Uncertainty & Learning
in Global Climate Analysis
vorgelegt von
Diplom-Physiker
Alexander Lorenz
aus Plauen
von der Fakultät VI – Planen Bauen Umwelt
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Wirtschaftswissenschaften (Dr. rer. oec.)
genehmigte Dissertation
Gutachter:
Prof. Dr. Ottmar Edenhofer
Prof. Dr. Hermann Held
Promotionsausschuss:
Prof. Dr. Volkmar Hartje (Vorsitz)
Prof. Dr. Ottmar Edenhofer
Prof. Dr. Hermann Held
Tag der wissenschaftlichen Aussprache: 14.02.2012
Berlin 2012
D83
3
Contents
Abstract 5
Zusammenfassung 7
Acknowledgements 9
1 Introduction 11
1.1 The Physics of Climate Change 11
1.2 The Impacts of Climate Change 13
1.3 The Economics of Climate Change Mitigation 15
1.4 Uncertainty and Learning in Global Climate Analysis 18
1.5 Thesis Outline 21
2 Climate Targets under Uncertainty: Challenges and Remedies 25
2.1 Introduction 27
2.2 Fixed Targets 29
2.3 Adjusting Targets 31
2.4 Conclusions 33
2.5 References 34
2.6 Supplement 36
3 How important is Uncertainty for the Integrated Assessment of Climate Change? 39
3.1 Introduction 41
3.2 How to measure the importance of Uncertainty and Perfect Learning? 44
3.2.1 The decision problem 44
3.2.2 Metrics for measuring the Importance of Uncertainty and Perfect Learning 46
3.2.3 A simple example: Quadratic Benefits and Costs of Mitigation 48
3.3 Importance of Uncertainty in MIND 50
3.3.1 The Model of Investment and Technological Development (MIND) 51
3.3.2 Importance of Uncertainty and Perfect Learning in MIND 51
4 Contents
3.3.3 The Marginal Costs - Benefit picture of MIND 55
3.3.4 Functional Dependencies within MIND 57
3.4 Changes in the Model Structure 60
3.4.1 Constant relative risk aversion η60
3.4.2 Exponential Damages 61
3.4.3 Linear Carbon Climate Response 62
3.5 Conclusion 64
3.6 Appendix 66
3.7 References 66
4 Anticipating Climate Threshold Damages 69
4.1 Introduction 71
4.2 Model and Methodology 73
4.2.1 Problem Formulation 73
4.2.2 Terminology 73
4.2.3 The Integrated Assessment Model MIND 74
4.2.4 Implementation of Learning about Climate Sensitivity and Damage Am-
plitude 75
4.2.5 Implementation of Learning about Threshold Damages 75
4.3 Results 76
4.3.1 Learning about Climate Sensitivity and Damage Amplitude 76
4.3.2 Learning about Threshold Damages 77
4.4 Conclusions 81
4.5 Appendix 81
4.6 References 82
5 Synthesis and Outlook 85
5.1 Formulating the Climate Problem under Uncertainty and Learning 86
5.2 Importance of Uncertainty for Global Climate Analysis 88
5.3 Importance of Anticipating Future Learning 89
5.4 General Outlook and further Research Questions 92
5
Abstract
Climate change, the 21st centuries challenge for cooperative human decision making, is sur-
rounded by large uncertainties concerning the scientific understanding of the climate system, of
climate change induced changes of natural and social systems and of the impacts of those changes
on human economic activities and human welfare in general. Parts of these uncertainties will be
resolved as science advances and new observations are made. This learning will allow refining the
decisions undertaken to cope with the climate problem.
This thesis is dedicated to examine the role of uncertainty and future learning in the formal assess-
ment of optimal global mitigation strategies for global warming. The central contributions of this
study are contained within three research articles.
The first article investigates the validity of the cost-effectiveness framework when applied to the
case of climate targets under uncertainty and future learning. The study highlights two major con-
ceptual problems of this formalism, namely the possibility of negative value of information and
infeasibility of the whole decision criterion. As a consequence an alternative decision framework
is proposed, the so-called cost-risk analysis, which avoids those conceptual problems but still re-
mains based on climate targets.
The second article is motivated by the clash between the general scientific intuition that epistemic
uncertainties about the climate system and climate damages should play a major role in determin-
ing optimal mitigation policies (and the resulting welfare gain compared to doing nothing) and
the results from the integrated assessment models that show only insignificant influence of those
uncertainties. We introduce a method of assessing the importance of uncertainty both in its impact
on optimal policy and in its impact on the welfare gain from acting upon climate change. We then
use a representation of the integrated assessment model MIND that allows linking the decomposed
value of climate policy to the structural form of the functions representing the climate cause-effect
chain, thereby understanding the negligible effect of uncertainty from the model structure. Finally
we propose some changes to the model structure that result in large impacts from including uncer-
tainty.
The third article investigates the circumstances under which the anticipation of future learning
about tipping-point-like threshold climate damages would be important for the determination of
near term mitigation decisions. We show that this is only the case if the learning occurs within
a narrow anticipation window. In this case far stronger near term mitigation is optimal to keep
the option open to avoid the threshold in case it turns out to lead to severe damages. The location
and width of this window is found to be sensitive to the DM’s flexibility to reduce emissions. If
reducing this flexibility in the MIND model, may this represent political or social barriers, the
anticipation window moves towards the present and broadens considerably, thereby increasing the
importance of including future learning into the analysis of climate change.
The articles are put into perspective by an introduction into the field that lays out the general
linking research questions and general conclusions.
6
7
Zusammenfassung
Der Klimawandel, als zentrale Herausforderung des 21. Jahrhunderts für globale Kooperation, ist
gekennzeichnet durch enorme Unsicherheiten im wissenschaftlichen Verständnis des Klimasys-
tems, klimainduzierter Veränderungen natürlicher und sozialer Systeme sowie der Folgen dieser
Veränderungen für menschliches Wirtschaften und die allgemeine Wohlfahrt. Teilweise werden
diese Unsicherheiten durch Fortschritte der Wissenschaft und neue Beobachtungen aufgelöst wer-
den können. Dieses zukünftige Lernen wird es ermöglichen, getroffene Entscheidungen zum Kli-
maschutz zu revidieren und an neue Situationen anzupassen.
Diese Dissertation widmet sich der Untersuchung der genauen Rolle dieser Unsicherheiten und
der Möglichkeit zukünftigen Lernens für die formale Analyse optimaler Vermeidungsstrategien
des Klimawandels. Die zentralen Beiträge dieser Arbeit sind in drei wissenschaftlichen Artikeln
dargelegt.
Der erste Artikel untersucht axiomatische Zielkonflikte bei der Anwendung der so genannten
"Kosten-Effektivitäts" Analyse auf Klimaziele unter Unsicherheit. Die Studie stellt zwei zen-
trale konzeptionelle Probleme dieses Formalismus fest, wenn man zusätzlich die Möglichkeit
zukünftigen Lernens einbezieht: die Möglichkeit, dass zusätzliche Information negativen Wert
zugeschrieben bekommt und die Möglichkeit der Unlösbarkeit des ganzen Entscheidungskriteri-
ums. Als Konsequenz wird ein alternatives Entscheidungskriterium vorgestellt, die sogenan-
nte "Kosten-Risiken" Analyse. Diese basiert immer noch auf der Angabe einzuhaltender Kli-
maschranken, vermeidet jedoch die benannten Probleme.
Die Motivation für den zweiten Artikel liefert der Widerspruch zwischen einer wissenschaftlichen
Intuition und den aktuellen Modellergebnissen. Die Intuition sieht einen starken Einfluss epis-
temischer Unsicherheiten auf die Bestimmung optimaler Vermeidungsstrategien und deren Ein-
fluss auf die Wohlfahrt (im Vergleich zum "Nichtstun"). Die Modelle hingegen zeigen nur einen
marginalen Einfluss dieser Unsicherheiten. Diese Studie entwickelt eine Methode, mit der man
die Wichtigkeit von Unsicherheit bestimmen kann, sowohl im Sinne des Einflusses auf die opti-
male Politik als auch im Sinne der Wohlfahrtsveränderung, die diese Politik hervorruft. Weiterhin
wird eine Darstellung des Modells MIND verwendet, die es ermöglicht, die vorgestellte Metrik
zur Messung der Wichtigkeit mit der Struktur der Funktionen zu verbinden, die die Kausalkette
des Klimawandels im Modell abbilden. Damit kann man die insignifikante Rolle von Unsicher-
heit direkt aus der Modellstruktur ableiten. Davon ausgehend testen wir einige, in der Literatur
diskutierte, Änderungen der Modellstruktur bezüglich ihres Einflusses auf die Wichtigkeit von
Unsicherheit.
Der dritte Artikel untersucht die Umstände unter denen die Antizipation zukünftigen Lernens über
Klimaschäden aus dem Überqueren eines sogenannten Tipping Points einen signifikanten Einfluss
auf die kurzfristige Vermeidungsstrategie ausübt. Wir zeigen, dass dies nur der Fall ist, wenn
das Lernen in einem engen "Antizipations-Zeitfenster" stattfindet. In diesem Fall ist eine strik-
tere kurzfristige Vermeidungsstrategie optimal, um die Option zu erhalten, im Falle schlechter
8 Zusammenfassung
Nachrichten über die Schäden des Tipping Points selbigen nicht zu überqueren. Die Lage und Bre-
ite dieses "Antizipations-Zeitfensters" ist stark abhängig von der Flexibilität, die Treibhausgase-
missionen zu reduzieren. Wenn man diese herabsetzt, beispielsweise um politische oder soziale
Barrieren zu repräsentieren, so bewegt sich das "Zeitfenster" näher zur Gegenwart und verbreit-
ert sich deutlich. Damit wird die Wichtigkeit von Antizipation für kurzfristige Entscheidungen
erhöht.
Eingefasst werden die Artikel von einer Einleitung in das generelle Forschungsumfeld, die
auch die zentralen, verbindenden, Forschungsfragen einführt, und einer Zusammenfassung, die
Schlussfolgerungen und weitere Forschungsschritte vorstellt.
9
Acknowledgements
I want to thank many colleagues from the Potsdam-Institute for Climate Impact Research from
whom I learned much and who contribute to an outstandingly open, stimulating and constructive
working environment from which I profited in many aspects.
Especially, I would like to thank Hermann Held, whose enthousiastic and impressive presentation
of the “inverse problems in climate science” inspired me to dive into the fascinating domain of
climate science. He offered me the opportunity to work with him and learn from him, first on a
masters project, and in continuation on this thesis. Without his scientific, intellectual, moral, and
financial support this thesis would simply not have been possible.
In equal intensity I want to thank Elmar Kriegler, for his collaboration, help, and profound advise
as well as for his continuous and increasing engagement in this PhD project, that helped me to
follow some of the many ideas to fuition. Without him, this thesis might have been possible but
would possibly not have been finished in its current form.
I would also like to thank Ottmar Edenhofer for supervising and examining this thesis as well as
for providing the whole research domain with the leadership that enables the constructive, positive
environment I could enjoy over the last four years.
With particular gratitude I want to mention my collaboration with Matthias Schmidt. With many
discussions, a continuous intellectual sparring, and an enviable motivation, he helped me to live
through the unavoidable downs of a PhD. Thanks for your collaboration.
Special thanks are due to Kai, Martin, Matthias, and Franziska for sharing an office with me, as
well as to Lavinia, Kristin, Odo and Islay for many relaxing counterpoints to the office.
I gratefully acknowledge funding by a scholarship of the German National Science Foundation,
from December 2009 until the finalization of this thesis.
And last, but surely not least, I want to thank Friederike, for your support, for your love, for
everything.
10
11
Chapter 1
Introduction
Anthropogenic climate change is one of the most prominent and most pressing global problems of
the unfolding century. While the basic principles of the cause-effect chain from human economic
activity to changing global climate conditions are well understood, large uncertainties remain in
the quantification of potential future evolution pathways of the climate system and the resulting
consequences for human civilization. However, during the last decades an enormous amount of
insights and evidence have been gathered and our understanding of the complex socio-economic-
climate system advances. Hence uncertainties and the expectation of future advances in system
knowledge are prominent features of the global climate problem.
This thesis aims at contributing to the assessment of the importance of these features of uncertainty
and learning for the integrated assessment of climate change mitigation policies. This will help
to clarify the justification of their prominence in the ongoing debate. This chapter introduces the
broader context of this thesis and clarifies its objectives. The basic physics of climate change is
reviewed in Section 1.1. Sections 1.2 and 1.3 introduce the economic approach to the analysis
of climate change. The issues rising from the recognition of uncertainties and learning about
key components of the combined socio-economic-climate system and the ways these problems
are approached in the literature are reviewed in Section 1.4. Within in this context, Section 1.5
introduces the objectives and structure of the thesis.
1.1 The Physics of Climate Change
The basic processes of anthropogenic global warming are well understood. A number of human
activities like pasturing land, cutting down forests, producing livestock, and especially the burning
of fossil fuels, leads to an increased flow of greenhouse gases (e.g. carbon dioxide, methane,
etc) into the atmosphere. This increases the atmospheric concentration of those gases above the
pre-industrial equilibrium level. As those greenhouse gases are transparent for incoming short
wave radiation from the sun but nearly opaque for the long wave heat radiation from the surface
of the earth a higher concentration of greenhouse gases perturbs the energy balance between the
incoming solar shortwave radiation and the outgoing longwave radiation which determines the
earth’s surface temperature. The resulting radiative forcing at the top of the atmosphere leads to
12 Introduction
an increase in GMT (until new equilibrium of the radiation balance). This basic concept of the
greenhouse effect was already described by Arrhenius (1896).
From the industrial revolution (1850) the atmospheric concentration of greenhouse gases (mea-
sured in CO2equivalents) has increased by about 50%, from 287 ppm (parts per million) to 433
ppm in 2004 (Solomon et al., 2007). Global emissions of carbon dioxide equivalent greenhouse
gases as of 2004 amounted to 49 Gt (Trenberth et al., 2007) and have been increasing by about
4% per year ever since. In 2009 the CO2-only emissions summed up to 29.5 Gt (compared to
27.7 Gt in 2005) (IEA, 2011). This led to an addition of 2 ppm of CO2in the atmosphere per
year on average between 2004 and 2009 (Stern, 2007), with a current atmospheric CO2concen-
tration of 389 ppm (as measured at Mauna Loa in September 2011, Conway & Tans, 2011). In the
same period (1850-2005), global mean temperature increased by 0.76±0.19 ◦C, at an accelerating
rate. The warming over the last 50 years is almost double that of the previous 100 years (Tren-
berth et al., 2007). Given our sound understanding of the thermodynamics of the climate system
it is very unlikely that those changes in temperature can be explained without external forcing
(like changes in solar irradiation, volcanic forcing, and anthropogenic activities). As temperature
has been strongly increasing when non-anthropogenic factors would be likely to have produced
a cooling effect, attribution studies find it very likely that it has been anthropogenic greenhouse
gas emissions that have caused most of the observed warming since the mid-20th century (p60,
Solomon et al., 2007).
During the last century, and especially within the last decades, a large number of changes in both
the global and regional parts of the climate systems have been detected: melting of glaciers all
over the world, changes in regional temperature and precipitation patterns, a rise in the global sea
level of about 25 cm, and an increase of 0.7◦C in the global mean temperature within the last
century (IPCC, 2007). Due to the large inertia of the climate system, stemming from the enor-
mous capacity of the oceans to store heat and the already large, accumulated athmospheric carbon
pool, this warming would continue for centuries even if greenhouse gas emissions were stopped
immediately. Following Ramanathan & Feng (2008), when keeping athmospheric CO2concen-
tration constant at the 2005 level (but reducing the emission of aerosols over the 21st century),
we would be committed by the current cumulative emissions to an expected global warming of
2.4(1.4−4.3)◦C above pre-industrial temperatures.
Although the thermodynamics of global warming are well understood, prognoses of future changes
in the different climate subsystems are inherently difficult, due to some less understood processes
that define the dynamics of the climate system. The sign and amplitude of different feedback
mechanisms like the water vapor feedback and the radiative forcing from changing cloud cover
are still highly uncertain. Additionally a large number of uncertain parameters in our representa-
tion of the climate system influence the cause-effect chain of global warming. Prominent examples
of uncertainties are the so-called climate sensitivity (Roe & Baker, 2007), the equilibrium change
in mean global surface temperature due to a doubling of atmospheric CO2concentration from the
pre-industrial level, and the transient climate response (Stott et al., 2006), that describes the im-
mediate temperature response to a 1% yearly increase in CO2concentration at the point where the
doubled concentration level is reached. Projections of global mean temperature within the 21st
The Impacts of Climate Change 13
century are also highly uncertain. In the absence of any climate policy, the IPCC predicts a global
warming of 4.0(2.4−6.4)◦C above pre-industrial mean temperatures to be reached by the year
2100 (Meehl, 2007).
The role of these uncertainties and potential future improvements in our knowledge is the central
matter of this thesis and will be introduced in more detail in Section 1.4.
1.2 The Impacts of Climate Change
A large number of changes in the global climate system are expected to accompany the rise in
global mean temperature or follow from it. Global sea levels are expected to rise by 0.18−0.59m
by the end of this century (IPCC, 2007). Newer estimates see the potential for even greater in-
creases (Vermeer & Rahmstorf, 2009). Due to varying regional compositions of the drivers of sea
level rise (SLR) its regional extent is also expected to differ significantly (Yin, 2009). Precipi-
tation will change both in amplitude and in the regional distribution (IPCC, 2007). Widespread
mass losses from glaciers and reductions in snow cover over recent decades are projected to accel-
erate throughout the 21st century, reducing water availability, hydropower potential, and changing
seasonality of flows in regions supplied by meltwater from major mountain ranges (IPCC, 2007).
One of the most important aspects of the impact of global warming is that for a given change in
global temperature, the regional changes can differ enormously. Thus a “moderate” global warm-
ing of about 2 ◦C above pre-industrial temperatures can lead to tremendous changes in different
climate subsystems and sensitive regions. The most severe changes in regional climates have been
observed in the Arctic, where the temperature change is two times as high as the global mean
(Trenberth et al., 2007). The minimum extent of the Arctic sea ice cover during summer has de-
creased by about 30% over the last decades (Stroeve, 2011). If this disaggregation is going to
continue at the same speed, the Arctic Ocean could be ice-free in summer by 2050. Addition-
ally, once the sea ice cover falls below a certain level of thickness, it becomes more vulnerable to
changes in regional weather conditions. Hence the volatility of sea ice cover increases and thus
the predictability of future changes decreases significantly. There are indications for the latidud-
inal position of the inner tropical convergence zone (ITCZ) changing in response to temperature
gradients between the northern and southern hemisphere (Broccoli et al., 2006). As temperature
increase in response to a rise in athmospheric greenhouse gas levels is expected to vary regionally,
this could mean that the ITCZ, as a band of highest precipitation, could move, thus regions with
once high levels of precipitation could dry up while others will encounter higher seasonal precipi-
tation events. This change will also lead to a corresponsing movement of the arid belt that follows
outside of the ITCZ.
A second important feature of global warming concerns the (partial) irreversibility of climate
change . Although the projections of global mean temperature are smooth over time, i.e. a linear
response to the forcing, this does not have to hold for regional changes. A number of climate
subsystems have been identified that can react in a strongly non-linear fashion to changes in the
overall forcing. Those so-called tipping elements (Lenton et al., 2008) were introduced to describe
14 Introduction
the components of the Earth system that can be switched – under particular conditions – into a
qualitatively different state by small perturbations. Thereby the term ‘tipping point’ is used to
refer to the critical point (in forcing and a feature of the system) at which such a transition is
triggered. Furthermore, most of these subsystems show hysteresis behavior, which means that
once such a tipping element reaches another stable state, a simple reversion of the forcing will in
general not suffice to switch the system back into its original state. Examples of these systems are
the thermohaline circulation in the north Atlantic, the Amazon Rainforest, and the South Asian
monsoon circulation. Another example is the Greenland ice sheet.
Some of these systems are characterized by the existence of a positive feedback mechanism that
causes the transition in the system, once the forcing passes the ‘tipping point’. In the case of
the Greenland ice sheet, increased melting due to higher temperatures leads to a lowering of the
thickness of the ice sheet. But by lowering the altitude of the maximum thickness, the surround-
ing temperature increases even more, which further exacerbates the melting process. Once the
thickness has fallen below a certain threshold, the total disaggregation of the ice sheet cannot be
stopped by simply bringing the temperatures back to normal.
