𝑐𝑐𝑖𝑖
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑒𝑒
𝑐𝑐𝑖𝑖 𝑑𝑑
−
2(𝑑𝑑 + 𝜀𝜀𝑐𝑐𝑒𝑒 ) 2(𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖 )
𝐸𝐸𝑈𝑈 𝑃𝑃
1 1
= ∙ − 𝑈𝑈
2 𝑐𝑐𝑏𝑏
1
𝑐𝑐𝑒𝑒
− 𝑐𝑐𝑒𝑒 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 − 𝑒𝑒 ∗ 2 � �1 + �
𝑐𝑐𝑖𝑖
2
𝑐𝑐𝑒𝑒
∗
∗)
(𝑒𝑒
𝑦𝑦
−
𝑒𝑒
+
𝑜𝑜𝑜𝑜𝑜𝑜
1
𝑐𝑐𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒
∗�
�
− 𝑑𝑑𝑒𝑒
−
2
2𝜀𝜀
𝐸𝐸𝑈𝑈 𝑀𝑀 = 𝑈𝑈
𝑖𝑖
1
𝑏𝑏 ∗ =
𝑐𝑐𝑏𝑏
𝑖𝑖
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝑑𝑑𝑐𝑐𝑖𝑖
−
2(𝑑𝑑 + 𝜀𝜀𝑐𝑐𝑒𝑒 ) 2𝑐𝑐𝑒𝑒 (𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖 )
𝜀𝜀𝜀𝜀𝑐𝑐𝑒𝑒
+
(𝑑𝑑 + 𝜀𝜀𝑐𝑐𝑒𝑒 )(𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖 )
𝑐𝑐𝑒𝑒
𝑑𝑑𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝑑𝑑𝑐𝑐
𝜀𝜀𝜀𝜀𝑐𝑐
∗
𝑖𝑖 = �
+ (𝑐𝑐 𝑖𝑖 ) − (𝑑𝑑+𝜀𝜀𝑐𝑐 )(𝑐𝑐𝑒𝑒
)
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑐𝑐𝑖𝑖
2(𝑑𝑑+𝜀𝜀𝑐𝑐𝑒𝑒
1
1
2𝑐𝑐𝑒𝑒 𝑒𝑒+𝑐𝑐𝑖𝑖
1
�
𝑒𝑒+𝑐𝑐𝑖𝑖)
𝑒𝑒
1
1
𝛼𝛼 = 𝑈𝑈 − 2 ∙ 𝑐𝑐 − 2 𝑐𝑐𝑒𝑒 2 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2 �𝑐𝑐 + 𝑐𝑐 � +
𝑏𝑏
𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 − 𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒∗
𝛽𝛽 = 1
𝑑𝑑𝑒𝑒
𝑐𝑐
𝑐𝑐 𝑑𝑑
𝑒𝑒
𝑖𝑖
𝜀𝜀𝜀𝜀𝑐𝑐𝑒𝑒2
𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒
𝛾𝛾 = − 2(𝑑𝑑+𝜀𝜀𝑐𝑐
− 2(𝑐𝑐 𝑖𝑖+𝑐𝑐 ) + (𝑑𝑑+𝜀𝜀𝑐𝑐
)
𝑒𝑒
𝑒𝑒
𝑖𝑖
𝑒𝑒 )(𝑐𝑐𝑒𝑒 +𝑐𝑐𝑖𝑖)
𝐸𝐸𝑈𝑈 𝑃𝑃
1 1
1
1 1
= ∙ − 𝑈𝑈 − 𝑐𝑐𝑒𝑒 2 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2 � + �
2 𝑐𝑐𝑏𝑏
2
𝑐𝑐𝑒𝑒 𝑐𝑐𝑖𝑖
𝑐𝑐𝑒𝑒
∗
∗)
(𝑒𝑒
𝑦𝑦
−
𝑒𝑒
+
−
𝑜𝑜𝑜𝑜𝑜𝑜
1
𝑐𝑐𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒
∗�
�
− 𝑑𝑑𝑒𝑒
−
2𝜀𝜀
2
𝑟𝑟
− 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )
𝑐𝑐𝑖𝑖
𝐸𝐸𝑈𝑈 𝑀𝑀 = 𝑈𝑈
The difference between the principal’s expected utility with emission reporting manipulation
and the initial one under negligence reveals opposed effects: a cost effect and a liability effect.