Besides the regional positive feedbacks which can lead to a tipping behavior in climate subsys-
tems, there also exist a number of positive feedbacks on a global scale. The greenhouse effect
of water vapor, as part of the overall greenhouse effect, is one of the potential, global positive
feedbacks. As the global temperature is increased by the greenhouse effect from airborne water
vapor, even more water enters the atmosphere and thereby further strengthens the greenhouse ef-
fect. However, the strength of this feedback is not known precisely as several complications occur
(Forster et al., 2007). The radiative effect from increasing water vapor content is decreasing by a
spectral saturation effect. A second example of a global feedback is the dependence of the ability
of the oceans to store CO2from the atmosphere on the global temperature itself. This ability is
influenced by temperature increases through a multitude of effects (see Fung et al., 2005): Warm-
ing reduces solubility of carbon, increases ocean stratification, reduces vertical mixing, and slows
the thermohaline circulation. This also impacts on the oceanic biological productivity. Partially
loosing the capacity of storing CO2in the oceans would further increase the greenhouse effect. Fi-
nally, there are some other sources of potentially drastic global positive feedbacks. Large amounts
of CO2and methane are stored in the permafrost soil of the northern hemisphere and even larger
amounts of methane are stored at the bottom of the marine continental shelf regions in the form of
methane hydrates. An increase in regional temperatures and oceanic temperature could lead to a
large scale release of those greenhouse gases into the atmosphere and lead to a drastic, so-called
‘run-away greenhouse effect’ of uncertain but tremendous amplitude (e.g. Keller et al., 2007;
Lenton, 2011). The bedeviling feature of these non-linear mechanisms is the inherent uncertainty
about the location of the tipping points and about the consequences of a switching for other parts
of the climate system. However there are some instances of abrupt changes in the past of the Earth
system found in paleo-climatic data, that might represent switches of tipping elements, and there
are also some methods for early warning about approaching a tipping point from observations
(Dakos et al., 2008; Held & Kleinen, 2004; Lenton, 2011).
Uncertainty and potentials for future learning prevail in each of the mechanisms for climate change
The Economics of Climate Change Mitigation 15
impacts, especially in the non-linear subsystems. Those uncertainties add another dimension to
the overall uncertainty about climate change and will also be tackled in this thesis.
1.3 The Economics of Climate Change Mitigation
The impacts of climate change on the earth system are not of concern per se. During its long his-
tory, the earth has experienced many drastic or abrupt changes within the climate system, like the
so-called snowball earth events (Hoffman et al., 1998) or major meteoric impact events (Alvarez
et al., 1980). Thus, from a very aggregated perspective one could say that abrupt and extreme
changes within the climate system are “quite normal”. Such events caused massive changes in
atmospheric composition, the global mean temperature, and regional conditions for life, resulting
in the extinction of 95% of all species. Anthropogenic climate change differs from those events:
The observed and expected changes in natural systems take place with unrecedented speed (100’s
rather than 100000’s of years). Human society has adapted to the relatively stable climate of the
past holocene and whenever the climate deviated slightly from this holocenic "stable state" soci-
eties suffered and whole civilisations broke down. Therefore the changes in the natural systems
due to anthropogenic climate change will have large impacts on human activities, health and con-
ditions for human life in general. Furthermore, as human activities are the cause of the current
changes in the climate system, and additional climate change could be avoided by stringent mit-
igation policies, the problem of anthropogenic climate change belongs to the realm of decision
problems. Economic theory provides tools for evaluation of climate induced damage as well as
for decision making in the climate context.
One of the major threats from climate change is not the change in the mean state of regional climate
variables but the rising probability of extreme events like droughts, floods, or storms. Both kinds
of impacts, changes in the mean state and a higher risk of extremes, will impact humans differently
across regions, income groups, age groups and other attributes. Sea level rise, for example, will
mainly impact coastal regions by increasing the risk for extreme flooding events that threaten
human life, health and infrastructure. However, other regions would be affected indirectly. As a
sea level rise of 0.5−1m would increase the number of people under risk of coastal flooding by
up to 200 million (NRC, 2011), large migration movements are expected to follow from climate
change impacts. A moderate increase of temperature would also lead to different impacts across
world regions, as the northern hemisphere (e.g. Russia, Scandinavia, or northern Germany) might
even benefit from increased crop yields while arid and warm regions like southern Europe or sub-
Saharan Africa will encounter drastic reductions in crop yields from reduced water availability
and soil erosion (Edenhofer et al., 2010). Welfare economics provides the means to evaluate the
impacts of climate change and aggregate them to a measure of losses to human welfare. However
such an evaluation poses several severe challenges. Firstly, the inherent uncertainties in predicting
all necessary aspects of climate change relevant to the damage assessment are immense. Secondly,
the incommensurability of different impacts poses an enormous problem for their aggregation, e.g.
when comparing losses in biodiversity to an x% decrease in agricultural productivity. And finally,
some very intricate normative issues arise, when aggregating climate damages over time, regions
16 Introduction
and over uncertain possible future states of the world (Stern, 2007; Nordhaus, 2008a; Dasgupta,
2008): how to value future damages against current welfare, how to value damages inflicted on
rich people relative to damages inflicted on poor people, and how to weigh the risk of catastrophic
damages?
Economic Theory provides two important insights for managing climate change. First, the climate
change phenomenon is described as a failure of markets to anticipate the social damages inflicted
by economic activities. This provides the economic rational for a regulating authority to inter-
vene and internalise the negative effects from climate change. The non existence of an effective
global market regulation body leaves this intervention to be negotiated between many sovereign
countries that are in a competitive situation on the global markets. This delivers an explanation
as to why global action to mitigate dangerous anthropogenic interference with the climate sys-
tem, as stipulated by UNFCCC, Art. 2, is so difficult to achieve although the potentially drastic
consequences of unmitigated climate change are provided by science. Second, it provides some
important concepts to analyze the trade-offs between current and future generations, between rich
and poor regions and between the costs of mitigation and the risk of unmitigated climate damage.
Climate change related damage caused by anthropogenic greenhouse gas emissions comprise a
market failure as each emitter of greenhouse gases only experiences a tiny part of the overall dam-
age from climate change. Thus the emitter does not anticipate the causal link between her actions
and the resulting negative implications. A large number of self interested actors that compete
on the global market would thus not agree on any non-trivial mitigation action. This situation is
equally referred to as a market failure, an externality, or a common good problem (Hardin, 1968).
The solution to a common good problem is that the governmental authority regulating the market
(and representing social interest) puts a price on the activity that creates the externality. Thereby
the external damages get internalized and the market once again efficiently implements the social
interest.
In the context of climate change this solution corresponds to the installation of a globally bind-
ing regime that puts a price on greenhouse gas emissions, either explicitly, by a carbon tax, or
implicitly, by a binding limit on absolute emissions combined with a trading scheme of carbon
emission permits (Weitzman, 1974). In the absence of further externalities, like technological
spillovers (Leimbach & Baumstark, 2010), and without uncertainty about mitigation costs and
damages, both instruments efficiently internalize the climate externality. The handling of multiple
externalities and asymmetric information would require additional policy instruments and repre-
sent a field of research on their own. The practical implementation of a globally binding climate
regime has not been successful up to now (COP 16, 2010). The lack of an authority regulating
the global market, the diversity of interests of the different major economic actors, the inherent
difficulty of including the interests of future generations, and the difference between those actors
who cause(d) climate change and those who will experience the damage are major obstacles for
such and agreement. However, at least the UN have collectively recognized the issue of climate
change mitigation by (loosely) committing to a target of constraining global warming to a 2◦C
increase from the pre-industrial global mean temperature (COP 16, 2010). Furthermore, unilateral
commitments for mitigation of greenhouse gas emissions have been piloted by several countries
The Economics of Climate Change Mitigation 17
(www.climateactiontracker.org). However, one can entertain some doubt as to whether those ac-
tions will suffice to counteract climate change.
Focusing on the trade-offs inherent to the question of the appropriate level of mitigation action,
economic theory provides two different approaches. The first one includes a full monetization
and aggregation of climate damages as well as mitigation costs and aims at a formal cost-benefit
analysis of mitigation actions (as done e.g. by Nordhaus, 1994 ,2008a). The second approach
refrains from monetizing damages and provides an assessment of climate related risks instead.
By fixing the risk exposure to a certain level, cost efficient mitigation policies can be derived
(Schneider & Mastrandrea, 2005; Held et al., 2009). While the choice of an optimal policy is
included in the first approach and has to be done ex post within the second approach, economists
use to emphasize that the main benefit of both methods lies in the exploration of the consequences
of alternative policy scenarios rather then delivering the one optimal solution (Weitzman, 2011).
When considering the problems in the assessment of damage and the dependence on normative
settings discussed above, a healthy warning about the use of formal cost-benefit analyses seems
more than appropriate.
The formal implementation of both approaches is done in so-called integrated assessment mod-
els (IAMs), which include a representation of the socio-economic and the climate system. This
way both the impact from economic activities on the climate system and the impact from climate
change on economic activity are captured within one modeling framework. The pioneering work
in this field is the DICE model by Nordhaus (1994). Integrated Assessment models vary in the
detail and methodology of the description of the single subsystems as well as in the choice of
normative settings. Large parts of controversies about differing results from IAMs can actually be
related to the choice of normative parameters (Stern, 2007; Nordhaus, 2006). What they have in
common though, is that they mostly only consider mitigation in terms of reductions in GHG emis-
sions (exceptions are e.g. Ingham et al., 2007, Bosello et al., 2010, who also consider adaptation).
Taking into account the whole chain of causes and effects of climate change, along the so-called
Kaya identity (see Waggoner & Ausubel, 2002, a comprehensive set of policy levers is thinkable
to reduce climate change impacts: Limiting population growth, per capita economic output, en-
ergy intensity of economic activity, carbon intensity of energy production (and other activities like
agriculture and transport), or capturing greenhouse gases at the production site, combined with
storing it (Carbon Capture and Storage), would decreases greenhouse gas emissions. Methods
like the enhancement of oceanic carbon uptake through iron fertilization (Boyd et al., 2007) and
the reduction of atmospheric carbon content through “air capture” (Lackner et al., 2008) are aim-
ing at managing the carbon cycle directly to avoid an increase in athmospheric concentration of
greenhouse gases. The atmospheric radiation balance could be influenced by emitting aerosols or
enhancing low level cloud cover by emission of sea salt particles, both of which have a cooling ef-
fect. Another possibility would be the installation of orbital “sun blocking” facilities, like mirrors.
While the carbon management is aimed at undoing the harm of anthropogenic carbon emissions,
the radiation management only aims at combating global warming and leaves out other effects, like
the acidification of the oceans (Keith, 2001). Formerly both groups of methods, carbon cycle and
radiation management have been summarized under the term “geo-engineering”. As these options
18 Introduction
come with potential side effects of highly uncertain types and magnitudes, from a risk perspective
they are to be seen as a measure of last resort rather than a primary policy option (Victor, 2009).
Finally there will be the necessity to cope with the impacts of climate change by adapting the the
changing natural systems and by building up resilience towards extreme events.
1.4 Uncertainty and Learning in Global Climate Analysis
As highlighted by the different examples above, uncertainty is a pervasive feature of global cli-
mate policy analysis, because it is inherent in all parts of the cause effect chain of the integrated
assessment of climate change: from future population change, over economic development, energy
intensity of economic activity, intensity in greenhouse gas emissions from the generation process
of final energy carriers, the carbon cycle, the temperature response to increases greenhouse gas
concentrations, and the impacts of a changing carbon content and temperature of the environment
on biological, social and economic systems. Our understanding of every single step of this process
chain is hampered by uncertainty.
Within the integrated assessment of climate change, these uncertainties influence both, the strin-
gency of optimal mitigation policy and the welfare implications from different possible mitigation
strategies. The amplitude and direction of these impacts determine the "importance" of accounting
for uncertainty and learning within the integrated assessment of climate change.
Uncertainty influences optimal climate policy in different ways. Decision makers (DMs) are gen-
erally modeled to be risk averse. This means that they dislike uncertainty, i. e. they are worse off
when given an expectation of different possible, but uncertain, outcomes instead of the outcome
for the expected value of the underlying parameter determining the outcomes. This attitude is rep-
resented by a concavity in the DM’s utility function, which describes how much welfare, or utility,
the DM derives from a given level of consumption of goods. If uncertainty is directly represented
on the level of consumption, the DM’s risk aversion is the only reason for disliking uncertainty.
However, if the DM is uncertain about a parameter that itself influences the consumption level, the
concavity of utility with respect to this uncertain parameter can also arise from a concavity in the
dependency of the consumption level on the uncertain parameter. For the optimal stringency of
climate policy the differential effect of uncertainty between different decisions is important: is the
uncertainty the DM dislikes greater or smaller when committing to ambitious mitigation instead
of following a business as usual scenario? The more ambitious the mitigation scenario, the less the
deviation of the climate system from its current state and the less the uncertainties about climate
change induced damages. On the other hand a strict mitigation scenario induces uncertainties in
the social costs of mitigation. It remains an empirical question which of the mentioned effects
dominates the analysis.
The uncertainties about the socio-economic system and the climate system might change over
time due to observations or improvements in scientific understanding of the underlying system.
Future learning, in general, involves two effects on optimal climate policy. After receiving new
information the DM might want to adjust her policy to the new state of knowledge. In the absence
Uncertainty and Learning in Global Climate Analysis 19
of a multi agent framework, when only one DM plays a “game against nature” she will always
be better off when she has the possibility to learn simply because there might be some cases
where adjusting the policy after learning leads to welfare gains against keeping to the default “no
learning policy”, but at least she can always simply chose the default policy so she cannot be worse
off (Gollier,2004).
In addition to the post-learning adjustments in the policy, the DM might also want to change his
near-term policy anticipating future learning. This might be due to irreversibility following the
decisions. For example the DM might want to strengthen her effort in near-term mitigation to
keep the option open to stay below a climate tipping point if she will learn about the location or
severity of such a threshold in the future. On the other hand she might want to decrease her effort
in the near term to avoid sunk costs of early mitigation in case she learns in the future that climate
change damage is not as severe as expected. This has been used as an argument to postpone
investments into mitigation of GHG emissions until more is known about the impacts of climate
change (for a theoretic presentation of the argument see (Baker, 2006).
Which side plays out to be dominant depends on the representation of flexibility and irreversibility
in decisions and on the representation of uncertainties and future learning possibilities, hence it is
an empirical question.
It is not only an empirical question whether uncertainty and learning lead to more or less strin-
gent short-term emission reductions but also how significant those adjustments of optimal climate
policy are in terms of additional welfare gains, compared to the overall net social benefit of acting
upon climate change. As the comprehensive handling of uncertainties and future learning possibil-
ities highly complicates the integrated assessment of optimal mitigation policy, it has to be asked
how much we would lose in terms of welfare when applying a suboptimal policy, derived from an
analysis that neglects uncertainty or learning.
The literature that is concerned with epistemic uncertainty and future learning can be divided into
studies on the effect of uncertainty and on the effect of learning. Thereby the focus is mostly on the
effect of uncertainty and learning on optimal (near term) decisions and less on the welfare effects
from the adjustments of optimal decisions due to the accounting for uncertainty and learning.
Within integrated assessment models (IAMs) the effect of uncertainty about the climate system
or about climate damage is found to only lead to small changes in the optimal stringency of mit-
igation policy compared to a situation where all uncertain parameters are fixed to their expected
value (Peck & Teisberg, 1993; Webster et al., 2008). This result changes when uncertainty about
the normative parameters of the decision framework is included (Pizer, 1997). This uncertainty
leads to strong changes in optimal policy (up to 30% in cumulated emissions). Another source
for high impacts of uncertainties is the inclusion of so-called “fat tailed distributions” of climate
response or climate induced damage, as investigated by Weitzman (2009). He showed that in this
case the existence of low probability, high impact events from climate change leads to significantly
higher mitigation efforts and can even dominate the cost-benefit analysis in the sense that, under
certain assumptions, society would be willing to spend almost all of its GDP to prevent a very
unlikely catastrophic future. Schmidt et al. (2011a) show that even without including fat tailed
distributions uncertainty can have a substantial effect on optimal policy, if the heterogeneity of the
20 Introduction
climate damage distribution across the global population is taken into account.
The question of whether the adjustment of an optimal policy to uncertainty leads to significant
changes in the welfare gain from acting upon climate change has received less attention (an ex-
ception is Pizer, 1997).
Chapter 3 builds upon this example and embarks on the issue of determining the importance of
explicitly accounting for uncertainty within the integrated assessment of climate change. The
benefit of accounting for uncertainty can be evaluated by comparing the outcome of a best guess
policy in a deterministic setting with an expected value maximizing policy under uncertainty.
Chapter 3 argues that accounting for uncertainty is important if it leads to significant changes in
optimal mitigation policy which in turn leads to significant changes in the net welfare benefit from
mitigation action. A framework is developed that allows to analyse this measure of the importance
of accounting for uncertainty within an integrated assessment model and to relate the importance
to the functional structure of the climate cause effect chain. The negligible magnitude of welfare
changes due to introduction of uncertainty found in many other studies is confirmed and related to
compensating factors from different steps within the cause effect chain.
Within the literature, the effect of future learning on optimal near-term mitigation decisions is
termed the so-called “anticipation effect”. Studies employing the expected utility maximization
framework found that this anticipation effect on near-term mitigation decisions is small (Peck &
Teisberg, 1993; Ulph & Ulph, 1997; Webster, 2002; Webster et al., 2008; O’Neill & Melnikov,
2008). Again, the welfare effect from this anticipative behavior has received less attention, which
is understandable in the absence of an effect on decisions. Within the risk management approach
mentioned above (Section 1.3), that maximises welfare under the constraint of limiting the risk of
crossing a climate treshold to a certain probability, a far higher impact of the anticipation of future
learning on near-term decisions and resulting welfare gains has been found (Webster et al., 2008;
Bosetti et al., 2008). However, as is shown in Chapter 2 (Schmidt et al., 2011b), adopting this
approach leads to axiomatic inconsistencies when learning is included. Keller et al. 2004 have
found that the impact of the anticipation of future learning strongly increases if uncertainty about
highly non-linear climate damage, e.g. from tipping elements in the climate system, are included.
While the theoretic literature agrees that a strongly non-linear relationship between the decisions
of a problem and the resulting welfare, together with irreversibilities, can lead to anticipation
effects (see e.g. Baker, 2002) a systematic analysis of the significance of these anticipation effects
is still missing.
Chapter 4 enters the discussion at this point and develops a framework for investigating the im-
portance of anticipation of future learning, both in terms of changes in optimal decisions and in
resulting welfare gains. Building upon the work of Keller et al. (2004) it first introduces a notation
of the so-called expected value of anticipation that results from a decomposition of the overall
expected value of future information into the welfare gain from pre-learning and post-learning
decisions adjustments. While the overall expected value of future information is important when
comparing the importance of the reduction of different possible uncertainties, the expected value
of anticipation measures the importance of explicitly including future learning into the analysis
of an optimal short-term policy. Chapter 4 shows that anticipation can become crucial both in
Thesis Outline 21
terms of necessary adjustments of pre-learning emissions and resulting welfare gains if learning
about an irreversible threshold is included. Conditions on the time of learning and the threshold
characteristic are determined, for which this is the case. They can be summarized as a narrow
“anticipation window”.
1.5 Thesis Outline
Summarizing the discussion above, this thesis contributes to the question of whether uncertainty
and learning play an important role in the integrated assessment of climate change by developing
a framework for testing the importance within complex models that allows to relate the magni-
tude of the welfare effect from uncertainty and learning to the functional model structure. The
framework is applied to investigate the circumstances under which anticipation of future learning
about tipping-point-like threshold damages leads to significant changes in the optimal near-term
mitigation policy and corresponding welfare improvements. Additionally the viability of different
decision frameworks for the investigation of optimal mitigation under uncertainty and learning is
investigated. This analysis is conducted within the three core chapters of the thesis (Chapters 2-4).
The Chapters will be outlined in the following and the author’s contributions to each of the single
articles will be mentioned.
Chapter 2: This chapter is to be seen as a methodological prerequisite for the central analysis
of this thesis as it investigates the acceptance of a growingly popular decision criterion for the
analysis of uncertainty and future learning.1Climate Targets are becoming ever more influential
as witnessed by the recent adoption of the 2◦C target by UNFCCC COP 15. As a consequence,
many studies limit themselves to finding least-cost solutions to achieve these targets in a cost-
effectiveness analysis. This article first argues that the 2◦C target, for instance, is only meant to be
met with a certain probability if uncertainty about global warming is taken into account. Meeting
it with certainty would simply be too costly or even impossible. Cost-effectiveness analysis for
the resulting probabilistic targets is then shown to imply major conceptual problems that prevent
the consideration of learning about uncertainty, which constitutes an essential part of the problem.
The article therefore proposes an alternative decision criterion that performs a trade-off between
aggregate mitigation costs and the probability of crossing the target. This criterion avoids the con-
ceptual problems of cost-effectiveness analysis and is still to some extent based on given climate
targets. This article has been published as “Schmidt, M.G.W., A. Lorenz, H. Held, E. Kriegler
2011. Climate Targets under Uncertainty: Challenges and Remedies. Climatic Change: Letters
104 (3-4): 783-791”. M.G.W. Schmidt conceived the idea for this research, performed the analy-
sis and wrote the article. The co-authors, and A. Lorenz in particular, contributed with extensive
discussions concerning all stages of the analysis as well as with several internal revisions of the
manuscript.
Chapter 3: A common result from the cost-benefit literature is that the inclusion of uncertainty
1The following description of Chapter 2 is taken from Schmidt (2011).
22 Introduction
about the climate system response to anthropogenic greenhouse gas emissions and the climate in-
duced damages has only small effects upon the stringency of optimal climate mitigation and the
resulting welfare gain from adjusting the policy to uncertainty. This negligibility of “normal” un-
certainty within the integrated assessment of climate change clashes with the common intuition,
at least of the authors, that uncertainty has to be of great importance. This study presents new
insights into the source of the negligible uncertainty effect. Unlike previous studies, we go beyond
general findings on sufficient conditions for a negligible uncertainty effect. Since such conditions
are not fulfilled by integrated assessment models, the magnitude of the uncertainty adjustment ef-
fect becomes an empirical question. We present a method to analyze the importance of uncertainty
by tracing it through the cause-effect chain from greenhouse gas emissions to temperature change
to induced climate damage to welfare implications. This allows us to explain the negligible un-
certainty effect as a result of compensating factors in the cause-effect chain. More concretely, we
introduce a decomposition of the overall benefit of climate policy into single components: The
benefit of the best guess policy, the re-evaluation of the best guess policy under uncertainty, and
the value of adapting the optimal policy to uncertainty. We extend the decomposition to the case
of perfect learning and analyze the relative importance of all components in the MIND model.
Additionally we project the complex integrated assessment model to an a-temporal marginal cost-
benefit picture. This allows us to connect the different components of the overall benefit of climate
policy to the functional form of the marginal benefits and costs. This understanding of the miss-
ing uncertainty effect allows to identify changes in the formulation of the climate cause effect
chain that would lead to significant impacts from uncertainty. Examples are more convex climate
damages (e.g. exponential damages) or a less concave (e.g. linear) response of temperature to
cumulated emissions. This article has been submitted to Climate Change Economics as “Lorenz,
A., E. Kriegler, H. Held, M.G.W. Schmidt. How important is Uncertainty for the Integrated As-
sessment of Climate Change?”. The research question for this article was jointly developed by all
four authors. A. Lorenz developed the article design, conducted the analysis and wrote the article.
The co-authors, and especially E. Kriegler, contributed with extensive discussions and with several
thorough internal reviews of revised versions of the manuscript.
Chapter 4: Uncertainty, and especially learning about strongly non-linear, tipping-elements-like
damages, might have a strong influence on near-term mitigation decisions. This article system-
atically investigates both, the changes in optimal near-term mitigation effort and the associated
welfare gain relative to the overall benefit from learning due to the anticipation of future learning
about the threshold damage. The analysis, conducted within the IAM MIND shows that learning
about threshold damage is of significance if and only if the learning happens within a specific, nar-
row, “anticipation” time window. In this case, the additional early mitigation effort keeps the op-
tion open to prevent crossing the threshold if the future learning reveals severe threshold damage.
Future learning has no significant effect on near-term policy otherwise. Within the “anticipation
window” the welfare gain from anticipating future learning is significant and contains nearly the
complete value of information. Learning is still valuable, but not its anticipation, if it happens ear-
lier (outside the anticipation window), if the DM is flexible enough to react after the information
Thesis Outline 23
has arrived, conditional on the message. Additionally, the article introduces some novel concepts
for the analysis of the separate welfare effect from anticipation of future learning. This article
has been accepted for publication in Environmental Modelling and Assessment as “Lorenz, A.,
M.G.W. Schmidt, E. Kriegler, H. Held. Anticipating Climate Threshold Damages.” The research
question and design for this article was developed jointly by all four authors. The analysis was
performed by A. Lorenz, who also wrote the main part of the article. M.G.W. Schmidt made sub-
stantial contributions to conceptualizing the results and rewriting the manuscript in several internal
revisions.
Chapter 5 summarizes the results from the different articles and draws some conclusions for the
importance of accounting for uncertainty and anticipating future learning in the decision process.
Requirements for further analysis and future research questions are given in an Outlook.
24 Introduction
25
Chapter 2
Climate Targets under Uncertainty: Challenges and Remedies∗
Matthias G.W. Schmidt
Alexander Lorenz
Hermann Held
Elmar Kriegler
∗This chapter has been published as Schmidt, M.G.W., A. Lorenz, H. Held, E. Kriegler 2011. Climate Targets under
Uncertainty: Challenges and Remedies. Climatic Change: Letters 104 (3-4): 783-791
26 Climate Targets under Uncertainty: Challenges and Remedies
Climatic Change (2011) 104:783–791
DOI 10.1007/s10584-010-9985-4
LETTER
Climate targets under uncertainty: challenges
and remedies
A letter
Matthias G. W. Schmidt ·Alexander Lorenz ·
Hermann Held ·Elmar Kriegler
Received: 16 June 2010 / Accepted: 22 October 2010 / Published online: 26 November 2010
© Springer Science+Business Media B.V. 2010
Abstract We start from the observation that climate targets under uncertainty
should be interpreted as safety constraints on the probability of crossing a certain
threshold, such as 2◦C global warming. We then highlight, by ways of a simple exam-
ple, that cost-effectiveness analysis for such probabilistic targets leads to major con-
ceptual problems if learning about uncertainty is taken into account and the target
is fixed. Current target proposals presumably imply that targets should be revised in
the light of new information. Taking this into account amounts to formalizing how
targets should be chosen, a question that was avoided by cost-effectiveness analysis.
One way is to perform a full-fledged cost-benefit analysis including some kind of
monetary damage function. We propose multi-criteria decision analysis including a
target-based risk metric as an alternative that is more explicite in its assumptions and
more closely based on given targets.
1 Introduction
Climate targets have been widely discussed since the United Nations Framework
Convention on Climate Change (UNFCCC 1992). More recently, the European
Union (European Council 2005) and the Copenhagen Accord (UNFCCC 2009)
adopted the 2◦C-target, which calls for limiting the rise in global mean temperature
with respect to pre-industrial levels to 2◦C.
Electronic supplementary material The online version of this article
(doi:10.1007/s10584-010-9985-4) contains supplementary material, which is available
to authorized users.
M. G. W. Schmidt (B
)·A. Lorenz ·E. Kriegler
Potsdam Institute for Climate Impact Research,
Telegraphenberg A31, 14473 Potsdam, Germany
e-mail: [email protected]
H. Held
University of Hamburg - Klima Campus, Bundesstr. 55,
Hamburg 20146, Germany
Introduction 27
784 Climatic Change (2011) 104:783–791
There are large uncertainties involved in climate change. Under probabilistic
uncertainty about climate sensitivity, for instance, a certain emissions policy leads
to a probability distribution on temperature increases. It is in general impossible or
at least very costly to keep the entire distribution below 2◦C, for instance. Therefore,
under uncertainty climate targets should rather be interpreted as safety constraints
on the probability of crossing a certain threshold such as 2◦C. Such probabilistic
targets have been studied amongst others in den Elzen and Meinshausen (2005),
Meinshausen et al. (2006,2009), den Elzen and van Vuuren (2007), den Elzen et al.
(2007), Keppo et al. (2007), Rive et al. (2007), Schaeffer et al. (2008).
The uncertainty surrounding climate change will at least partly be resolved in the
future, which is called “learning”. Uncertainty about climate sensitivity, for instance,
will be reduced by future advances in climate science. This will change the probability
of crossing a certain threshold for a given policy. But it will also allow to adjust
climate policy. Since there are irreversibilities and inertia both in the climate system
and the economy, it is not only important to adapt to new information but also to
choose an anticipating near-term climate policy that provides flexibility to adapt to
future information. There is an extensive literature on whether such a policy is more
or less stringent. For an overview of the theoretical and the integrated assessment
literature see Lange and Treich (2008) and Webster et al. (2008), respectively.
Cost-effectiveness analysis (CEA) determines climate policies that reach a given
climate target at minimum costs. It takes targets as (politically) given and does not
answer the question of what an optimal target should be in the light of the available
information. In Section 2we highlight that CEA for fixed probabilistic targets
leads to major conceptual problems if learning is taken into account. Therefore,
and because it is presumably part of current policy proposals anyway, we have to
take into account that targets will be adjusted to new information. This demands
formalizing how targets are determined based on the available information and by
balancing costs and benefits in a broad sense. This is discussed in Section 3. Hence,
the condensed message of this letter is that learning is an important part of the
climate problem, and that if learning is taken into account, it is not a viable option to
just perform CEA for a given climate target but necessary to formalize how targets
should be determined.
More precisely, in Section 2we highlight that a decision maker performing CEA
for a fixed probabilistic target might be worse off with learning than without and
consequently reject to learn. Furthermore, we show that she can also be unable to
meet even the probabilistic interpretation of her target due to learning. We do this by
using results from the literature on decision making under uncertainty and a simple
example. Both problems are strong arguments for not using CEA for probabilistic
targets if learning is considered.
In Section 3, we discuss ways to take the adjustment of targets to new information
into account. One way is a full-fledged cost-benefit analysis (CBA) including a mon-
etary climate damage function. CBA applied to the climate problem has numerous
detractors. A main point of criticism is that CBA “conceal[s] ethical dilemmas“ (Azar
and Lindgren 2003) and difficult value, equity, and subjective probability judgments
concerning climate impacts. Alternative approaches based on a precautionary or
sustainability principle in turn do not have a clear formalization. As a middle ground,
we explore multi-criteria decision analysis based on a trade-off between aggregate
mitigation costs and a climate target based risk metric such as the probability of
crossing the target threshold.
28 Climate Targets under Uncertainty: Challenges and Remedies
Climatic Change (2011) 104:783–791 785
2 Fixed targets
Exemplarily, we consider a temperature target and uncertainty about climate sen-
sitivity denoted by θ, but analogous results hold for any probabilistic target. The
target consists of a temperature threshold Zof global warming, e.g. Z=2◦C,
and a maximum acceptable threshold exceedance probability Q. We will also call
the threshold exceedance probability the “risk” and Qthe “risk tolerance”. We
denote the vector of greenhouse gas emissions over time by E(t), the resulting
temperature trajectory by [T(E, θ)](t), and aggregate mitigation costs not including
any climate damages by C(E).C(E)can also be a utility function of costs. The
risk as a functional of emissions is given by R(E)=Rdθf(θ)2(Tmax(E, θ) −Z),
where f(θ) is the probability density function, 2(·)is Heaviside’s step-function, and
Tmax(E, θ) =maxt[T(E, θ)](t)is the maximum temperature. If yet nothing is learned
about the uncertainty, CEA for the probabilistic target reads as
min
EC(E),
s.t.R(E)≤Q.(1)
Costs are minimized such that the probability of crossing the threshold, or the risk,
is no larger than Q. Due to the constraint on a probability, such a problem is called
a chance constrained programming (CCP) problem (Charnes and Cooper 1959). For
an extensive numerical investigation of this problem see Held et al. (2009). The
equivalence to Value-at-Risk constrained problems is shown in Section 1 of the
Supplement.
In order to include learning, we consider a simple so called act-learn-act frame-
work. That means the decision maker first decides on emissions before learning,
denoted by E1(t), t≤tl. At time tlwith probability qmshe receives a signal or
message mthat is correlated with θ, and she updates her prior probability distribution
f(θ) and risk metric R(E)to a posterior distribution f(θ|m)and risk Rm(E)=
Rdθf(θ|m)2(Tmax(E, θ) −Z)according to Bayes’ rule. Subsequently she decides
on emissions after learning, denoted by Em(t), t>tl, which in general depend on the
message that has been received. A dynamic extension of CCP then reads as
min
E1(X
m∈M
qmmin
Em©C(E1,Em)ª),
s.t.Rm(E1,Em)≤Q,∀m∈M,(2)
Hence, expected costs are minimized such that the posterior probability of crossing
the threshold is no larger than Qfor all messages m. Equation 2is not the only way
to extend CCP to an act-learn-act framework. An alternative formulation is obtained
by constraining the expected value of the probability of crossing the threshold across
all messages, i.e. Pm∈MqmRm(E1,Em)≤Q. This alternative is also discussed below.
A similar problem to Eq. 2was studied in O’Neill et al. (2006). For the special case
of Q=0, where the target has to be met with certainty, it was studied in Webster
et al. (2008), Johansson et al. (2008), and Bosetti et al. (2009). Q=0is problematic
because it is likely to be infeasible if the upper tail of the probability distribution of
climate sensitivity is taken into account. Schaeffer et al. (2008), for instance, report
Fixed Targets 29
786 Climatic Change (2011) 104:783–791
a non-zero probability of crossing 2◦C even if greenhouse-gas concentrations were
stabilized at current levels. And even if Q=0were feasible, it would lead to very
high mitigation costs and arguably does not correspond to current target proposals.
Webster et al. (2008), for instance, report a cost-effective carbon tax of more than
$250/ton from 2040 on for the 2◦C target.
For Q6= 0, i.e. if the threshold doesn’t have to be avoided with certainty, CCP as in
Eq. 2leads to conceptual problems. A decision maker performing CCP can be worse
off with learning than without, and therefore reject to learn if possible. Most people
would say this is unacceptable for a normative decision criterion, better information
should be valuable. The benefits from learning can be measured by the expected
value of information, EVOI =Pm∈MqmC(El
1,El
m)−C(Enl), where El
1,El
mand Enl
are optimal emissions before, after, and without learning, respectively. Hence, the
EVOI is simply the difference in expected costs (or utility) between the case with
and the case without learning. The possibility of a negative EVOI in CCP was first
noted by Blau (1974) for a linear program and clarified in Hogan et al. (1981,1984).
Details of these papers were criticized by Charnes and Cooper (1975,1983), but a
rigorous analysis confirming the problem has been provided by LaValle (1986). In
Section 2 of the Supplement, we show that CCP violates the independence axiom of
von Neumann and Morgenstern, and we cite results that show that this necessarily
leads to the possibility of a rejection of learning.
Here we construct a simple example for providing an intuition why the EVOI can
be negative. We assume that climate sensitivity θcan take only three values with
equal probability, θ=2,3,4◦C. We also assume that if the threshold is avoided for
a certain value of climate sensitivity, it is also avoided for all lower values. Finally,
we assume Q=50%. We now compare the case without learning with the case of
immediate perfect learning where the true value of θis revealed at tl=0, i.e. before
any decisions have to be made. The case of partial learning, where the posterior dis-
tributions are non-degenerate, is discussed in Section 3 of the Supplement. There are
three policy options: Stay below the threshold for (I) only θ=2◦C, (II) θ=3◦C (and
hence also θ=2◦C), (III) θ=4◦C. (I) is the cheapest and least stringent, (III) the
most expensive and stringent alternative. Without learning, policy (II) is the cheapest
alternative with admissible risk of 1/3. With learning, the choice depends on the true
value of θ. If θ=2◦C, (I) is the cheapest admissible alternative, if θ=3◦C it is (II),
and if θ=4◦C it is (III). We have EVOI=(1/3C(I)+1/3C(II)+1/3C(III))−
C(II). It is negative if abatement costs are sufficiently convex in emissions reductions
so that C(I)+C(III) > 2C(II).
We have argued that climate targets under uncertainty probably cannot or should
not be met with certainty. A second conceptual problem is that if learning is taken
into account, even the resulting probabilistic targets can generally not be met. This
was first noted for a generic linear CCP problem by Eisner et al. (1971, they call
Eq. 2“conditional-go approach”). If, for instance, the threshold could not be avoided
for θ=4◦C in our simple example, it would be possible to limit the probability of
crossing the threshold to 50% without learning but not in the “bad” learning case
where θ=4◦C is revealed as the true value. More generally, under perfect learning
any probabilistic target with a threshold that cannot be avoided with certainty in the
prior becomes infeasible. Perfect learning is not a bad approximation in the long run,
and, as mentioned before, most thresholds such as 2◦C arguably cannot be avoided
with certainty given current information. If the probabilistic target is infeasible in
30 Climate Targets under Uncertainty: Challenges and Remedies
Climatic Change (2011) 104:783–791 787
some learning cases, it is unclear how to perform CCP. Infeasibility could be avoided
by relaxing the target threshold from 2◦C to 3 or 4◦C, for instance. But the problem
of a negative EVOI would persist as long as a chance constraint is applied. Besides, it
would mean that the 2◦C target can not be considered, which is problematic in itself.
Intuitively, what drives the results above is (i) the fact that the set of feasible (or
target complying) emissions trajectories changes depending on what is learned and
(ii) that the benefits of target compliance are not taken into account in the objective
function. If the optimal policy without learning, i.e. (II) in our example, were feasible
in all learning cases, neither infeasibility due to learning nor a negative EVOI would
be possible. The latter is because choosing (II) in all learning cases would guarantee
the same expected costs as without learning. And if sufficient benefits and not only
the costs of choosing (III) instead of (II) if θ=4◦C is revealed were taken into
account in the objective function, the EVOI would be positive despite a change in
the set of feasible trajectories. In Section 3we discuss how to include the benefits in
the objective function.
The feasible emissions trajectories change because the probabilistic target is fixed
and independent of what is learned and because the corresponding chance constraint
was put on each individual posterior distribution. As mentioned before, CCP in an
act-learn act framework could alternatively be formulated with a constraint only on
the expected value of the probability of crossing the threshold across the different
learning cases. Eisner et al. (1971) call this a “total probability constraint”, and
LaValle (1986) an “ex ante constraint”. In this formulation the same trajectories are
feasible with learning as without and the problems do not occur (see also LaValle
1986). But specifically this would mean that not reducing emissions at all if θ=4◦C
is learned and staying below the threshold in the other two learning cases would
be an admissible strategy. The expected probability of crossing the threshold would
only be 1/3. I would also be the cheapest feasible strategy because it implies the least
emissions reductions. It is a questionable recommendation, though, not to reduce
emissions at all after learning θ=4◦C only because the probability of crossing the
target would have been zero if something else had been learned. In decision theory
it would be called a violation of consequentialism (e.g. Machina 1989).
The problems of CCP are known since the 1970s, and CCP is still widely used
in many different areas from aquifer remediation design (Morgan et al. 1993) to air
quality management (Watanabe and Ellis 1993). If learning about uncertainty and
adjustment to new information can safely be neglected for a given problem, then
CCP can be a satisfactory and intuitive decision criterion under uncertainty. This is
the case if either little is learned, or if the EVOI is not of interest and the system
is flexible enough so that anticipation of learning is not important. In the climate
problem, though, learning and system inertia play an important role and should be
taken into account in determining climate policy. Therefore, CCP, in our view, is not
a suitable option.
3 Adjusting targets
In the preceding section we held the probabilistic target, i.e. the temperature thresh-
old Zand the risk tolerance Q, fixed and independent of what is learned, and
we did not include any benefits from target compliance in the objective function.
Adjusting Targets 31
788 Climatic Change (2011) 104:783–791
Current policy proposals, such as the 2◦C target arguably assume that targets will
be adjusted to new information in the future. The Copenhagen Accord explicitly
mentions the “consideration of strengthening the long-term goal referencing various
matters presented by the science” (UNFCCC 2009). In this section we discuss how
to adjust targets and how to avoid the problems of CCP by including the benefits of
target compliance in the objective function and by balancing costs and benefits in a
broad sense.