The cost effect consists on the one hand of the savings through higher emission levels so that
the costs for emission reduction decrease. On the other hand, the principal has to bear the costs
for the manager’s emission reporting manipulation activity. These costs exceed the savings so
that all in all the cost effect is negative. On the contrary, the manager’s manipulation reduce the
EPI 𝑦𝑦 so that the costs for future damages for the firm decrease. Which of both effects domi-
nates depends on two aspects: if 2𝑑𝑑 > 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 holds, i.e. if the incremental future damage is
48
sufficiently high, the liability effect dominates and the overall effect will be positive 28. If 2𝑑𝑑 ≤
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 holds, i.e. if the incremental future damage is sufficiently low, the liability effect will
still dominate as long as the GHG emission report is sufficiently precise, i.e. 𝜀𝜀 ≤ 𝜀𝜀̃. Above this
threshold, the reports are so imprecise that the liability effect does not dominate any longer.
Thus, the overall effect becomes negative 29. The mathematical derivation follows.
Proof of 𝑬𝑬𝑼𝑼𝑷𝑷 𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘 𝑹𝑹.𝑳𝑳. < 𝑬𝑬𝑼𝑼𝑷𝑷 𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘 𝒊𝒊 if (a) 𝟐𝟐𝟐𝟐 > 𝒆𝒆𝒎𝒎𝒎𝒎𝒎𝒎 𝒄𝒄 or (b) 𝜺𝜺 ≤ 𝜺𝜺�:
The difference between the principal’s expected utility without reputation loss and the one in
the initial model without the manager’s possibility of manipulation is:
𝛥𝛥𝐸𝐸𝐸𝐸 𝑃𝑃 =
1 1
1
𝑐𝑐𝑒𝑒
∙ − 𝑈𝑈 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 − (𝑒𝑒 ∗ + 𝛥𝛥𝛥𝛥)2 ) �1 + �
2 𝑐𝑐𝑏𝑏
2
𝑐𝑐𝑖𝑖
𝑐𝑐
∗
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − (𝑒𝑒 + 𝛥𝛥𝛥𝛥) + 𝑒𝑒 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − (𝑒𝑒 ∗ + 𝛥𝛥𝛥𝛥)�
1
𝑐𝑐𝑖𝑖
� (A 15a)
− 𝑑𝑑 (𝑒𝑒 ∗ + 𝛥𝛥𝛥𝛥) � −
2
2𝜀𝜀
1 1 𝑐𝑐𝑒𝑒
1 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 ∗
− � ∙ − (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 ∗ )2 − 𝑑𝑑𝑒𝑒 ∗ � −
��
2 𝑐𝑐𝑏𝑏 2
2
2𝜀𝜀
where 𝑒𝑒 ∗ is the individual optimal emission level in the initial negligence model
𝑒𝑒 ∗ = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑑𝑑
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
−
2𝑐𝑐𝑒𝑒 2(𝜀𝜀𝑐𝑐𝑒𝑒 + 𝑑𝑑 )
(A 15b)
and 𝑒𝑒 ∗ + 𝛥𝛥𝑒𝑒 is the individual optimal emission level in the manipulation model.
𝑒𝑒 ∗ + 𝛥𝛥𝛥𝛥 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
28
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
𝑑𝑑𝑐𝑐𝑖𝑖
−
2(𝑑𝑑 + 𝜀𝜀𝑐𝑐𝑒𝑒 ) 2𝑐𝑐𝑒𝑒 (𝑐𝑐𝑒𝑒 + 𝑐𝑐𝑖𝑖 )
(A 15c)
This result is intuitive: If the possible consequences of being liable are serious, the firm benefits from manipu-
lating the indicator which would convict them to be guilty.