One possibility is to assume that climate targets and optimal climate policy can
be derived by a full-fledged CBA including a monetary climate damage function.
As mentioned in the introduction, this kind of CBA has numerous critics. One of
their main points is that by combining all damages in a monetary damage func-
tion, including loss of life, biodiversity, and the damages resulting from the highly
uncertain disintegration of the West Antarctic Ice Sheet, for instance, CBA rather
“conceals[s] ethical dilemmas” (Azar and Lindgren 2003) and difficult value, equity,
and subjective probability judgments than highlighting them to decision makers (see
the discussion in Azar and Lindgren 2003). Besides, it would be useful to have a
decision criterion that is at least to some extend based on politically given climate
targets.
As a consequence of the problems of CCP, Bordley and Pollock (2009) suggest
in an engineering context to specify an additional target threshold for the costs and
then to minimize the probability of crossing either threshold. Jagannathan (1985)
uses a simple trade-off between costs and threshold exceedance probability in order
to avoid a negative EVOI. Applied to the climate context, a linear form reads as
min
E1nPm∈Mqmmin
Em©wC(E1,Em)+Rm(E1,Em)ªo,(3)
The normative parameter wdetermines the trade-off between costs and risk. It
equals the per centage points of risk increase that would be accepted in ex-
change for a unit decrease in costs. We will call Eq. 3cost-risk analysis (CRA).
CRA can be seen as a weighted multi-criterion decision analysis or also as a
CBA in a broader sense. In contrast to CCP, the benefits, namely the reduction
of risk, are now included in the objective function. The trade-off is assumed to
be linear in order to have an equivalence to the expected utility maximization
maxE1©Pm∈MqmmaxEm©Rf(θ|m)A(E1,Em, θ)ªªwith A(E1,Em, θ) = −(wC(Em)+
2(Tmax(E1,Em, θ) −Z)). The conceptual problems encountered for CCP therefore
cannot occur (see also Section 2 of the Supplement). Jagannathan (1987) suggests to
consider non-linear trade-offs as well, but we could not find a convincing non-linear
form of the trade-off that is still equivalent to an expected utility maximization (see
also LaValle 1987).
Mastrandrea and Schneider (2004,2005) develop a risk management framework
based on the probability of exceeding a threshold of “dangerous anthropogenic inter-
ference” (UNFCCC 1992) as risk metric. But they only report different risk levels for
different stabilization targets and do not formalize the final trade-off between costs
and risk, which becomes necessary if learning is included in the analysis. This could be
done in CRA. Schneider and Mastrandrea (2005) also propose a more sophisticated
risk metric that better represents the temperature path dependence of risk. It is based
on the concept of maximum exceedance amplitude (MEA: by how many Kelvin the
target threshold is exceeded) and the concept of degree years (DY: the area above
the threshold between the temperature trajectory and the threshold). The expected
32 Climate Targets under Uncertainty: Challenges and Remedies
Climatic Change (2011) 104:783–791 789
value of some function 8of MEA and DY could also be used as a risk metric in
CRA, Rm(Em)=Rf(θ|m)8 (MEA(E1,Em, θ), DY(E1,Em, θ)).
The main difference between CRA and standard CBA is that the former makes
the necessary trade-offs between mitigation costs and impacts (risks) on a more
aggregate level, directly in the objective function, and thereby more explicitly and
to some extent based on given targets. Thus, the main difference is the framing of
the decision. The main difficulty of CRA, as of most multi-criteria decision analyses,
is that it is hard for decision makers to specify the value of the trade-off parameter w,
i.e. to value a probability of crossing a threshold in terms of costs, for instance. But
we would argue that at least for non-market and highly uncertain impacts, it might
still be easier to specify and more practical than a monetary climate damage function.
More specifically, the following combination of standard CBA and CRA might
better suit the climate problem than a pure CBA, CRA or CEA. Market-damages,
whose value can be estimated by observing markets without significant externalities,
are included over a damage function, which in turn is included in the cost metric
C(E). Non-market impacts like loss of life and public goods, impacts from highly
uncertain climate tipping-points, as well as wider societal impacts like migration and
conflict are included over an aggregate, climate target-based risk metric R(E). As
highlighted before, valuing these impacts is inherently difficult, and there is no way
around some kind of multi criteria decision analysis. Instead of mixing the value
judgments concerning these impacts with market impacts in a monetary damage
function as in standard CBA, an aggregrate trade-off between a target-based risk
and aggregate mitigation costs might be a more practical framing of the problem.
4 Conclusions
Climate targets such as the 2◦C target probably cannot or are not supposed to be
met with certainty. They should rather be interpreted as probabilistic targets. Cost-
effectiveness analysis (CEA) for such targets constitutes a chance-constrained pro-
gramming (CCP) problem. Transferring results from the literature to the climate
context, we have highlighted that CCP can imply a negative expected value of infor-
mation, which most people would consider normatively unsatisfactory. Furthermore,
even a probabilistic interpretation of relevant targets, such as the 2◦target, becomes
infeasible if learning is taken into account, so that it is unclear how to perform CCP
at all. Consequently, and because it is arguably part of the current target proposals,
we have discussed how to avoid the problems by adjusting climate targets to new
information and by balancing benefits and costs in a broad sense. A prominent way to
do this is cost-benefit analysis (CBA) including a monetary climate damage function.
But specifying such a damage function is notoriously difficult and controversial. We
took the problems of both CBA and CEA as motivation for asking, whether there is a
middle-ground between a full-fledged CBA and CEA. Partly based on previous
suggestions in the literature, we discussed a combination of a damage function for
market impacts and a more aggregate target-based risk metric for non-market and
highly uncertain catastrophic impacts as a promising candidate.
Acknowledgements M.G.W.S. was supported by the EU project GEO-BENE (No. 037063). E.K.
was supported by a Marie Curie International Fellowship (MOIF-CT-2005-008758) within the 6th
European Community Framework Programme. Helpful comments by two anonymous reviewers are
acknowledged.
Conclusions 33
790 Climatic Change (2011) 104:783–791
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References 35
Climate Targets under Uncertainty: Challenges and Remedies
Supplement
Matthias G.W. Schmidt ·Alexander Lorenz ·Hermann
Held ·Elmar Kriegler
1 Value-at-Risk
A probabilistic target is essentially equivalent to a limit on the Value-at-Risk (VaR) in finance. The x%-VaR,
or VaR at the x%confidence level, of a financial position equals the x-percentile of the distribution of the
uncertain losses of the position. In other words, with x%certainty, losses will be smaller than the x%-VaR.
Hence, we can formulate the probabilistic target as a constraint on the VaR in the distribution of maximum
temperature: The (1 −Q)-VaR has to be smaller or equal than a given threshold, such as 2◦C.
2 Violation of the Independence Axiom
We shortly introduce some basic decision theoretic terminology and formulate CCP as a preference relation on
simple lotteries. Subsequently, we show that CCP does not fulfill the independence axiom by von Neumann and
Morgenstern. There is an extensive literature on the consequences of relaxing the axioms of von Neumann and
Morgenstern. We shortly review one result that shows that the possibility of a rejection of learning encountered
in the main text follows from violation of the independence axiom.
A simple lottery describes an uncertain outcome. It is defined by the set of possible outcomes with their
respective objective or subjective probability. For the climate example without learning every emissions path
can be assigned a simple lottery. This lottery is defined by the vector of relevant outcomes, here maximum
temperature and mitigation costs, and the probability (density) for these outcomes. So we denote lotteries
by LE,f := {(Tmax (E, θ), C(E)) , f(θ)}. In a mixed lottery, the outcomes of a first stage lottery are again
lotteries. We denote the mixture of two lotteries L1and L2with mixing probability βby βL1+ (1 −β)L2.
The ordering of simple lotteries implied by CCP as in Eq. (1) in the main text is akin to a lexicographic
ordering. Lexicographic orderings consist of a hierarchy of orderings like a lexicon: words with the same first
letter are ordered according to the second letter and so on. The primary ordering (≻1) in CCP is according to
whether the probabilistic target is met or not. It strictly prefers all emissions plans that meet the target over
Matthias G.W. Schmidt ·Alexander Lorenz ·Elmar Kriegler
Potsdam Institute for Climate Impact Research
Telegraphenberg A31, 14473 Potsdam, Germany
E-mail: schmidt@pik-potsdam.de, Tel. +49-311-2882566
Hermann Held
University of Hamburg - KlimaCampus
Bundesstr. 55, 20146 Hamburg, Germany
36 Climate Targets under Uncertainty: Challenges and Remedies
2 Matthias G.W. Schmidt et al.
plans that do not (L1≻1L2⇔(R(L1)≤Q)∧(R(L2)> Q), where ∧is the logical AND). But unlike for
typical lexicographic orderings, the primary ordering in CCP does not lend itself to a definition of indifference
as “none of the two lotteries is strictly preferred to the other” (L1≃1L2⇔(¬(L1≻1L2)) ∧(¬(L2≻1L1)),
where ¬is the logical NOT). Such a definition would imply indifference between all emissions plans that meet
the target and all of those that do not. When applying the secondary ordering in CCP, i.e. preference of the less
costly plan over the more costly one (L1≻2L2⇔C(L1)< C(L2)), to these two indifference classes, it will
produce a sensible ordering of the plans that meet the target. But it would identify the business-as-usual case
with zero emissions reductions as the preferred strategy among those that miss the target. This would be clearly
unsatisfactory. In this sense CCP preferences can be regarded as incomplete and indifference in the primary
ordering is limited to plans that meet the target (L1≃1L2⇔(R(L1)≤Q)∧(R(L2)≤Q). The primary and
secondary ordering in CCP allow differentiating between plans meeting and violating the target, and between
plans that all meet the target, but not between plans that all miss the target. Alternatively to having an
incomplete primary ordering, one could assume indifference in the primary ordering between plans that don’t
meet the target and apply a different, more satisfactory secondary ordering than cost minimization to these
plans. However, the incompleteness is not necessarily problematic, because it still allows for the formulation of
an overall preference relation
L1≻L2⇔(L1≻1L2)∨((L1≃1L2)∧(L1≻2L2)) (1)
that has the desirable properties of asymmetry (L1≻L2⇒ ¬(L2≻L1)) and negative transitivity (¬(L1≻
L2)∧ ¬(L2≻L3)⇒ ¬(L1≻L3)(Kreps, 1988)). In particular, it allows identifying a choice set of most
preferred strategies. However, the target infeasibility due to learning discussed in the main text has shown
that the choice set of CCP can become useless, i.e. will indiscriminately include all available strategies, if no
strategy can meet the target.
CCP as in Eq. (1) violates both the continuity and the independence axiom by von Neumann and Mor-
genstern. We only discuss the latter here. Independence is violated because the chance constraint cannot be
formulated as a set of separate, or independent, constraints for each state of the world. The avoidance of
the threshold in one state of the world, via the chance constraint, has an influence on the need to avoid the
threshold in other states of the world. More formally, independence would be fulfilled if for any three lotteries
L1, L2, L3and for all β∈(0,1] we had
L1≻L2⇒ {βL1+ (1 −β)L3≻βL2+ (1 −β)L3}(2)
So independence means that the preferences are not changed by mixing the same lottery L3into two given
lotteries L1and L2. This is not the case for CCP because of the primary ordering according to the chance
constraint. E.g., it is possible that R(L2)< R(L1)< Q < R(L3),C(L2)> C(L1)and (βR(L2) + (1 −
β)R(L3)) < Q < (βR(L1) + (1 −β)R(L3)), i.e. both L1and L2fulfill the chance constraint but L2is less risky
and gives higher costs than L1.L3does not fulfill the constraint and βis chosen such that the mixed lottery
of L2and L3fulfills the constraint, whereas the mixed lottery of L1and L3does not. We then have L1≻L2
and βL1+ (1 −β)L3≺βL2+ (1 −β)L3, which shows non-independence.
Supplement 37
Climate Targets under Uncertainty: Challenges and Remedies 3
The possibility of a decision maker being worse off with learning than without that we encountered in the
main text follows from violation of the independence axiom. Wakker (1988) proves the following consequence:
¬Independence ∧Correct anticipation of future decisions ∧Consequentialism
⇒Information can make decision maker worse off. (3)
So if the antecedents are fulfilled including violation of the independence axiom, then the receipt of additional
information can make the decision maker worse off. We have already shown that CCP violates the independence
axiom. Future decisions are also anticipated correctly. It is correctly anticipated that after receipt of a message,
the target will have to be met based on the updated posterior information. More critical is the assumption of
consequentialism. Consequentialism intuitively means that only current and future payoffs have an influence
on current decisions. Past outcomes and foregone options have no influence on current decisions. CCP as in
Eq. (2) of the main text is consequentialist because the chance constraint is applied to every single posterior,
and forgone risk in other learning cases is not taken into account.
3 Partial Learning
1 3 5 7 9
0
0.2
0.4
0.6
0.8
1
1.2
θ / °C
Probability Density
Prior
1
2
3
Posteriors:
Fig. 1 Information structure with prior distribution and
three posterior distributions. The vertical dashed lines in-
dicate the medians.
One might object to the simple example in the
main text that the EVOI only becomes negative
because we consider perfect learning. Under per-
fect learning the posterior risk has to be reduced
to zero and not only 50%. So the target stringency
is effectively increased by learning. But firstly, per-
fect learning is probably not unrealistic in the long
run, so the decision criterion should be able to han-
dle it. Secondly, the same problems occur for par-
tial learning, where the uncertainty is only reduced
from a prior to a non-degenerate posterior distri-
bution. Consider the prior and posterior distribu-
tions shown in Fig. 1. If maximum temperature is
monotonic in climate sensitivity θ, i.e. if the target
is met for θ1it is met for all θ≤θ1, then we can
translate the risk tolerance into a maximum value of θ, for which the target threshold has to be avoided. This
value, of course, depends on what is learned. It decreases from about 3◦C to about 2◦C in the “good” learning
case (posterior 1) and increases to about 4◦C in the bad case (posterior 3). These are the same values for θas
in the perfect learning example in the main text. Hence, we would get the same negative EVOI.
References
D. Kreps. Notes on the theory of choice. Westview Press, 1988.
P. Wakker. Nonexpected utility as aversion of information. Journal of Behavioral Decision Making, 1:169–175,
1988.
38 Climate Targets under Uncertainty: Challenges and Remedies
39
Chapter 3
How important is Uncertainty for the Integrated Assessment of
Climate Change? ∗
Alexander Lorenz
Elmar Kriegler
Hermann Held
Matthias G.W. Schmidt
∗This chapter is under revision for Climate Change Economics as Lorenz, A., E. Kriegler, H. Held, M.G.W. Schmidt.
How important is Uncertainty for the Integrated Assessment of Climate Change?
40 How important is Uncertainty for the Integrated Assessment of Climate Change?
How important is Uncertainty for the Integrated
Assessment of Climate Change?
Alexander Lorenz∗†
, Elmar Kriegler∗, Hermann Held‡
, Matthias G.W. Schmidt∗
Abstract
We investigate the importance of explicitly accounting for uncertainty in the as-
sessment of optimal global climate policy. The benefit of accounting for uncertainty
can be evaluated by comparing the outcome of a best guess policy in a deterministic
setting with an expected value maximizing policy under uncertainty. We apply the
approach to the case of uncertainty about the temperature response to greenhouse gas
emissions and climate induced damage in the integrated assessment model MIND. In
accordance with the literature we find that the welfare gain from adjusting climate
policy to uncertainty is negligible. We use a decomposition of the uncertainty adjust-
ment effect to explain its negligible magnitude in the standard setting as a result of
compensating factors in the climate cause-effect chain from emissions to damage to
welfare implications. We demonstrate several changes in the model structure, such
as exponential climate damage and linear climate-carbon response that can lead to
a significant welfare gain from explicitly accounting for uncertainty in climate policy
analysis.
1 Introduction
Global Climate Change Analysis is surrounded by large uncertainties about key parameters
in the socio-economic system and the climate system. The uncertainties arise from imper-
fect knowledge about the dynamics of the subsystems, from internal short term dynamics
or stochasticity and from the long time lag between cause and effect within the climate
system (Tol, 1999). Analysts use models that integrate the socio-economic system and
the climate system to assess long term strategies to mitigate climate change (Nordhaus,
∗A. Lorenz (corresponding author), Elmar Kriegler, Matthias G.W. Schmidt
Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany
E-mail: lorenz@pik-potsdam.de Tel.: +49-331-2882562
†now at
Environmental Change Institute, School of Geography and the Environment
University of Oxford, South Parks Road, Oxford, OX1 3QY
E-mail: [email protected]x.ac.uk
Tel.:+44-1865-275894
‡H. Held
University of Hamburg & KlimaCampus Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
also at
Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany
1
Introduction 41
1994). A key question concerning uncertainty and climate change assessment is whether
the analysts should explicitly account for uncertainty in their integrated assessment models
to capture the effect of uncertainty on decisions? An initial intuition would be, that under
uncertainty about the climate response to greenhouse gas emissions and resulting climate
damage the risk of very high damage beyond the mean damage is introduced. Assuming a
right-skewed damage distribution and some degree of risk aversion, one would be willing to
undergo stronger emission reductions than in the best-guess case to better insure against
high damage scenarios. Insuring against those high damage scenarios would also become
more valuable than simply reducing average climate damage, thus we would expect climate
policy to become more valuable under uncertainty compared to the best-guess case.
Over the last two decades following the pioneering work of Nordhaus [1994], many
contributions have been made to check the intuition described above, and to answer the
question how explicitly accounting for uncertainty changes the optimal climate policy (e.g.
Keller et al., 2004; Wirl, 2007; Heal & Kristrï¿œm, 2002, and references therein) . Some
studies (e.g. Pizer, 1997) also investigated the welfare effect from introducing uncertainty.
A common result has emerged from cost-benefit analyzes with integrated assessment models
that optimize the trade-off between mitigation costs of reducing greenhouse gas emissions
and climate change induced damage costs (e.g. see Nas, 1996): Although the optimal
climate policy might change significantly due to the introduction of uncertainty, the welfare
gain associated with adjusting the policy to uncertainty is negligible (Pizer, 1997). One
exception is the work of Weitzman [2010] who shows that the effect of uncertainty can
become significant and even dominating if fat tailed probability distributions for the climate
damage are considered. Under certain conditions, such fat tailed climate damage can lead
to unbounded expected welfare losses.
Parts of the uncertainties about the climate system and climate change induced damage
may be resolved by future learning, both passively via observing the state of the coupled
socio-economic-climate system (Kelly & Kolstad, 1999) and actively via investing into new
measurements and leveraging paleo-information (Lorenz et al., 2009). This would allow to
adapt mitigation policy to new information. The welfare implications of the possibility to
learn and adjust to new information has also gained much attention (e.g. see Nordhaus
& Popp, 1997; Karp & Zhang, 2006; Ingham et al., 2007). Here a general finding is that
adapting climate policy to new information can lead to significant welfare gains.
What is the explanation for the negligible effect of uncertainty in cost-benefit analyzes
of climate policy? As pointed out in the literature, uncertainty has no effect on decisions
and welfare if the model is (nearly) linear in the uncertain parameters (see e.g. Baker,
2006; Lange & Treich, 2008). Furthermore the marginal utility of the model also needs to
be non-linear in the uncertain parameter for the optimal decisions to be effected, as they
are determined by the trade-off between marginal benefits and costs. For the importance
of future learning for short-term actions Webster [2002] has argued that so-called “cross-
period interactions” in utility are a necessary condition for future learning to affect short-
term decisions. However the existence of such a dependency of future utility on today’s
2
42 How important is Uncertainty for the Integrated Assessment of Climate Change?
decisions is not sufficient for a strong influence of future learning, as different cross-period
interactions could lead to contradicting influences from future learning on today’s decisions
that might cancel each other out. However these theoretical findings do not explain why
uncertainty has nearly no effect on welfare within the more complex integrated assessment
models, as those models do incorporate a number of (partly strong) nonlinearities as well
as cross-period interactions: the welfare function, the temperature response to carbon
emissions, the damage function, etc.
This study presents new insights into the source of the negligible uncertainty effect.
Unlike previous studies, we go beyond general findings on sufficient conditions for a neg-
ligible uncertainty effect. Since such conditions are not fulfilled by integrated assessment
models, the magnitude of the uncertainty adjustment effect becomes an empirical question.
We present a method to analyze the importance of uncertainty by tracing it through the
cause-effect chain from greenhouse gas emissions to temperature change to induced climate
damage to welfare implications. This allows us to explain the negligible uncertainty effect
as a result of compensating factors in the cause-effect chain.
More concretely, we introduce a decomposition of the overall benefit of climate policy
into single components: The benefit of the best guess policy, the re-evaluation of the best
guess policy under uncertainty, and the value of adapting the optimal policy to uncertainty.
We extend the decomposition to the case of perfect learning and analyze the relative
importance of all components in the MIND model. Additionally we project the complex
integrated assessment model to an a-temporal marginal cost-benefit picture. This allows us
to connect the different components of the overall benefit of climate policy to the functional
form of the marginal benefits and costs.
Applying this methodology to the integrated assessment model MIND we can identify
the following reasons for the negligible effect of accounting for uncertainty in the model:
First, the overall benefit of climate policy is constrained by the saturation of the emis-
sions to temperature change relationship compensating for the non-linearity in the climate
damage function and by the consumption smoothing property of the welfare function.
Second, this assessment does not change significantly under uncertainty because the addi-
tional marginal welfare benefit of reducing a unit of emissions becomes only significant for
large mitigation effort, where mitigation costs are already high. Third, the welfare gains
from adjusting the mitigation policy under uncertainty are limited because of the strongly
increasing mitigation costs.
Using this understanding of the relationship between the model formulation, the shape
of the marginal benefits and costs, and the components of the benefit of climate policy, we
introduce several changes in the model structure to create a significant welfare effect from
uncertainty. These changes include a sensitivity analysis with respect to the parameter of
constant relative risk aversion, a switch towards exponential damage and an implementa-
tion of a linear climate response to cumulative carbon emissions as proposed by Metthews
et al. [2009].
The paper is structured as follows: Section 2 introduces the general climate decision
3
Introduction 43
problem under uncertainty and (perfect) learning and describes the decomposition of the
overall benefit of climate policy into the single components determining the benefit of cli-
mate policy under uncertainty. The decomposition is illustrated within a simple analytical
model of climate change with quadratic costs and benefits of mitigation. In Section 3
the framework is applied to the Model of Investment and Technological Development
(MIND). The importance of introducing uncertainty is compared to the overall benefit
of acting against climate change and to the benefits from perfect learning. The sensitiv-
ity of the results towards changes in normative parameters is evaluated. The marginal
cost-benefit picture of the model as well as the functional dependency along the climate
cause-effect chain is presented to investigate the origin of the negligible welfare effect from
uncertainty. In Section 4 several changes to the model structure are investigated with
respect to their influence on the importance of uncertainty: changes in the parameter of
constant relative risk aversion, exponential damage, and linear climate carbon response.
Section 5 concludes.
2 How to measure the importance of Uncertainty and Perfect
Learning?
2.1 The decision problem
First, we formulate the general decision problem incorporating uncertainty and (perfect)
learning in its simplest version. We only consider one decision period. The principle
agent (DM) decides upon a set of decision variables x, like investments, emission control
rates, etc. The decisions might also represent a whole time path of single decisions x(t).
Depending upon the decisions and upon the state of the world (SOW) the DM derives
an overall welfare U(x, θ). Uncertainty about the SOW is represented by a probability
distribution π(θ).
Technically speaking we model an open loop optimal control problem and thus neglect
the effect of changes in available information over time. The problem now is to maximize
the overall expected utility V(x, π):
max
xV(x, π) = max
xX
j
π(θj)U(x, θj). (1)
Second, we introduce the cases of the DM’s information structure relevant for the
climate change example. The random variable θrepresents the uncertain magnitude of the
temperature response to greenhouse gas emissions and of climate change induced damage.
The DM’s knowledge, or belief, about the values of the uncertain climate response and
damage is represented by the probability distribution function π(θ). The general case of
uncertainty is simply denoted by π. The degenerate case, where the DM is certain about
4
44 How important is Uncertainty for the Integrated Assessment of Climate Change?
θtaking the value θjis defined via:
πj≡π
1θ=θj
0else
. (2)
Two special cases of the degenerate distribution are the case of no climate damage at all,
denoted by π0, and the case of certainty about θtaking its expected value, denoted by ¯π:
π0≡π
1θ= 0
0else
,¯π≡π
1θ=Pjπ(θj)θj
0else
. (3)
Third, utilizing the decision framework and the special instances of information struc-
ture, we define the following four policy scenarios, relevant for measuring the importance
of uncertainty and perfect learning:
The No-Control Case (NC): We define the policy case of “no-control” as the opti-
mal policy in the absence of any climate damage: ˆx0≡arg maxxV(x, π0). The rationale
behind this definition stems from the difference between the non cooperative and the co-
operative solution of a decentralized market economy. In such an economy a no-control
behavior is caused by the imperfect cooperation of a large number of decision makers. In
a competitive setting each decision maker only anticipates her own small share of global
climate induced economic damage leading to an almost total neglect of the climate problem
in individual decisions. Thus the climate problem is called an externality to the market.
In a fully cooperative setting the single actors would optimize their combined welfare and
thus correctly anticipate global warming. Within the model, we simulate the lack of co-
operation by making the DM ignorant towards climate change. Within our setting the
no-control case is not only suboptimal due to the lack of mitigation efforts, but addition-
ally the savings rate cannot be adjusted to the observed climate damage. Thus the benefit
from internalizing climate damage is slightly exaggerated. However, this error is small, as
the savings rate adjustment due to climate damage only becomes significant for very high
levels of climate damage (D≫50%).
The Best-Guess Case: We define the optimal policy under certainty about climate
response and damage by ˆx1≡arg maxxV(x, ¯π). This is the common approach to take the
expected value of the uncertain parameters as best guess values.
The Uncertainty Case: We define the optimal decision under uncertainty about
climate response and damage by ˆx2≡arg maxxV(x, π).
The Perfect Information Case: For comparing the importance of uncertainty
and learning we consider the case of immediate perfect learning. The information about
the true state of the world is revealed before any decision has been made. Any other
implementation of partial or later learning leads to less benefits of information thus we
consider the limiting case of what can be gained by learning about the SOW. We define
the optimal decision under perfect information by ˆx3(θj)≡arg maxxV(x, πj). From the ex
5
How to measure the importance of Uncertainty and Perfect Learning? 45
ante perspective there is no longer a single optimal policy, but a set of policies conditional
on what has been learned before deciding. The expected utility over all possible messages
that might have been revealed is given by W≡Pkq(k)V(ˆx3(θj), πkj), whereby in the
case of consistent learning the probability qkof getting message πkj is simply equal to the
probability πjof the revealed state θj.
2.2 Metrics for measuring the Importance of Uncertainty and Perfect
Learning
Figure 1: Welfare levels for the combinations of
policy scenarios [ˆx0,ˆx1,ˆx2,ˆx3(θ)] and information
structures [π0,¯π, π]that are relevant for defining
importance metrics. Also shown are relevant wel-
fare differences, measured in ∆CBGE.
In this section we use the nomenclature de-
fined above to introduce metrics that mea-
sure the different components of the overall
benefit of climate policy separately. Com-
bining the above defined policy scenarios
[ˆx0,ˆx1,ˆx2,ˆx3(θ)] with the possible assump-
tions of how the world reacts to the policy
decisions, represented by information struc-
tures [π0,¯π, π], leads to 12 possible out-
comes in terms of expected utility V(or W
in case of perfect learning), from which only
7outcomes are of further interest. Those
are depicted schematically in Fig.1.
The welfare differences between those
cases, measured as changes in certainty and
balanced growth equivalents (∆CBGE, see
Appendix 5), can be used as metrics for the
importance of the different effects of uncer-
tainty in welfare terms:
The relevant measure for the im-
portance of climate policy in a best-
guess world is the net benefit of re-
acting to climate change, i.e. chang-
ing from ˆx0to ˆx1, BCP(ˆx1,¯π) =
∆CBGE [V(ˆx1,¯π), V (ˆx0,¯π)]. This benefit
of climate policy is usually small compared
to the mitigation costs within a cost-efficiency framework, as it already combines the ben-
efits from reducing climate damage with the costs of mitigation.
Introducing uncertainty has two effects: First, the valuation of policies ˆx0and ˆx1
changes. If, for all x, the (expected) utility V(x, ¯π)is concave (convex) in the uncertain
parameter θ, the expected utility for ˆx0and ˆx1in the uncertain case (π) will be smaller
(larger) than in the best-guess case (¯π) (Fig. 1 shows the case of concave utility). As the
6
46 How important is Uncertainty for the Integrated Assessment of Climate Change?
benefit of climate policy BCP(x, ¯π)is defined as difference between two levels of (expected)
utility the behavior of BCP(x, ¯π)for a switch between the best-guess and the uncertainty
case depends on the curvature of the marginal (expected) utility in the uncertain parameter
θ. If, for all x, the marginal (expected) utility is convex (concave) in θ, then the increase
(decrease) of (expected) utility due to uncertainty in θcompared to the best guess ¯
θis
smaller (larger) for ˆx1than for ˆx0. Hence the benefit of adopting the optimal climate
policy from the best guess world increases (decreases) when evaluated in the uncertain
world: BCP(ˆx1, π)>(<)BCP(ˆx1,¯π). The difference between the benefit from the best
guess policy in the uncertain world and the certain world is denoted by ∆BCP(ˆx1) =
BCP(ˆx1, π)−BCP(ˆx1,¯π).
Second, when not only evaluating the solution of the best guess world under un-
certainty but explicitly maximizing the expected utility, the optimal climate policy will
change (from ˆx1to ˆx2). This possibility of adjusting climate policy to uncertainty leads
to an increase in overall expected utility, denoted as Benefit of Adjusting to Uncertainty,
BOAU ≡BCP(ˆx2, π)−BCP(ˆx1, π).
Taking both effects of uncertainty into account, the overall benefit from optimally
responding to climate change under uncertainty BCP(ˆx2, π)can be divided into three
parts:
BCP(ˆx2, π)≡BCP(ˆx1,¯π) + ∆BCP(ˆx1) + BOAU . (4)
A common measure for the “strength” of this adjustment effect is the absolute or relative
change in optimal decisions itself, i.e..∆ˆx≡(ˆx2−ˆx1)/ˆx1(e.g. see Tol et al., 1999). We
argue that the comparison of optimal policies is insufficient to decide upon the importance
of including uncertainty as even a large ∆xdoes not necessarily has to correspond to a
large BOAU. To assess the importance of uncertainty and of optimizing expected utility,
we compare the single contributions of Eq. 4 normalized to their sum.1.
Introducing (perfect) learning allows the DM to adjust her policy to the received signal.
As a limiting case of early learning, the optimal policy can be chosen conditional on the
perfect knowledge about the respective state of the world, ˆx3= ˆx3(θ). The expectation
for the overall welfare is now not taken over uncertain states of the world, but over the
possible messages leading to certain states of the world. To be consistent with the ex ante
knowledge, the distribution over the messages has to be identical to the distribution over
the SOW in the uncertain case. The benefit of perfect learning is measured by comparing
the expected utility with and without learning. The Value of Perfect Information is defined
via: VPI(π)≡BCP(ˆx3(θ), π)−BCP(ˆx2, π). The relative importance of perfect learning
can be compared to the importance of maximizing under uncertainty and the importance
of reevaluating the optimal best guess policy by dividing the overall benefit of acting upon
climate change under perfect learning into the components:
BCP(ˆx3(θ), π)≡VPI(π) + BCP(ˆx1,¯π) + ∆BCP(ˆx1) + BOAU , (5)
1As done by Pizer [1997], who uses the relative measure BOAU/BCP(ˆx2, π)to assess the importance
of optimizing under uncertainty.
7
How to measure the importance of Uncertainty and Perfect Learning? 47
and then comparing the normalized contributions of the single effects.
2.3 A simple example: Quadratic Benefits and Costs of Mitigation
We apply a simple analytical model of costs and benefits for climate change mitigation to
illustrate the three components of the overall benefit to act upon climate change in the case
of uncertainty. We will also use the simple setting to illustrate the connection between the
different welfare effects from including uncertainty and the marginal expected benefits and
costs of mitigation. Later this “marginal cost-benefit” picture will be used to understand
the finding of small effects from uncertainty in the more complex integrated assessment
model MIND.
Let xdenote the level of abatement of greenhouse gas emissions relative to the no-
control case in a simple one period framework. The abatement leads to benefits B(x, θ)
due to reduced climate damage and comes with costs of mitigation C(x). The dependence
of Bon the state of the world (SOW) θdenotes the uncertainty in climate damage and
benefits of mitigation. The decision problem is to maximize the net benefits of mitigation:
max
xU=EθB(x, θ)−C(x). (6)
Trivially, the optimal abatement level in the no-control case is ˆx0= 0. To be able to
derive the optimal policies in the best guess and uncertainty case, ˆx1,ˆx2and resulting
welfare changes analytically, we choose quadratic benefits and costs of mitigation, i.e.
B(x, θ) = a1x2+f(θ)x+c1and C(x) = a2x2+b2x+c2, similar to the model used by
Karp & Zhang [2006].We assume the benefits and damage to vanish for zero mitigation:
c1=c2= 0. The SOW is a normally distributed random variable: θ=N(µ, σ2). For
simplicity we assume f(θ)to take the form f(θ) = θ2. By solving the first order condition
for xthe optimal solutions for the best guess and the general case of uncertainty read:
ˆx1=1
2
µ2−b2
a2−a1
,ˆx2=1
2
µ2+σ2−b2
a2−a1
. (7)
Fig. 2 shows the marginal benefits and costs for the best guess and the uncertain case.
Presented this way, the benefits of climate policy can be illustrated simply as areas between
the curves for marginal costs and benefits. For our functional setup, the three components
from Eq. 4, normalized to the overall benefit of climate policy under uncertainty, are
fully determined by the “strength” of the uncertainty effect on decisions, i.e. by ∆ˆx:=
σ2/(µ2−b2). The relative contributions of the components are shown in Fig. 3. For
amplitudes of the relative change in optimal policies due to uncertainty ∆ˆxthat would be
considered significant, e.g. 10%, the welfare gain from adjusting the optimal policy is much
lower (1%), nearly the whole benefit of acting upon climate change is already realized by
choosing, and reevaluating, the best-guess climate policy. Only very large adjustments of
optimal policy ∆ˆx > 45% lead to a significant (>10%) contribution to the overall benefit
of climate policy.
8
48 How important is Uncertainty for the Integrated Assessment of Climate Change?
mitigation effort
marg. benefits and costs [utility units]
ˆx0ˆx1ˆx2
BCP (ˆx1,¯π)
BCP (ˆx2, π)
BCP (ˆx1, π)−BCP (ˆx1,¯π)
BOAU
C(x)
B(x, Eθθ)
EθB(x, θ)
Figure 2: Marginal benefits and costs of mitiga-
tion for quadratic benefits and costs.
Value of
best guess policy
Reevaluation
of best guess policy
Adapting optimal
policy
0.2
0.4
0.6
0.8
1.0
Σ2
Μ2-b2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Part
Figure 3: Three components of the overall benefit
(value of best-guess policy in darker Grey, reeval-
uation of best-guess policy under uncertainty in
lighter Grey, value of adapting optimal policy to
uncertainty in darker green) of acting upon climate
change under uncertainty depending on the ampli-
tude of uncertainty σ2/(µ2−b2).
The optimal policy can also be derived graphically within the (marginal) cost bene-
fit picture. The optimal best guess policy is determined by the intersection between the
marginal costs and the marginal benefits for the expected value of the uncertain parame-
ter. The optimal policy under uncertainty is determined by the intersection between the
marginal costs and the expected marginal benefits. The size of the three components of
the overall benefit of climate policy under uncertainty is determined by the slope and cur-
vature of the marginal benefits and costs and by the slope and curvature of the difference
between the expected marginal benefits in the uncertainty case and the marginal benefits
in the best guess case, called the marginal risk premium MRP.
Assuming simple polynomial costs C(x) = xαand benefits B(x) = xβ, the slope and
curvature of the marginals is determined by αand β. For the costs of mitigation the
common assumption is to take α > 2, thus the marginal costs are convex increasing in the
mitigation effort. For the benefits of mitigation the situation is not clear. Depending on
β, three regimes would be possible: For 0< β < 1the marginal benefits are decreasing,
for 1< β < 2they are concave increasing and for β > 2they are convex increasing. Fig. 4
shows examples from the different regimes and the resulting benefits of best guess climate
policy, here the marginal benefits are normalized by B′(x, β) = βx(β−1)/(β·0.5(β−1))such
that the marginal benefits coincide at x= 0.5for all β. Clearly the benefit of climate
policy decreases with increasing β.
Introducing uncertainty changes the marginal benefits. Commonly one assumes the
benefits to be convex in the uncertain parameter θ, thus the marginal benefits increase due
to uncertainty. However the slope and curvature of the marginal risk premium MRP ≡
Piπ(θi)B(x, θi)−B(x, Piπ(θi)θi)are not clearly determined. The benefit from adapting
9
How to measure the importance of Uncertainty and Perfect Learning? 49
marginal benefits
marginal costs
Β=3/4
Β=3/2
Β=2
Β=3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.5
1.0
1.5
2.0
Figure 4: Marginal costs and benefits of mitiga-
tion for different slopes and curvatures of marginal
benefits. The cost and benefit functions are nor-
malized to reach the same value at x= 0.5..
marginal costs
expected marginal
benefits
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.5
1.0
1.5
2.0
Figure 5: Expected marginal costs and benefits of
mitigation for different slopes and curvatures of the
marginal risk premium (MRP). The benefit of the
best guess policy is marked as darker Grey area,
the benefit of reevaluating the best guess policy
is marked as green area and the benefit of adapt-
ing the policy to uncertainty is marked as orange
area. The different shadings result from the differ-
ent MRP realizations.
the policy to uncertainty (BOAU) is strongly influenced by the MRP. Fig. 5 shows a case
of convex increasing marginal benefits with different possible MRP functions. The BOAU
increases the more convex increasing the MRP. If the curvature of the MRP itself does not
depend on the mitigation level x, then the increasing BOAU with increasing convexity of
the MRP is contrasted with a decrease in the reevaluation of the benefit of climate policy.
3 Importance of Uncertainty in MIND
Why does accounting for uncertainty about the climate response and the climate dam-
age change the results in standard applications of integrated assessment models of climate
change only to a small degree? In this section we investigate this question by replacing
the simple analytic model by the more complex integrated assessment model MIND, and
applying the decomposition of the benefit of climate policy presented above. For this pur-
pose, we introduce uncertainty and immediate perfect learning about climate sensitivity
and the severity of climate change induced damage into the MIND model (Section 3.1).
First, we reproduce the findings in the literature (e.g. Pizer, 1997; Manne, 1995) that
explicitly including uncertainty has a small influence on the benefit of climate policy (Sec-
tion 3.2). We then interpret the MIND model in the “cost-benefit” picture from Section 2.3
(Section 3.3) and resolve the functional dependencies between the decision variables and
the resulting marginal benefits and costs of climate change mitigation to shed light on the
prerequisites for a significant effect of uncertainty (Section 3.4 ).
10
50 How important is Uncertainty for the Integrated Assessment of Climate Change?
3.1 The Model of Investment and Technological Development (MIND)
We employ the Model of Investment and Technological Development (MIND) (Edenhofer
et al., 2005) in its stochastic version presented by Held et al. [2009]. Additionally we
include learning as introduced to the model by Lorenz et al. [2011]. MIND is a model
in the tradition of the Ramsey growth model and similar to the well-known DICE model
(Nordhaus, 1993). The version we use differs from the classical Ramsey model in two
major aspects: Firstly, the production sector depends explicitly on energy as production
factor, that is provided by a crudely resolved energy sector. The energy sector contains
(i) fossil fuel extraction, (ii) secondary energy production from fossil fuels, and (iii) re-
newable energy production. The macroeconomic constant-elasticity-of-substitution (CES)
production function depends on labor, capital and energy as input factors. Secondly, tech-
nological change is modeled endogenously in two ways. The DM can invest into research &
development activities to enhance labor and energy efficiency. Additionally, productivity
of renewable and fossil energy producing capital increases with cumulative installed capac-
ities (learning-by-doing). We assume welfare to be an inter-temporally separable isoelastic
utility function of per capita consumption with a constant relative risk aversion η= 1.5
that is changed for the sensitivity study later on. It takes the form:
U(c(I, s)) =
te
X
t0
L(t)·1
1−η"[c(I, s)](t)
L(t)1−η
−1#e−ρtdt , (8)
where I= (IK, IR&D, IFossil, IRenewables)is the vector of investment flows in the different
sectors over time, sis the unknown state of the world, ρis the pure rate of social time
preference taken to be 0.01/yr, and L(t)is an exogenously given population scenario.