29
If the indicator gets too imprecise, the firm cannot rely on the manipulations to save them from liability. High
values of 𝜀𝜀 go hand in hand with high emission levels which in turn enhance the expected future damage costs.
The liability effect does not outweigh the cost effect anymore and the overall effect becomes negative.
49
Moreover, we assume for simplifying the analysis:
𝑐𝑐𝑒𝑒 = 𝑐𝑐𝑖𝑖 = 𝑐𝑐
(A 15d)
Inserting these relations in (A 15a), simplifying and rearranging yields:
𝑑𝑑 2
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑐𝑐
1
𝑑𝑑
�
𝛥𝛥𝐸𝐸𝐸𝐸 =
− 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 − (𝑐𝑐𝑐𝑐 + 𝑑𝑑 ) � −
��
8𝑐𝑐 (𝑐𝑐𝑐𝑐 + 𝑑𝑑 )
𝜀𝜀
2 2𝑐𝑐𝑐𝑐
𝑃𝑃
(A 15e)
This expression is greater than zero if
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑐𝑐
1
𝑑𝑑
− 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 − (𝑐𝑐𝑐𝑐 + 𝑑𝑑 ) � −
�>0
𝜀𝜀
2 2𝑐𝑐𝑐𝑐
(A 15f)
because 𝑑𝑑, 𝑐𝑐 > 0 per definitonem.
Rearranging (A 15f) yields:
𝑑𝑑
𝑑𝑑 2
𝜀𝜀 2 + 2𝜀𝜀 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � − 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 + � � < 0
𝑐𝑐
𝑐𝑐
(A 15g)
Obviously, the left side of this expression raises with 𝜀𝜀. Consequently, it reaches it’s highest
value for 𝜀𝜀 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
�𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
Rearranging:
𝑑𝑑
𝑐𝑐
𝑑𝑑
because 𝜀𝜀 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − . Inserting this in (A 15g) leads to:
𝑐𝑐
𝑑𝑑 2
𝑑𝑑
𝑑𝑑
𝑑𝑑 2
2
− � + 2 �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − � �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � − 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � � < 0
𝑐𝑐
𝑐𝑐
𝑐𝑐
𝑐𝑐
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 < 2𝑑𝑑
(A 15h)
(A 15i)
If this relation holds, 𝛥𝛥𝐸𝐸𝐸𝐸 𝑃𝑃 > 0 holds even for the greatest possible value of 𝜀𝜀.
But also in the case that 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 ≥ 2𝑑𝑑 is valid, 𝛥𝛥𝐸𝐸𝐸𝐸 𝑃𝑃 > 0 can hold. Because the left side of
(A 15g) raises with 𝜀𝜀 there will be a certain 𝜀𝜀̅ above which the left side of (A 15g) exceeds zero.
Then, (A 15g) does not hold anymore and as a consequence 𝛥𝛥𝐸𝐸𝐸𝐸 𝑃𝑃 becomes negative. This
threshold can be determined via setting the left side of (A 15g) to zero:
𝑑𝑑
𝑑𝑑 2
2
𝜀𝜀̅ + 2𝜀𝜀̅ �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � − 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � � = 0
𝑐𝑐
𝑐𝑐
2
(A 15j)
50
Solution approach for quadratic equations:
𝜀𝜀̅1,2
Rearranging:
𝑑𝑑
𝑑𝑑 2
𝑑𝑑 2
2
�
= − �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � ± �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � + 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − � �
𝑐𝑐
𝑐𝑐
𝑐𝑐
(A 15k)
𝑑𝑑
𝑑𝑑
𝜀𝜀̅1,2 = − �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � ± �3𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 + 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝑐𝑐
𝑐𝑐
Because 𝜀𝜀 ≥ 0 and 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑑𝑑, 𝑐𝑐 > 0 the second solution drops:
𝑑𝑑
𝑑𝑑
𝜀𝜀̅ = − �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � + �3𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 + 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝑐𝑐
𝑐𝑐
Checking whether this solution is allowed in terms of the domain of 𝜀𝜀 ≥ 0:
𝑑𝑑
𝑑𝑑
− �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � + �3𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 + 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ≥ 0
𝑐𝑐
𝑐𝑐
�3𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 + 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚
𝑑𝑑
𝑑𝑑
≥ 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 +
𝑐𝑐
𝑐𝑐
𝑑𝑑
𝑑𝑑 2
3𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 ≥ �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + �
𝑐𝑐
𝑐𝑐
2
𝑑𝑑
2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 ≥ � �
𝑐𝑐
which holds for every 𝑒𝑒𝑚𝑚𝑚𝑚𝑥𝑥 , 𝑑𝑑, 𝑐𝑐 because we assumed 𝑐𝑐𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 > 𝑑𝑑.