Investments are related to the global consumption [c(I, s)](t)via the budget constraint:
Ynet(t, s) = [c(I, s)](t) + X
n
In(t, s), c(I, s)≥0,(9)
with the Gross World Product (GWP) Ynet net of climate related damage. Ynet is related
to gross GWP over Ynet =Ygross ·DF, where DF is a multiplicative damage factor defined
by the damage function (see Roughgarden & Schneider, 1999):
DF(T) = 1
1 + a·Tb.(10)
3.2 Importance of Uncertainty and Perfect Learning in MIND
The uncertainties about climate sensitivity and climate damage are described by proba-
bility distribution functions. The information about climate sensitivity CS is modeled by
a log-normal distribution by Wigley & Raper [2001]: ¯π(CS) =LN (0.973,0.4748). The
uncertainty about climate damage is taken to influence the amplitude aof the damage
factor, but not the exponent b, which is taken as constant b= 2. The distribution over ais
derived from a normal distribution over the parameter a′in DF(T)∗= 1/[1+(T/a′)2], with
11
Importance of Uncertainty in MIND 51
−2
−1.5
−1
−0.5
0
Uncertainty Best guess
x0
x0
x1
x1
x3 x2 x0
x0
∆CBGE [%]
Figure 6: Welfare levels, measured as changes in
CBGE relative to no-control, for the different sce-
narios with and without uncertainty. Shown are
the results for η= 1.5.
2020 2040 2060 2080 2100
0
5
10
15
20
25
time
CO2 Emissions [GtC]
x0
x1
x2
x3(θ)
Figure 7: Optimal emission pathways for the dif-
ferent information settings. Shown are the solu-
tions for risk aversion η= 1.5.
a′=N(18,5). This choice of the mean is near to the best guess case by Nordhaus [2008]
(a= 0.0028 vs. our a= 0.0030). The uncertainty range is inspired by the distribution by
Gerst et al. [2010], who chose a=N(0.0028,0.0013), but due to the inverse distribution,
higher damage values are favored by our distribution. For the numerical implementation
we draw samples of size nfrom the distributions according to a scheme related to descrip-
tive sampling (see Saliby, 1997). The uncertainty space is divided into nhypercubes. Each
hypercube icarries a chosen probability weight wiand is represented by the expected value
of the parameters on this hypercube. For simultaneous uncertainty about both climate sen-
sitivity and damage, each dimension is sampled with four equi-probable points which are
combined to only four learning paths for the perfect learning comparison according to the
descriptive sampling scheme (instead of 16 learning paths with a full factorial design). We
tested the influence of the low sampling size on the results by complementing the analysis
with a sampling of 10 ×10 samples in CS and afor the case of uncertainty vs. best guess.
The results did not change significantly.
Fig. 6 shows the welfare changes, relative to the no-control case, for the different sce-
narios with and without uncertainty within the MIND model. First, the benefit from
acting upon climate change is small relative to the net costs due to the existence of climate
change. In other words only a small part of the climate change induced welfare losses can
be countered by mitigation policy. This observation stays the same in the uncertain set-
ting, although the best guess climate policy leads to higher benefits against the no-control
policy with uncertain damage. The welfare benefit from adapting the optimal policy is
nearly invisible, whereas the welfare gain from perfect learning is significant. This finding
compares to only small changes in optimal emission pathways from the best guess to the
uncertain policy against strong changes in case of perfect learning (see Fig. 7)..
In a study close to this one, Pizer [1997] investigated the effect from explicitly including
uncertainty into the DICE model by Nordhaus [1994]. He not only considered uncertainty
12
52 How important is Uncertainty for the Integrated Assessment of Climate Change?
about the socio-economic and the climate system but also about the normative parameters
of risk aversion and pure rate of time preference. He found that the uncertainty about the
normative parameters by far dominates the uncertainties about the socio-economic system.
We perform a sensitivity study of the uncertainty components towards the parameter of
constant relative risk aversion. The necessary scenarios for optimal climate policies under
best guess, uncertainty and perfect learning have been evaluated for 8different values of η.
The resulting changes in the partition of the benefit of optimal climate policy are depicted
in Fig. 8 for the case of uncertainty and in Fig. 10 for the case of perfect learning. The
changes in optimal decisions between the best guess and the uncertainty case are depicted
in Fig. 9.
From Fig. 8 a clear ordering of the different components of the overall benefit of climate
policy emerges: The main part of the overall benefit of climate action can be realized by
simply taking the optimal best guess policy. However, reevaluating this best guess policy in
an uncertain information setting significantly increases the benefit. The changes in optimal
decisions between the best guess and the uncertainty setting (see Fig. 9) are at least partly
significant, e.g. a >5% change in cumulative carbon emissions for the next two centuries.
But the resulting welfare effect from this adjustments (BOAU) is insignificant for the whole
range of η, thus the explicit incorporation of uncertainty into the optimization only plays
a minor role. From Fig. 10 one can see that the contribution from perfect learning by far
dominates the contribution from adapting to uncertainty. With increasing η, the value of
perfect learning even dominates all other components. Summarizing the numbers indicates
that within MIND uncertainty about the climate response to anthropogenic carbon emis-
sions and about climate induced damage is not important for the assessment of optimal
climate change mitigation whereas perfect learning clearly changes the picture.
An additional interesting feature of the sensitivity study with respect to relative risk
aversion ηis the rapidly decreasing overall benefit of climate policy for increasing η. For
values of η > 2the benefit from acting upon climate change gets lower than 0.01% of
change in CBGE consumption. This can be explained with the dual role of the parameter
η. Within the expected utility framework employed in most studies of optimal global miti-
gation assessment the parameter ηrepresents both, the DM’s constant relative risk aversion
and her aversion to fluctuations of consumption over time. With increasing risk aversion,
the DM reacts with stricter policies to minimize the uncertainty in climate impacts. But
with increasing aversion to consumption fluctuations over time, within an overall growing
economy the DM’s incentive to shift consumption from the future towards the present
becomes stronger. Within MIND, obviously the second effect is stronger, as the mitigation
effort decreases with η, and therefore the benefit from acting upon climate change also
decreases. A separation of both roles of ηcan be achieved within a normative satisfying
setting by Treager (2009, and references therein). This is left for future studies.
13
Importance of Uncertainty in MIND 53
02468
0
20
40
60
80
100
η
[%]
10-5
10-4
10-3
10-2
10-1
100
101
∆ CBGE[%]
BCP(x1)
dBCP
BOAU
BCP(x2)
135
Figure 8: The three contributions (benefit of best
guess policy in darker green, reevaluation of best
guess policy in lighter green, benefit of adapting
policy in yellow) normalized to the overall benefit
of climate policy under uncertainty, calculated for
different values of constant relative risk aversion η
(the three small values for η≤2are η=.5; .75; 1.5).
Also shown is the overall benefit of climate policy
itself on the right axes.
123456
−20
−15
−10
−5
0
5
η
[%]
rdab
inresex
infossil
inrenew
Figure 9: Relative changes in decision variables
∆ˆx(investments in R&D to increase energy effi-
ciency (rdab), investments in extraction of fossil
energy resources (inresex), investments in the cap-
ital stock of fossil energy carriers (infossil), and in-
vestments in the capital stock of renewable energy
carriers (inrenew)), cumulated over the full time
horizon (2010-2200), from the optimal best guess
strategy ˆx1to the optimal strategy under explicit
inclusion of uncertainty ˆx2.
0 2 4 6 8
0
20
40
60
80
100
η
[%]
10−5
10−4
10−3
10−2
10−1
100
101
∆ CBGE [%]
BCP(x1)
dBCP
BOAU
VPI
BCP(x2)
BCP(x3)
Figure 10: The four contributions (benefit of best
guess policy in darker green, reevaluation of best
guess policy in lighter green, benefit of adapting
policy in yellow, value of perfect information in
purple) normalized to the overall benefit of climate
policy under perfect learning, calculated for differ-
ent values of constant relative risk aversion η(the
three small values for η≤2are η=.5; .75; 1.5).
Also shown are the benefits of climate policy for
the case of uncertainty (lighter green triangles) and
perfect learning (darker green triangles).
14
54 How important is Uncertainty for the Integrated Assessment of Climate Change?
3.3 The Marginal Cost - Benefit picture of MIND
In the following we apply the marginal cost-benefit picture from Section 2.3 to the MIND
model to understand the reasons for the small welfare effect from explicitly including un-
certainty about the climate response and climate induced damage. Therefore we interpret
the welfare benefit from choosing the optimal policies ˆx1,ˆx2instead of the no-control pol-
icy ˆx0as composition of mitigation benefits B(x, θ)and mitigation costs C(x), analogue to
Sec. 2.3. For any climate policy xwe define B(x, π)≡ {[U(x, π)−U(x, π0)] −[U(ˆx0, π)−U(ˆx0, π0)]}
as the difference in the welfare impacts due to the existence of climate induced damage
between the policies xand ˆx0. Thereby πindicates a world with climate damage, and π0
indicates a world without climate damage. We define mitigation costs C(x)of policy x
as: C(x)≡[U(ˆx0, π0)−U(x, π0)]. This is the loss in welfare for choosing a suboptimal
policy xinstead of the optimal policy ˆx0in a world without climate damage (π0). Simple
calculus shows, that this choice actually delivers the desired composition for any policy x:
B(x, π)−C(x) = {[U(x, π)−U(x, π0)] −[U(ˆx0, π)−U(ˆx0, π0)]}
−[U(ˆx0, π0)−U(x, π0)]
=U(x, π)−U(ˆx0, π).
Using this composition, the problem of finding the optimal climate policy ˆxfor a
given information setting (Eq. 1) can be rewritten as maximizing the difference between
mitigation benefits and costs. This can be recast in an a-temporal cost-benefit picture
by identifying the intersection of the marginal benefits (dB(x, π)/dx) and marginal costs
(dC(x)/dx). The intersection point on the x-axis corresponds to the optimal policies ˆx1
(using B(x, ¯π)or ˆx2(using B(x, π)). Numerically the benefits and costs for a given policy
xare calculated by evaluating welfare differences as relative changes in CBGEs (see Sec. 5).
To be able to inspect the cost-benefit picture visually we additionally need to project
the multi-dimensional decision variable xon a single-dimensional quantity. Thereby we
loose the exact equivalence between the welfare picture and the cost-benefit picture. The
goal is to choose a projection x→˜xthat approximates the welfare effect of uncertainty
with high accuracy and allows an interpretation of the small amplitude. We achieve the
one-dimensional projection by introducing a constraint on cumulative emissions in a setting
without climate damage. For a constraint above 3165GtC the no-control policy emerges.
With decreasing levels of admissible cumulative emissions, the DM reacts by adjusting the
investments into the different energy technologies (see Fig. 11). With increasing stringency
of the constraint on cumulative emissions the investments into R&D in energy efficiency
increase, as well as the investments into carbon free renewable energy. Contrary, the invest-
ments in carbon intensive fossil energy carriers and the corresponding resource extraction
sector decrease.
Introducing a dense sampling in the cumulative emissions constraint and evaluating
the resulting policies in settings with and without uncertainty allows to construct the cost-
15
Importance of Uncertainty in MIND 55
0 500 1000 1500
0
100
200
300
400
Trillion1995USD
R&Denergyefficiency
renewables
resourceextraction
fossils
mitigationeffort
Figure 11: Aggregated investments (NPV, dis-
counted with 5%) into the energy system depending
on the required mitigation effort (in GtC cumula-
tive reduction relative to the no-control in which
3165 GtC are emitted over the period 2010-2200).
0 500 1000 1500
0
1
2
3
4
5x 10−4
x1x2
mitigation effort [GtC] (zero equals 3165 GtC)
[%∆CBGE /GtC]
65%
32%
4%
Figure 12: Marginal costs (lower black line)
and (expected) marginal benefits (upper black and
black dashed line) of mitigation within the MIND
model. The Grey curves are the raw data from
MIND, the black curves are polynomial fits. Mit-
igation effort is parametrized by a decreasing con-
straint on cumulative emissions. The optimal poli-
cies from the “correct” optimization are shown as
vertical dashed lines. The constant relative risk
aversion ηis set to 2, this can be compared to the
other values investigated in Sec. 4.1.
benefit picture for the MIND model (see Fig. 12). The marginal benefits for the best
guess case are derived by fixing all uncertain parameters to their expected value while the
expected marginal benefits are derived by applying the cost-benefit decomposition to the
expected utility. The fluctuations in the Grey curves, that represent the raw data from
the model can be explained by the limited temporal resolution of the model (5yrs). When
optimizing under a binding constraint with increasing stringency (such as the constraint
on cumulative emissions), the timing of the mitigation effort to stay below the constraint
can only be adjusted within this limited temporal resolution. This leads to small jumps in
the overall welfare and thus also in marginal welfare and in marginal benefits and costs.
The bold black lines are polynomial fits to the raw data.
Analogue to Section 2.3 the optimal mitigation effort for the best guess and the un-
certainty setting can be obtained as intersections between the marginal costs and the
(expected) marginal benefits of mitigation. The different contributions to the overall ben-
efit of acting upon climate change can be visualized as areas between the benefit and cost
curves. The pie chart in the upper left corner shows their relative contributions to the
overall benefit of climate policy. A comparison between the optimal values of cumulative
emissions derived from the marginal picture and those derived from the “correct” welfare
optimization shows the “error” of the approximation. The optimal level of mitigation in
cumulative emissions is represented within an 4% error, while the welfare effects of uncer-
tainty are overestimated by up to 5%.
16
56 How important is Uncertainty for the Integrated Assessment of Climate Change?
Nevertheless the cost-benefit picture allows to identify reasons for the negligible uncer-
tainty effect. First, the overall value of acting upon climate change is constraint due to the
convex increasing functional form of both (expected) marginal benefits and marginal
costs. The combination of these functional forms lead to a very small area between both
curves and thus, a small overall benefit of climate policy. This result is somewhat counter
intuitive as one would assume climate damage to be convex increasing in temperature and
temperature more or less linearly connected with mitigation effort and thus would expect
decreasing marginal benefits. The reason for the counter-intuitive result from the MIND
model is discussed further below. If marginal benefits were concave increasing in the miti-
gation effort or even decreasing, for fixed intersection points ˆx1and ˆx2, the overall benefit
of climate policy would increase. However, the same is not true for the value of adapting to
uncertainty (BOAU) that is represented by the darker green “triangle” within Fig. 12. For
fixed ˆx1and ˆx2more concave marginal benefits would increase the BOAU, but even more
so the other components of the BCP, thus the relative importance of explicitly including
uncertainty would even decrease.
Second, the marginal risk premium, that is the difference between expected marginal
benefits under uncertainty and marginal benefits for the expected parameter values, only
increases linearly in the mitigation effort and with a small slope. Together with the strongly
convex increasing marginal costs, this leads to a relatively small difference between the two
optimal policies ˆx1and ˆx2. To increase the BOAU, one would need to either increase the
slope of the MRP or even better, increase the convexity of the MRP in the mitigation
effort. Both measures would lead to a larger difference between the marginal benefits in
the best guess and the uncertainty case and thus to a larger difference between ˆx1and ˆx2.
However, increasing only the slope of the MRP would also increase the reevaluation effect
of the best guess policy (lighter Grey area) and thereby limiting the relative importance
of the BOAU.
3.4 Functional Dependencies within MIND
To find an explanation for both results, the slope and curvature of marginal benefits and
the small MRP, we apply the marginal representation to the single steps in the climate
cause-effect chain. The absolute and marginal functional form of the individual elements
of the chain are shown in Fig. 13. This allows us to investigate in detail how the slope and
curvature are determined in the integrated assessment model MIND.
The (cumulative) emissions lead to a rising concentration of greenhouse gases in the
atmosphere and increasing radiative forcing. The maximum forcing reached for different
levels of mitigation effort is shown in panel a. The maximum total forcing is concave
increasing in cumulative emissions and concave decreasing in mitigation effort respectively.
This can be explained by the saturation effect represented by the logarithmic relation from
concentration to forcing. With increasing atmospheric concentration of carbon dioxide,
the frequency band in which CO2absorbs the outgoing radiation saturates, thus a further
17
Importance of Uncertainty in MIND 57
0 500 1000 1500
4
5
6
a
[W/m2]
max(ftot)
−4
−2
0
2
x 10−3
[W/(m2 GtC) ]
max forc
marg. max forc
0 500 1000 1500
3
3.5
4b
[°C]
−1.5
−1
−0.5
0
x 10−3
[°C/GtC]
max(T)
0 500 1000 1500
3
4
5
6
7
c
[% net GDP]
−3
−2
−1
0
x 10−3
[1/GtC]
max DMG
0 500 1000 1500
499.3
499.4
499.5
499.6
d
trl USD
−15
−10
−5
0
5
x 10−4
[trl USD/GtC]
NPV GDP
0 500 1000 1500
0.7
0.8
0.9 e
∆CBGE [%]
Welfare Damages
mitigation effort [GtC] (zero equals 3165 GtC)
−2
−1.5
−1
−0.5
x 10−4
[%/GtC]
0 500 1000 1500
0
0.05
0.1
0.15
f
∆CBGE [%]
mitigation effort [GtC] (zero equals 3165 GtC)
0
1
2
3
4
x 10−4
[%/GtC]
Welfare Benefits
best guess
expected value
Figure 13: Functional dependencies of individual components in the cause-effect chain of
climate change on mitigation effort measured in terms of cumulative emissions reductions from
the BAU emissions of 3165 GtC in the period 2010 −2200 :(a) maximum radiative forcing,
(b) maximum temperature change, (c) damage in %of net GDP for the maximum temperature
change, (d) net present value (NPV) of gross output including mitigation costs, but excluding
climate damage, (e) welfare equivalent damage measured in %∆CBGE, and (f) welfare benefits
measured in %∆CBGE. Shown are the functions (continuous lines) and the marginal functions
(dashed lines). For those quantities that depend on the uncertain SOW the best guess value is
shown in darker Grey and the expected value of the uncertain setting is shown in darker green.
The original model data are shown in lighter Grey, where as the functional dependencies are
polynomial fits of the data.
18
58 How important is Uncertainty for the Integrated Assessment of Climate Change?
increase in concentration leads to less and less additional radiative forcing. The same
concave behavior, increasing in cumulative emissions and decreasing in mitigation effort,
occurs for the maximum temperature increase max(T), shown in panel b. The global mean
temperature reacts to changes in radiative forcing. The overall temperature response is
determined by the amplitude (climate sensitivity) and time scale (ocean diffusivity) of a
simple impulse response model. The climate damage that is incurred by the maximum
temperature change, measured in %of net GDP, is shown in panel c. Panel dshows the
net present value of gross output excluding climate damage, aggregated over time by an
endogenous discount rate ρt≡δ+η·gt, where δis the pure rate of time preference, ηis
the rate of constant relative risk aversion and gtis the endogenously determined growth
rate of consumption. The gross economic output is concave decreasing in the mitigation
effort, leading to convex increasing mitigation costs in GDP terms, which are derived by
subtracting the gross output curve from the gross output of the no-control case x= 0.
Multiplying the damage factor, DF = 1/(1 + D), where Dare the net GDP damage
from panel c, with the gross GDP in each time step gives the time series of net GDP
that constrains the investment decisions and consumption level via a budget equation.
Thus both the costs from mitigation (as seen in the gross GDP) and the climate damage
lower the consumption level and thus the welfare. The welfare equivalent damage for the
different mitigation scenarios, shown in panel e, is derived by evaluating the difference in
CBGE between a case with the damage factor DF as above and a case without damage
(but with mitigation costs), where DF = 1. Formally the welfare damage is given as
∆CBGE(V(x, π0), V (x, π/¯π)) or in loose notation as U(x, π0)−U(x, π/¯π). Normalizing
the welfare damage to the no-control case delivers welfare benefits from mitigation, shown
in panel f. Formally this normalization is done by subtracting the welfare damage for
the no-control case, leaving us with the definition of benefits from Section 3.3 U(x0, π0)−
U(x0, π)−(U(x, π0)−U(x, π)) = B.