2
To sum up, there are two conditions under which 𝛥𝛥𝐸𝐸𝐸𝐸 𝑃𝑃 > 0 holds:
(a) 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 < 2𝑑𝑑 or
𝑑𝑑
𝑑𝑑
(b) 𝜀𝜀 < 𝜀𝜀̅ = − �𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + � + �3𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 2 + 2𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 .
𝑐𝑐
Results under Negligence, with Reputation Loss
𝑐𝑐
The Lagrange function equals:
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑒𝑒 + 𝑖𝑖 + 𝜀𝜀
� − 𝑟𝑟𝑟𝑟
2𝜀𝜀
1
1
+ 𝜆𝜆 �𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝛾𝛾 − 𝛾𝛾𝛾𝛾 − 𝑐𝑐𝑏𝑏 𝑏𝑏2 − 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)2
2
2
1
2
− 𝑐𝑐𝑖𝑖 𝑖𝑖 − 𝑈𝑈� + µ(𝛽𝛽 − 𝑐𝑐𝑏𝑏 𝑏𝑏) + 𝜐𝜐�𝛾𝛾 + 𝑐𝑐𝑒𝑒 (𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒)�
2
+ 𝜑𝜑(−𝛾𝛾 − 𝑐𝑐𝑖𝑖 𝑖𝑖 )
𝐿𝐿 = 𝑏𝑏 − 𝛼𝛼 − 𝛽𝛽𝛽𝛽 − 𝛾𝛾𝛾𝛾 + 𝛾𝛾𝛾𝛾 − 𝑑𝑑𝑑𝑑 �1 −
(A 16a)
51
The results of this optimization problem are included in Table A 6. As one would expect, the
Lagrange multipliers as well as the business as usual activity in equilibrium 𝑏𝑏∗ and the corresponding compensation parameter 𝛽𝛽 remain unaffected. To ensure positive values of the manager’s emission reporting manipulation activity in equilibrium, again one condition has be fulfilled:
𝑐𝑐𝑒𝑒
𝑑𝑑
𝑟𝑟
𝑑𝑑 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑖𝑖 + 𝜀𝜀 + 𝑐𝑐𝑒𝑒
< �
�
𝑐𝑐𝑖𝑖 𝑐𝑐𝑒𝑒
2𝜀𝜀
This is always the case because for 𝜀𝜀 ≤ 𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
greater than 1. Thus, if the assumption of
𝑟𝑟
𝑐𝑐𝑖𝑖
<
𝑑𝑑
𝑐𝑐𝑒𝑒
𝑑𝑑
𝑐𝑐𝑒𝑒
(A 16b)
the expression in brackets is always
holds, equation (A 16b) is always fulfilled.
The introduction of reputation loss in this model does not change the fact that more precise
environmental indicators induce lower emission levels 𝑒𝑒 ∗ but higher emission reporting manip-
ulation levels 𝑖𝑖 ∗ in equilibrium 30. Again, in correspondence to the results under strict liability,
the absolute value of 𝛾𝛾 is reduced and thereby also the value of the fixed salary parameter 𝛼𝛼.
The manager’s expected utility is pushed to her reservation utility. Given that the expected
reputation loss is not that crucial that any emission reporting manipulation is deterred, i.e.
𝑑𝑑
𝑐𝑐𝑒𝑒
𝑟𝑟
𝑐𝑐𝑖𝑖
<
, the difference between the principal’s expected utility with emission reporting manipulation
and reputation loss and her expected utility with emission reporting manipulation but without
reputation loss is always negative. That means, in a situation when the manager can manipulate
the GHG emission report, the firm does not want the public to get to know this. This results
seems to be intuitive. Moreover, for every 𝑟𝑟 ≥ 0 the difference between the principal’s expected
30
because
𝜕𝜕𝑒𝑒 ∗
𝜕𝜕𝜕𝜕
> 0 and
𝜕𝜕𝑖𝑖 ∗
𝜕𝜕𝜕𝜕
<0
52
utility with emission reporting manipulation and reputation loss and the one in the initial negligence model is also always negative. Hence, if there was a chance that the public can punish
the firm for manipulated emission reports through reputation loss, e.g. if there is an adequate
verification process, the firm would be better off with a system where the manipulation of GHG
emission reports is impossible per se.
A8: Linear Capacity Constraint
Introducing a linear capacity constraint, that limits the manager’s maximum working effort, 𝜅𝜅,
which has to be shared between the two tasks through adding the following constraint to the
optimization problem:
𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑒𝑒 + 𝑏𝑏 ≤ 𝜅𝜅
(A 17a)
Adding this constraint to the Lagrange function (A 5b) or (A 6b) through the Lagrange-Multiplier φ leads in deed to a reaction function regarding the tasks 𝑏𝑏 and 𝑒𝑒:
Table A 7: Results under Introducing a Linear Capacity Constraint
Strict Liability
(1) Reaction Function
(2) First Derivative of
the Reaction Function
(3) b’s reaction on 𝜅𝜅
(binding linear capacity
constraint)
(4) e’s reaction on 𝜅𝜅
(binding linear capacity
constraint)
(5) 𝑐𝑐𝑒𝑒 < 𝑐𝑐𝑏𝑏 (binding linear capacity constraint)
(6) 𝑐𝑐𝑒𝑒 > 𝑐𝑐𝑏𝑏 (binding linear capacity constraint)
𝑏𝑏(𝑒𝑒) =
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑐𝑐𝑒𝑒 (𝑒𝑒−𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 )+𝑑𝑑+1
𝑐𝑐𝑏𝑏
𝑐𝑐
= 𝑐𝑐𝑒𝑒
𝑏𝑏
𝑏𝑏(𝜅𝜅) = −
𝜕𝜕𝜕𝜕
𝜕𝜕𝜅𝜅
= − 𝑐𝑐
𝑐𝑐𝑒𝑒
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜅𝜅
𝜕𝜕𝜕𝜕
𝜕𝜕𝜅𝜅
= 𝑐𝑐
𝑐𝑐𝑏𝑏
𝑒𝑒 −𝑐𝑐𝑏𝑏
𝜕𝜕𝑒𝑒
< 0, 𝜕𝜕𝜅𝜅 < 0
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
> 0, 𝜕𝜕𝜅𝜅 > 0
𝜕𝜕𝜕𝜕
−𝑐𝑐𝑏𝑏 𝜅𝜅+𝑑𝑑+1
𝑐𝑐𝑒𝑒 −𝑐𝑐𝑏𝑏
𝜕𝜕𝜅𝜅
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜅𝜅
𝜕𝜕𝜕𝜕
𝜕𝜕𝜅𝜅
𝑐𝑐𝑏𝑏
𝑐𝑐
𝑑𝑑
𝑏𝑏
𝑑𝑑
1
− 𝑐𝑐 �2 −
𝑏𝑏
1
𝑏𝑏
𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
2𝜀𝜀
𝑒𝑒
1
+ 𝜀𝜀 � + 𝑐𝑐
𝑏𝑏
2(𝑐𝑐𝑒𝑒𝜅𝜅𝜅𝜅+𝜀𝜀)+𝑑𝑑(2𝜅𝜅−2𝑒𝑒𝑚𝑚𝑚𝑚𝑥𝑥 +𝜀𝜀�𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜 −1�)
𝑐𝑐 𝜀𝜀+𝑑𝑑
= (𝜀𝜀(𝑐𝑐 𝑒𝑒+𝑐𝑐
𝑏𝑏
𝑒𝑒(𝜅𝜅) =
𝜕𝜕𝜅𝜅
𝑐𝑐𝑒𝑒(𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −𝑒𝑒)
= − 𝑐𝑐𝑒𝑒 − 𝑐𝑐 ∙ 𝜀𝜀