The two most interesting features in the cost-benefit picture of MIND are the positive
slope and the convex curvature of the marginal benefits of mitigation. Concerning the
slope of the marginal benefits in welfare, the explanation can already be found in panels
band c. The marginals of maximum (Panel c) and welfare equivalent damage (Panel
e) are decreasing in the mitigation effort implying increasing marginal benefits. As can
be seen from a comparison of panel c and e, this behavior is not a result of the welfare
evaluation of climate damage (although it is strengthened by it), but already present in
the marginal of the maximum damage. Since the maximum damage is convex increasing
with rising temperature, their marginal is increasing with temperature as well. The fact
that their marginal is decreasing when plotted against increasing mitigation effort instead
of temperature, due to concave instead of convex decreasing maximum damage, points to
the fact that the concavity found in the temperature response to mitigation dominates
the convexity of damage in rising temperature. Thus, we find that the saturation of the
emissions to temperature change relationship over-compensates the non-linearity in the
climate damage function, leading to increasing instead of decreasing marginal benefits of
19
Importance of Uncertainty in MIND 59
mitigation, and thus limiting the overall benefit of the best guess climate policy (the dark
Grey area in Fig. 2).
The convexity of marginal welfare benefits in mitigation effort however does not orig-
inate from the combination of maximum temperature with the damage function, but
emerges from the welfare valuation of climate damage. This can be seen by comparing
the convex decreasing marginal of maximum damage (panel c) to the concave decreas-
ing marginal of welfare equivalent damage (panel e). Hence, the influence of the welfare
function, i.e. of the normative parameters of constant relative risk aversion ηand pure
rate of time preference ρdetermines the curvature of the marginal benefits. Comparing the
marginal benefits for the best guess case and the case of uncertainty, it can be seen that the
convexity increases when accounting for uncertainty, implying a convex increasing MRP.
Thus, the additional marginal welfare benefit of reducing a unit of emissions under uncer-
tainty grows with increasing mitigation effort. This works against a large contribution of
re-evaluating the best guess climate policy under uncertainty (DBCP; light Grey area in
Fig. 2), and favors a larger relative contribution of adjusting the mitigation policy under
uncertainty (BOAU; orange area in Figure 2). However, due to the strongly increasing
marginal mitigation costs, the welfare gains from adjusting the mitigation policy remain
small.
Another important observation in the cause-effect chain is the small influence of un-
certainty about climate sensitivity and the correlated time lag of the climate response
(panel b). The expected maximum temperature increase for uncertain cs is slightly shifted
towards lower levels, which in itself is counter intuitive, as uncertainty about cs has an
asymmetric upper tail. The explanation for this could be the correlation between climate
sensitivity and the time scale of climate response due to the observations of 20thcentury
global mean temperature. Higher values of climate sensitivity are connected to longer time
scales of temperature response, thus the temperature increase will only show later. In
combination with the limited time horizon due to discounting this limits the influence of
high cs values. In addition, the marginal maximum temperature increase shows nearly no
change from best guess to the expected case. Thus the uncertainty about cs can not lead
to a change in marginal benefits due to uncertainty.
4 Changes in the Model Structure
Which assumptions about the climate cause-effect chain would lead to a significant welfare
gain from adapting the optimal policy to uncertainty? In this section we investigate several
changes in the model structure and their influence on the cost-benefit picture and the
BOAU.
20
60 How important is Uncertainty for the Integrated Assessment of Climate Change?
0 500 1000 1500 2000
0
0.5
1
1.5
x 10−3
x1
x2
mitigation effort [GtC] (zero equals 2959 GtC)
[%∆CBGE/GtC]
75%
25%
< 1%
0 500 1000 1500 2000
0
1
2
3
4
5x 10−4
x1x2
mitigation effort [GtC] (zero equals 3090 GtC)
[%∆CBGE/GtC]
68%
30%
3%
0 200 400 600 800 1000
0
0.2
0.4
0.6
0.8
1x 10−5
x1x2
mitigation effort [GtC] (zero equals 3298 GtC)
[%∆CBGE/GtC]
60%
35%
5%
Figure 14: Sensitivity of the marginal cost-benefit picture of MIND with respect to changes in the
parameter of constant relative risk aversion η. Shown are the pictures for η= 0.75(left), η= 1.5(middle),
η= 3(right). The legend is equivalent to Fig. 12.
4.1 Constant relative risk aversion η
We have shown in Section 3.4 that the curvature of the welfare function, represented by
the parameter of constant relative risk aversion η, strongly influences the curvature of the
marginal benefits of mitigation. We have also shown (Section 2.3) that the curvature of
the marginal benefits strongly influence the overall benefit of climate policy. We use these
dependencies and investigate the relative importance of adjusting the optimal policy to
uncertainty depending upon the parameter of constant relative risk aversion. The changing
cost-benefit pictures of MIND are shown in Fig. 14 for values of ηbetween 0.75 and 3.
The effects of the curvature of the welfare function are manifold: The first, and most
important one, is a scaling effect. As already shown in Fig. 8, the overall net benefit
of climate policy strongly decreases with increasing η. This effect can be explained by
the dual role of η. It does not only represent risk aversion, but also the DM’s aversion
towards inter temporal fluctuations in consumption. If this aversion is high, the DM
prefers a smooth, constant consumption stream over a fluctuating, increasing one. In
a growing economy with consumption growth in the future the decision maker prefers
to delay mitigation as it would require to divert consumption into early investments in
carbon free energy technologies. This effect can already be seen in the baseline cumulative
emissions (the number given as label below the figures). Hence the net benefits from
mitigation, i.e. reduced climate damage minus mitigation costs, are lower for a high η
as mitigation reduces early consumption. The second effect concerns the curvature of the
marginal benefits. This effect is directly evident: as discussed in Sec. 2.3, the exponent η
of the welfare function obviously directly determines the curvature of the marginal welfare
depending on consumption.
In combination both effects result in increasing absolute overall benefits of climate
policy and increasing absolute welfare gains from adjusting climate policy to uncertainty
with decreasing η. However, the relative contribution of the BOAU to the benefit of
climate policy decreases with η.
21
Changes in the Model Structure 61
4.2 Exponential Damage
Unlike Weitzman [2010] who focused on the potential fat tails of the distribution on climate
sensitivity and climate damage we are searching for a setting in which a strong impact of
uncertainty on optimal mitigation efforts also occurs for thin tailed distributions. As stated
before a stronger increase, and convexity, in the marginal risk premium in welfare terms
for rising mitigation effort would lead to a stronger BOAU.
First , we replace the standard quadratic formulation of the damage function by an
exponential formulation:
DFe=1
1 + k·exp(T
l)−k. (11)
We choose the parameters kand lsuch that the exponential damage in net GDP,
k·exp(T/l)−kequal the standard formulation at T= 3°for the best guess case. We
assume a normally distributed lwith l=N(2.2571,0.61). Together with k= 0.01, this
choice leads to a best guess marginal damage function which is nearly identical to the
standard best guess marginal damage function used in the previous section. However, the
expected marginal damage function is far more convex in temperature than in the quadratic
case. Thus the marginal risk premium in net GDP damage increases more strongly. The
resulting difference in the marginal of the maximum damage is shown in rows aand b
of Fig. 15 together with the identical maximum temperature functions and the resulting
cost benefit pictures. The change towards exponential damage shows several interesting
effects: First, as we have chosen identical best guess marginal damage, the optimal policy
in the best guess case also does not change, but the higher marginal risk premium leads
to an increased ˆx2. The welfare benefit from adapting the policy to uncertainty increases
significantly and now contributes 13% to the overall benefit of climate policy (instead of 4%
in the quadratic case). The increase in the relative contribution of the BOAU is dampened
by the fact that the effect of reevaluating the best guess policy under uncertainty is also
strongly increasing. This is due to the fact, that the expected marginal benefits do not only
increase more strongly than before but are also shifted upwards over the whole domain.
Second, the change towards exponential damage leads to at least partly increasing expected
marginal damage in net GDP. However this shift in the slope of the marginal damage is not
strong enough to be reflected in the expected marginal benefits, it “gets lost” through the
convolution with the welfare function. Finally, compared to the case of quadratic damage,
the overall value of climate policy more than doubles for the assumption of exponential
damage, with more than two thirds of the benefit due to taking uncertainty into account.
4.3 Linear Carbon Climate Response
Finally, we replace the climate module by a linear relationship between cumulative carbon
emissions and increase in global mean temperature that has been found by Matthews et
al. [2009] within an ensemble of state of the art climate models. By introducing the
22
62 How important is Uncertainty for the Integrated Assessment of Climate Change?
0 500 1000 1500
2.5
3
3.5
4
4.5
a
[°C]
−1.4
−1.2
−1
−0.8
−0.6
x 10−3
[°C/GtC]
max(T)
0 500 1000 1500
2
4
6
8
10
12
[% net GDP]
−6
−4
−2
x 10−3
[1/GtC]
max DMG
0 500 1000 1500
0
1
2
x 10−4
[% GtC]
x1x2
65%
32%
4%
0 500 1000 1500
2.5
3
3.5
4
4.5
b
[°C]
−1.4
−1.2
−1
−0.8
−0.6
x 10−3
[°C/GtC]
max(T)
0 500 1000 1500
2
4
6
8
10
12
[% net GDP]
−6
−4
−2
x 10−3
[1/GtC]
max DMG
0 500 1000 1500
0
1
2
x 10−4
[% GtC]
x1x2
31%
56%
13%
0 500 1000 1500
2.5
3
3.5
4
4.5
c
[°C]
−1.4
−1.2
−1
−0.8
−0.6
x 10−3
[°C/GtC]
max(T)
0 500 1000 1500
2
4
6
8
10
12
[% net GDP]
−6
−4
−2
x 10−3
[1/GtC]
max DMG
0 500 1000 1500
0
2
4
x 10−4
[% GtC]
x1x2
63%
35%
3%
0 500 1000 1500
2.5
3
3.5
4
4.5
d
[°C]
mitigation effort [GtC] (zero equals 3165 GtC)
−1.4
−1.2
−1
−0.8
−0.6
x 10−3
[°C/GtC]
max(T)
0 500 1000 1500
2
4
6
8
10
12
[% net GDP]
mitigation effort [GtC] (zero equals 3165 GtC)
−6
−4
−2
x 10−3
[1/GtC]
max DMG
0 500 1000 1500
0
2
4
x 10−4
mitigation effort [GtC] (zero equals 3165 GtC)
[% GtC]
x1x2
38%
58%
4%
Figure 15: Maximum temperature change, maximum climate damage in net GDP, and
marginal cost benefit picture for four different structural model settings: (a) standard cli-
mate module and quadratic damage function, (b) standard climate module and exponential
damage function, (c) linear climate carbon response and quadratic damage function, and
(d) linear climate carbon response and exponential damage function. The functional rela-
tions are shown in darker Grey for the best guess case and in darker green for the expected
value of the uncertainty case. The dashed lines represent the marginal functions. The fluc-
tuating lines in lighter Grey are the original model data. The smooth lines are polynomial
fits to the data. The legend for the cost benefit pictures is analogous to Fig. 12.
23
Changes in the Model Structure 63
so called carbon climate response (CCR) parameter, the relationship between global mean
temperature change (relative to pre industrial) ∆Tand cumulative carbon emissions reads:
∆T(t) = CCR ·
t′
X
t0
e(t′), (12)
where eare globally aggregated carbon emissions. Within the model ensemble, Matthews
et al. [2009] found a carbon climate response of CCR = 1.5[1.0−2.1]°C/TtC. The val-
ues in square brackets mark the 5and 95 quantile. We choose a log-normal distribution
for the CCR with CCR =LN(log(1.461),0.23), which gives the best fit to the quantiles
and 1.5°C as expected value. The resulting maximum temperature, maximum net GDP
damage and the cost benefit pictures are shown in rows cand dof Fig. 15. In row c
the linear climate carbon response is combined with quadratic damage and in row dwith
exponential damage from Sec. 11. Considering the maximum temperature response, the
difference between the best guess case and the expected case under uncertainty nearly van-
ishes. This is clear, as the uncertain parameter CCR now enters linearly into the function,
thus the expectation operator only acts on the parameter itself. Thus the uncertainty in
the climate system is now irrelevant for the mitigation problem. But even more interesting
is the change in the best guess maximum temperature function itself. It declines more
strongly in the mitigation effort than before. This leads to a stronger difference between
the best guess and expected damage. The changed curvature of the climate response leads
to convex decreasing net GDP damage. However, the increasing slope of the marginals
is changed back to decreasing marginal welfare equivalent damage further downstream by
the welfare function. Hence the marginals in the welfare benefit are still increasing, but
less convex than before. Compared to the standard climate module, the overall benefit of
climate policy increases strongly (by a factor of X), but the individual contributions of the
three components remain largely unchanged. In particular, the welfare contribution from
adapting the optimal policy to uncertainty is still negligible.
Combining the linear climate carbon response with exponential damage amplifies the
distinct features of the two cases. The convexity in expected net GDP damage gets strong
enough to “survive” the convolution with the welfare function leading to initially decreas-
ing expected marginal benefits in welfare terms. This further increases the benefit from
reevaluating the best guess policy. The BOAU stays small.
Summarizing the results from this Section, the functional formulation of the building
blocks of the climate cause-effect chain (temperature response, climate induced damage and
the aggregated welfare function) and especially their marginals determine the strength of
the effect of including uncertainty. Thereby the non-linearities in the temperature response
and the damage function partly compensate each other. Everything else equal, the impor-
tance of adjusting policies to uncertainty becomes the more important the more convex
the damage function, the lower the risk aversion parameter and the lower the concavity of
the maximum temperature in cumulative emissions.
24
64 How important is Uncertainty for the Integrated Assessment of Climate Change?
5 Conclusion
We applied a decomposition of the overall benefit of acting upon climate change into its
single components to measure the importance of uncertainty and perfect learning within
the integrated assessment model MIND.
Uncertainty influences both, the optimal mitigation policy and the expected utility of dif-
ferent policies. Including uncertainty explicitly is important, if it leads to a significant
change in the optimal policy (before any potential future learning) that in turn leads to
a significant change in the benefit gained from acting upon climate change. Uncertainty
might also be considered relevant, but would not have to be included explicitly into the
optimization framework, if it significantly changed the assessment of the benefit of climate
policy compared to the best guess case (reevaluation effect), even though the optimal cli-
mate policy (before learning) did not change significantly.
Within the MIND model the reevaluation effect is dominating the welfare gain from ad-
justing the policy under uncertainty, while perfect learning is dominating both of these
effects. Overall, the welfare effect of accounting for uncertainty is rather small, which fur-
ther corroborates the findings in the literature.
To understand the origin of these findings, we projected the complex MIND model to an
a-temporal marginal cost-benefit picture and resolved the functional relationship between
the single steps of the climate cause-effect chain. Thereby we located the origin of the
negligible welfare gain from adapting the optimal policy to uncertainty. This benefit of
anticipating uncertainty (BOAU) is only of significant size if uncertainty leads to non-
linear shifts in the marginal benefits of mitigation, that would lead to a convex strongly
increasing marginal risk premium with increasing mitigation effort.
In the standard model setting with a quadratic damage function and a zero-dimensional
climate-carbon response box model, this behavior was constrained by the saturation of
the emissions to temperature change relationship compensating for the non-linearity in the
climate damage function and by the consumption smoothing property of the welfare func-
tion. Thus for seeing a significant influence from including uncertainty one has to consider
alternative model settings that induce a strongly convex increasing marginal risk premium
(MRP).
Two such changes in the model setup, an exponential climate damage function and a linear
climate carbon response have been implemented. We showed that those changes to the
model structure indeed can lead to a strongly convex increasing MRP and a significant
uncertainty effect.
The other feature that constrains the importance of including uncertainty is the strongly
increasing marginal mitigation cost curve in the model. Thus a change in the convexity of
marginal mitigation costs, especially reducing the strong increase for higher levels of mit-
igation would also lead to more significant uncertainty effects. This is of special interest
as it emphasizes the combined importance of the modeling of mitigation options and the
impact and damage formulation for the overall importance of uncertainty for the integrated
25
Conclusion 65
assessment of climate change.
These results come with the usual caveats. The employed integrated assessment model
MIND, although more complex than quasi-analytical cost-benefit models and the com-
monly used DICE model, still includes a strongly simplified representation of the cause-
effect chain of climate change.The representation of uncertainty and learning had to be
constrained to a few sample points and to the limiting case of perfect learning, and we
only investigated the effect from a single information setup.
Thus this study should not be seen as an attempt to find a conclusive answer to the question
whether accounting for uncertainty and learning is important for the assessment of climate
policy. Rather, we present an approach to decompose and trace the uncertainty effect in
complex integrated assessment models, which we believe will prove useful to improve our
understanding about the effect of structural model assumptions on the significance of the
uncertainty effect.
A Comparing Welfare across different scenarios
As (expected) utility is only defined up to an affine transformation, we use differences in the
certainty and balanced growth equivalents (CBGE), as presented by Anthoff & Tol [2009],
to compare different scenarios. The certainty equivalent of an uncertain consumption
outcome is an amount of consumption the DM would demand instead of a distribution
of outcomes to get the same expected utility. The same principle works for the balanced
growth equivalent: here the consumption path, that possibly varies over time, is replaced
by a path consisting of an initial consumption level that growth over time with a constant
growth rate αand gives the same utility. If one is only interested in relative changes in
the CBGE between different scenarios, the measure is independent of the growth rate α.
Thus the relative change in CBGE, denoted by ∆CBGE, can be interpreted as fraction of
consumption the DM would be willing to pay, now and forever, to switch from a scenario
with lower CBGE to the other scenario. Formally the ∆CBGE for isoelastic utility reads:
∆CBGE [EU1, EU2]≡
EU1
EU21−η−1for η6= 1
exp EU1−EU2
PT
t0Pt(1+ρ)t−1for η= 1 . (13)
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68 How important is Uncertainty for the Integrated Assessment of Climate Change?
69
Chapter 4
Anticipating Climate Threshold Damages ∗
Alexander Lorenz
Matthias G.W. Schmidt
Elmar Kriegler
Hermann Held
∗This chapter has been published as: Lorenz, A., Schmidt, M.G.W., Kriegler, E., Held, H 2011. Anticipating
Climate Threshold Damages. Environmental Modeling and Assessment. 17 (1-2): 163-175.
70 Anticipating Climate Threshold Damages
Environ Model Assess
DOI 10.1007/s10666-011-9282-2
Anticipating Climate Threshold Damages
Alexander Lorenz ·Matthias G. W. Schmidt ·
Elmar Kriegler ·Hermann Held
Received: 31 July 2010 / Accepted: 1 April 2011
© Springer Science+Business Media B.V. 2011
Abstract Several integrated assessment studies have
concluded that future learning about the uncertainties
involved in climate change has a considerable effect on
welfare but only a small effect on optimal short-term
emissions. In other words, learning is important but
anticipation of learning is not. We confirm this result in
the integrated assessment model “model of investment
and technological development” for learning about cli-
mate sensitivity and climate damages. If learning about
an irreversible threshold is included, though, we show
that anticipation can become crucial both in terms of
necessary adjustments of pre-learning emissions and
resulting welfare gains. We specify conditions on the
time of learning and the threshold characteristic, for
which this is the case. They can be summarized as a
narrow “anticipation window.”
Keywords Epistemic uncertainty ·Learning ·
Anticipation ·Value of information ·
Value of anticipation ·Threshold damages
A. Lorenz (B
)·M. G. W. Schmidt ·E. Kriegler ·H. Held
Potsdam Institute for Climate Impact Research,
14412 Potsdam, Germany
e-mail: [email protected]
A. Lorenz
Environmental Change Institute,
School of Geography and the Environment,
University of Oxford, South Parks Road,
Oxford, OX1 3QY, UK
H. Held
University of Hamburg & Klima Campus Hamburg,
Bundesstr. 55, 20146 Hamburg, Germany
1 Introduction
Climate change poses a formidable global problem.
Climate impacts may occur over a wide range of sectors,
countries and time. Moreover, the regions most vulner-
able to the impacts differ from those responsible for
the largest parts of emissions. Although climate science
has gained a profound understanding of the elementary
processes underlying climate change, big uncertainties
about its magnitude and implications remain. These sci-
entific uncertainties will be reduced in the future, and it
will be possible to adjust climate policy accordingly.1
Investments in mitigation of greenhouse gas emissions
are at least partially sunk or irreversible, respectively.