𝑏𝑏(𝜅𝜅) =
𝑐𝑐𝑏𝑏 −𝑐𝑐𝑒𝑒
𝑏𝑏 −𝑐𝑐𝑒𝑒
𝑏𝑏(𝑒𝑒) =
𝜕𝜕𝜕𝜕
𝑐𝑐𝑒𝑒 𝜅𝜅−𝑑𝑑−1
𝑒𝑒(𝜅𝜅) = 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −
𝜕𝜕𝜅𝜅
Negligence
2(𝜀𝜀(𝑐𝑐𝑏𝑏 +𝑐𝑐𝑒𝑒)+𝑑𝑑)
𝑒𝑒 )+𝑑𝑑)
𝜀𝜀(−2𝑐𝑐𝑏𝑏 𝜅𝜅+𝑐𝑐𝑏𝑏 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 +2𝑐𝑐𝑒𝑒 𝑒𝑒𝑚𝑚𝑚𝑚𝑚𝑚 −𝑑𝑑+2)+𝑑𝑑𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜
= 𝜀𝜀(𝑐𝑐
−𝑐𝑐𝑏𝑏 𝜀𝜀
2(𝜀𝜀(𝑐𝑐𝑒𝑒 +𝑐𝑐𝑏𝑏 )+𝑑𝑑)
𝑒𝑒 +𝑐𝑐𝑏𝑏 )+𝑑𝑑
𝜕𝜕𝜕𝜕
> 0, 𝜕𝜕𝜅𝜅 < 0
𝜕𝜕𝜕𝜕
> 0, 𝜕𝜕𝜅𝜅 < 0
53
In principle, all pairs of (𝑏𝑏, 𝑒𝑒) that satisfy these reaction functions are possible results of how
the manager could split her maximum working effort. But by defining the parameters of the
compensation contract 𝛼𝛼, 𝛽𝛽, 𝛾𝛾 the principal can influence the manager which of these possible
pairs she has to choose. The case of sufficient capacity is trivial because the principal would
induce her expected utility maximizing efforts (𝑏𝑏∗ , 𝑒𝑒 ∗ ). Inserting the results from Table A 7,
line (1) and (2), into the principal’s expected utility function (8) or (12) and subsequently maximize it, leads to this insight in a mathematical way.
A binding capacity constraint leads to individually optimal working efforts that depend on the
maximum available working effort 𝜅𝜅 in accordance with line (3) and (4) of Table A 7. As lines
(5) and (6) show, the relation of the cost parameters 𝑐𝑐𝑒𝑒 and 𝑐𝑐𝑏𝑏 determine which task will be
preferred if the maximum available capacity changes under strict liability. E.g. if 𝑐𝑐𝑒𝑒 < 𝑐𝑐𝑏𝑏 and
the maximum available capacity is reduced, the manager will shift her effort to the emission
reducing action because this task is cheaper for her and vice versa. Under negligence, both
actions will be reduced or expanded simultaneously if the maximum available capacity, 𝜅𝜅, is
reduced or expanded. The reduction of the GHG emissions in this case lowers the probability
of getting liable and thereby induces more utility to the principal. Thus, in equilibrium the manager will not reduce the working effort for one task unilaterally, if 𝜅𝜅 is reduced, but she will
reduce the effort for both tasks simultaneously.
This brief analysis shows that a binding linear capacity constraint links both tasks of the manager through the maximum available working effort 𝜅𝜅. In equilibrium, the resulting working
efforts for each task then depend on this 𝜅𝜅. We could include this interaction into the analysis’
of our paper but that would shift the attention away from the initial research question – how
does the introduction of environmental liability affect the quality of GHG emission reporting –
to the question of how the manager splits her working effort in different scenarios.
54