The combination of uncertainty, learning about uncer-
tainty and irreversibility makes it interesting to study
the effect of anticipation of future learning on optimal
near-term climate policy. Important questions in this
context are: Should society wait for better information
about the climate system and climate damages before
committing to mitigation measures or should it miti-
gate preemptively? Does anticipation of future learning
yield significant welfare increases?
A theoretical literature has established theorems
about the sign of the anticipation effect, i.e., the effect
of anticipation of future learning on optimal short-term
decisions. In very simple two-period models, a Bayesian
decision maker (DM) is characterized by a goal
1We will assume that learning eventually reveals the true values
of parameters. For interesting examples, where new informa-
tion might narrow the uncertainty around a false value, see
Oppenheimer et al. [30] and Kriegler [19].
Introduction 71
A. Lorenz et al.
function U(x1,x2,s), where sis the state of the world,
and the decision variables xt,t∈ {1,2}denote direct
consumption of a generic good, emissions of a pollu-
tant, or investment decisions. The DM first chooses x1,
then gets some message ycontaining information about
the uncertain s, and finally chooses x2. The question
under consideration is: In which direction does the
optimal first period decision x1change depending on
the informativeness of y? The most general answer to
this question has been given by Epstein [9], who showed
that it depends on the properties of the 2nd-period value
function j(x1, π) ≡maxx2PsπsU(x1,x2,s), where πsis
the probability of s. More information (in the sense
of Blackwell [4]) unambiguously, i.e., independent of
the specific form of the information structure (in the
sense of Marschak and Miyasawa [23]), leads to a lower
optimal level of x1if and only if ∂j/∂x1is convex
in πs. One strand of the literature applies Epstein’s
condition in simple analytically solvable models (see,
e.g., Kolstad [18]; Gollier et al. [12]). In more complex
models, though, Epstein’s condition is of limited value
for two reasons: Firstly, it is hard to apply because it
is difficult to determine the convexity of the marginal
value function in πs. Therefore, Baker [2] and and
Salanie and Treich [34] have recently provided neces-
sary and sufficient conditions for the primitives of the
model, i.e., U(x1,x2,s)instead of j(x1, π), for being
able to decide upon the anticipation effect unambigu-
ously: Uhas to be separable in s, which means that U
has to be linear in some function g(s). Unfortunately,
most integrated assessment models do not belong to
this class; thus, further investigation and imposition of
more structure on the model and information setup will
be necessary to come to a satisfactory answer.
The integrated assessment literature has therefore
focused on explicitly calculating optimal short-term
decisions under learning in more complex numerical
models. A few studies have investigated the effect of
learning under a climate target O’Neill et al. [28],
Bosetti et al. [5], Johansson et al. [14], and parts of
Webster et al. [40]. The latter, e.g., find that antic-
ipation of learning about climate sensitivity leads to
significantly stronger short-term emission reductions
under a strict targets. However, Schmidt et al. [36]
argue that this effect results from a disputable interpre-
tation of climate targets as targets that have to be met
with certainty. Investigations of the anticipation effect
in cost-benefit analysis include Peck and Teisberg [31],
Yohe and Wallace [42], Kelly and Kolstad [17], Leach
[22], and parts of Webster et al. [40]. See Lange and
Treich [21] for a review. These studies have shown that
learning has generally a small effect on optimal short-
term decisions, whereas the question of the welfare gain
due to anticipatory changes in pre-learning decisions
was not addressed.
Here, we confirm this result in the integrated assess-
ment model “model of investment and technological
development” (MIND) for two key uncertainties of the
climate problem, namely climate sensitivity and climate
damages. We find considerable values of information
but insignificant gains from anticipating learning. We
then focus on the question whether the anticipation of
learning about a tipping point-like irreversible thresh-
old damage is important. This was already done with a
different model and somewhat different focus by Keller
et al. [16]. We advance on this analysis by investigating
the welfare gain from anticipation, by using a different
integrated assessment model, and by performing addi-
tional sensitivity analysis. We find that the anticipation
of learning about threshold damages can lead to sig-
nificant welfare gains if learning takes place in a specific
“anticipation window,” which depends on the threshold
under consideration and the flexibility of the decision
maker to reduce emissions. Thereby, the largest welfare
gain due to anticipation does in general not result from
the largest anticipatory change of near-term emissions.
The paper is structured as follows: Section 2shortly
introduces the problem formulation, the terminology of
the expected value of anticipation, and the integrated
assessment model MIND. The results from learning
about climate sensitivity and smooth climate damages
are presented in Section 3.1. Section 3.2 focuses on
learning about irreversible, tipping point-like threshold
damages and includes the main results. Section 4con-
cludes with potential implications for climate policy. A
table of the nomenclature we will use is shown on the
right.
Nomenclature
BAU Business as usual
BOCP Benefit of climate policy
(C)BGE (Certainty) and balanced growth equivalents
CEVOI Conditional expected value of information
DM Decision maker
EVOA Expected value of anticipation
EVOI Expected value of information
(E)VPI (Expected) value of perfect information
MIND Model of investment and technological
development
RnD Research and development
72 Anticipating Climate Threshold Damages
Anticipating Climate Threshold Damages
2 Model and Methodology
2.1 Problem Formulation
We introduce learning, i.e., the change of information
available to the DM over time, in its simplest possible
form. The overall time horizon is split into a first period
before and a second period after a one-time updating of
information at learning point tlp. A strategy consists of
first period decisions (investments) x1=I(t), t0<t≤
tlp and second period decisions x2(y)=I(y)(t), tlp <
t≤T, which are conditional on messages y. The prob-
lem of the decision maker is now to maximize the
outcomes of the chosen strategy in terms of an inter-
temporally separable, aggregated expected utility.
The learning between the two periods can formally
be described by the concept of an information struc-
ture. The terminology follows Marschak and Miyasawa
[23] as presented in Jones and Ostroy [15]. We denote
states of the world and messages, or observations, by
s∈Sand y∈Y, respectively. Let πand qbe prior
probability vectors on Sand Y, respectively. Let πy
be a posterior probability vectors on Safter receipt of
message yand 5the matrix whose columns are the
πy. If the learning is consistent, which is ensured by
applying Bayes’ rule to update the prior probabilities,
it holds
πs=X
y
qyπy
s.(1)
Therefore, we will shortly denote the information struc-
ture by the tuple (5, q).
Using this notation, the recursive optimization prob-
lem reads:
max
x1X
s
πsu1,s(x1)+X
y
qymax
x2X
s
πy
su2,s(x1,x2,y)
=: EU(5, q), (2)
where u1,s(·)and u2,s(·)are the vectors of utility in
period 1and 2, respectively, with elements equal to
utility for a specific state of the world s. We solve
the problem numerically in the equivalent, but more
convenient, sequential form
max
xy
1,xy
2PyqyPsπy
s¡u1,s¡xy
1¢+u2,s¡xy
1,xy
2¢¢,
s.t.xj
1=xk
1,∀j6= k.(3)
Here, the constraint ensures that only second period
decisions can be tailored to the messages.
2.2 Terminology
We will distinguish between a “no learning” case, rep-
resented by an information structure with posterior
distributions equal to the prior distribution, and a
“learning” case in which the probability distribution
narrows between the two time periods due to the
received messages y. We will further distinguish two
learning cases: Either the DM anticipates future learn-
ing before it happens or not. Learning has both an
effect on optimal pre- and post-learning decisions, i.e.,
x1and x2, both of which have a positive effect on
welfare. The pre-learning adjustments are due to the
anticipation of future learning, whereas post-learning
adjustments can be made even if the learning is not
anticipated. This is shown schematically in Fig. 1.
We now introduce several concepts that separate the
effect of anticipated and non-anticipated learning. The
benefits from adjusting post-learning decisions to new
information for given first period decisions can be mea-
sured by the conditional expected value of information
(CEVOI). Formally
CEVOI(x1, 5, q)≡V(x1;5, q)−V(x1;π, 1), (4)
where V(x1;5, q)is the so-called value function,
namely the optimal second period utility for given
first period decisions and information structure
(5, q),V(x1;5, q)=Pyqymaxx2Psπy
su2,s(x1,x2).
V(x1;π, 1)is the value function without learning.
Fig. 1 Schematic plot of optimal emissions over time under
different information scenarios and for two learning paths
Model and Methodology 73
A. Lorenz et al.
The anticipatory adjustment of first period decisions
to future learning can be measured by the expected
value of anticipation (EVOA):
EVOA(5, q)≡X
s
πsu1,s(x∗
1)+V(x∗
1, 5, q)
−ÃX
s
πsu1,s(x′
1)+V(x′
1, 5, q)!, (5)
where x∗
1and x′
1denote the optimal first period deci-
sions with and without learning, respectively.
The overall wealth benefits from future learning
can be measured by the expected value of information
(EVOI). It is defined as the difference between ex-
pected utility with and without learning
EVOI(5, q)≡EU(5, q)−EU(π, 1)
=X
s
πsu1,s(x∗
1)+V(x∗
1, 5, q)
−ÃX
s
πsu1,s(x′
1)+V(x′
1, π, 1))!
=CEVOI(x′
1, 5, q)+EVOA(5, q), (6)
The EVOI could be used to decide about the imple-
mentation of a certain observation campaign or scien-
tific program providing certain information. The EVOI
would therefore be compared to the implementation
costs. The relevance of anticipatory changes in short-
term policy as part of the overall benefits from infor-
mation can be measured by the ratio EVOA/EVOI.
CEVOI, EVOA, and EVOI are defined as differ-
ences in expected utility, which are not invariant with
respect to linear affine transformations of utility. To
obtain this invariance, we use the concept of balanced
growth equivalents (BGE) due to Mirrlees and Stern
[24]. The BGE is defined as an initial level of consump-
tion γsuch that the balanced growth path c(t)=γ·
exp(αt)yields the same expected utility as the original
consumption path. Since we consider uncertainty and
learning, we use the certainty equivalent BGE (CBGE)
defined by Anthoff and Tol [1], where the certainty
equivalent is with respect to the uncertain state of the
world and the learning paths. For constant relative risk
aversion η, the relative change in CBGE is:
1CBGE =γ (EU)−γ (EU′)
γ (EU′)
=
£EU
EU′¤1
1−η−1η6= 1
exp ³EU−EU′
PT
t=0Lt(1+ρ)−t´−1η=1,(7)
where EU and EU′are expected utility with and with-
out learning, respectively, and the other denominations
are population Ltand a discount factor due to impa-
tience (1+ρ)−t. It can easily be shown that relative
changes in CBGE are independent of the growth rate
α(Anthoff and Tol [1]). Intuitively, a 1% reduction
inCBGE, for instance, can be interpreted as a perma-
nent loss of consumption of 1%.
2.3 The Integrated Assessment Model MIND
We use the MIND (Edenhofer et al. [7]).2We use
the version from Held et al. [13] and add anticipated
learning about uncertainty (see Section 2.1), but we
leave out carbon capturing and sequestration (CCS) for
tractability. Edenhofer et al. [7] and Held et al. [13]
perform cost-effectiveness analysis for a given climate
target. We have shown elsewhere (Schmidt et al. [36])
that cost-effectiveness leads to conceptual problems if
learning about uncertainty is taken into account. There-
fore, we perform cost-benefit analysis.
MIND is a model in the tradition of the Ramsey
growth model and similar to the well-known DICE
model (Nordhaus [26]). The version we use differs from
the classical Ramsey model in three major respects:
Firstly, the production sector depends explicitly on en-
ergy as production factor that is provided by a crudely
resolved energy sector. The energy sector contains (a)
fossil fuel extraction, (b) secondary energy production
from fossil fuels, and (c) renewable energy production.
The macroeconomic constant-elasticity-of-substitution
(CES) production function depends on labor, capital,
and energy as input factors. Secondly, technological
change is modeled endogenously in two ways. The so-
cial planner can invest into research and development
activities to enhance labor and energy efficiency. Ad-
ditionally, productivity of renewable and fossil energy
producing capital increases with cumulative installed
capacities (learning by doing). Thirdly, a simple en-
ergy balance model is used to translate global CO2
and SO2emissions3to radiative forcing and changes
in global mean temperature (Petschel-Held et al. [32];
Kriegler et al. [20]). SO2emissions are coupled to CO2
emissions with an exogenously declining ratio of sulfur
per unit CO2representing desulfurization. Radiative
2Modified model versions feature an endogenous carbon captur-
ing and sequestration (CCS) module (Bauer [3]), a more elabo-
rate carbon cycle and atmospheric chemistry module (Edenhofer
et al. [8]), and parametric uncertainty (Held et al. [13]).
3The emissions are induced by (a) endogenous consumption
of fossil fuels and (b) exogenous CO2emissions from land-use
change (SRES A1T).
74 Anticipating Climate Threshold Damages
Anticipating Climate Threshold Damages
forcing from other greenhouse gases and aerosols is
included as exogenous scenario (see Held et al. [13]).
We assume welfare to be an inter-temporally separa-
ble isoelastic utility function of per capita consumption
with a constant relative risk aversion of η=2. It takes
the form:
U(c(I,s)) =
te
X
t0
L(t)·1
1−η
×"µ[c(I,s)](t)
L(t)¶1−η
−1#e−ρtdt,(8)
where I=(IK,IR&D,IFossil,IRenewables)is the vector of
investment flows in the different sectors over time, sis
the unknown state of the world, ρis the pure rate of
social time preference taken to be 0.01/year, and L(t)is
an exogenously given population scenario. Investments
are related to the global consumption [c(I,s)](t)via the
budget constraint:
Ynet(t,s)= [c(I,s)](t)+X
n
In(t,s) , c(I,s)≥0,(9)
with the gross world product (GWP) Ynet net of cli-
mate related damages. Ynet is related to gross GWP
over Ynet =Ygross ·DF, where DF is a multiplicative
damage factor defined by the damage function (see
Roughgarden and Schneider [33]):
DF(T)=1
1+a·Tb.(10)
For some of the results, we will limit the flexibility
of the decision maker in MIND in one of two ways.
First, we introduce a maximum flexibility in emissions
changes 1Emax/year as the maximum possible relative
emissions change in one year both upward and down-
ward. This inflexibility is assumed to originate from
processes that are not included in the model MIND,
such as political or societal constraints. Second, we
limit the use of different mitigation options in MIND
and particularly renewable energy and investments in
energy efficiency. This increases the costs for emission
reductions and thus lowers the flexibility in emission
reductions. The influence of these two different kinds
of inflexibility on the value of learning and anticipation
is investigated.
2.4 Implementation of Learning About Climate
Sensitivity and Damage Amplitude
We now consider a perfect learning case, i.e., messages
yreveal the true state of the world. We focus on
uncertainty about climate sensitivity CS, defined as
equilibrium temperature change for a doubling of at-
mospheric CO2concentration from pre-industrial level,
and on uncertainty about the climate damage parame-
ters aand bin Eq. 10. We consider learning about
climate sensitivity and damages separately as well as the
combined effect of learning about both uncertainties
simultaneously. The time of arrival of new information
is varied between early (tlp =2030), intermediate (tlp =
2050), and late learning (tlp =2070). The uncertainties
are described by probability distribution functions that
are given explicitly in Appendixes 1and 2. For the
numerical implementation, we draw samples of size n
from the distributions according to a scheme related to
descriptive sampling (see Saliby [35]). The uncertainty
space is divided into nhypercubes. Each hypercube
icarries a chosen probability weight wiand is repre-
sented by the expected value of the parameters on this
hypercube. Thereby we do not choose an equiprobable
spacing but choose a few central sampling points that
carry the main part of probability and complement
them by some points at the outer margin of probability.
This technique of explicitly sampling the 1st and 99th
percentile allows us to account for the low-frequency
high-impact events in the tails of the distributions. For
the implementation of learning about single uncertain-
ties, we choose a sampling size n=5. For the simulta-
neous learning about both uncertainties, each dimen-
sion is sampled with four equiprobable points which
are combined to only four learning paths according to
the descriptive sampling scheme (instead of 16 learning
paths with a fully factorial design).
2.5 Implementation of Learning About
Threshold Damages
Keller et al. [16] have found significant changes in
emissions due to anticipation of learning if a highly non-
linear irreversible threshold is included in the analysis.
More specifically, they considered a possible shutdown
of the North Atlantic thermohaline (THC) circulation
(Broecker [6]). We add to this study by focusing on the
welfare benefits from anticipation, i.e., the EVOA, by
using MIND as a model featuring endogenous technical
change, and by performing a sensitivity analysis with
respect to learning time, flexibility in emissions reduc-
tions, threshold temperature, and damages.
Hence, in addition to the damage function in Eq. 10
by Nordhaus [25], we consider explicit tipping point-
like threshold damages. Similar to Keller et al. [16],
who considered a threshold in atmospheric CO2con-
centration depending on climate sensitivity, we as-
sume that the temperature T0, at which the threshold
occurs, is known, but the resulting damages DFthresh
are uncertain. The damages are added to Nordhaus’s
Model and Methodology 75
A. Lorenz et al.
damage factor DF leading to output net of damages,
Ynet =Ygross ·DFthresh. We assume that the threshold is
irreversible, i.e., if it has been crossed, the threshold
damages continue to be incurred even if temperature
returns to values below the threshold. This can be
expressed formally as
DFthresh(t,In,t,s)=1
1+a·Tb+Dthresh(s)·ξ(t,In,t,s),
(11)
where Dthresh(s)is the amount of damages in the uncer-
tain state of the wold sand ξ(t,xt,s)indicates whether
the threshold was crossed before time tin the state sfor
given decisions up to time t,In,t.ξis defined as
ξ(t,In,t,s)=1−
t
Y
t′=t0
[1−2(T(t′,In,t,s)−T0)], (12)
and equals one if the threshold was crossed in the past
and zero if not. Here, 2is Heavyside’s step function.
For simplicity, we only consider perfect learning
about the threshold-damage amplitude Dthresh, which
can only take two values, Dthresh = [Dx,0]. Damage
Dthresh =Dxoccurs with probability pand damage
Dthresh =0with 1−p, such that the expected dam-
age EDthresh =1.5% of net GDP is in accordance with
empirical estimates for the expected impact of a THC
shutdown by Tol [37]. We calculate the EVOI and
the EVOA for different threshold temperatures T0,
threshold damages Dx(where pis adjusted such that
expected net damages are unchanged, whereas the
expected gross damage factor DFthresh changes), and
learning points tlp.
3 Results
3.1 Learning About Climate Sensitivity
and Damage Amplitude
The welfare benefits from learning about climate sen-
sitivity and standard climate damages, measured by the
EVOI, are listed in Table 1. Learning about damages
leads to an increase in CBGE of about 0.1% for early
learning. When asking for the importance of including
learning into the analysis of optimal climate policy, this
value might best be compared to the overall benefit
of climate policy (BOCP). The BOCP is the welfare
difference between BAU and optimal policy measured
in CBGE. It amounts to 0.12% CBGE in case of un-
certain climate sensitivity and 0.14% CBGE in case
of uncertain damages. Including learning about dam-
ages increases the BOCP by (21.8–64.5)%for late and
early learning, but learning about climate sensitivity by
only 1.75-4.95%. Hence, learning about damages can
substantially increase the benefits from climate policy.
Learning about climate sensitivity is less valuable by
roughly an order of magnitude.
Simultaneous learning about both uncertainties
strongly increases the EVOI, e.g., up to 0.45% for early
learning. That relates to an increase of the BOCP by
up to 347%. Hence, learning multiplies the benefits
from climate policy if both parameters are uncertain.
States of the world characterized by extreme values in
both parameters imply very high damages. These can
be mitigated after learning without having to spend the
associated costs in all states of the world.
Also shown in Table 1is the proportion of the EVOI
that is obtained by anticipatory changes in pre-learning
decision, i.e., the ratio EVOA/EVOI (see Section 2.2).
We see that it is generally small (<2%). The welfare
benefits from anticipating future learning about dam-
ages or climate sensitivity is negligible.
The result that learning implies only very small anti-
cipatory changes in optimal pre-learning decisions
in cost-benefit analysis was already found in other
integrated assessment models (see, e.g., Ulph and Ulph
[38]; Nordhaus and Popp [27]; Webster [39]; O’Neill
and Melnikov [29]; Webster [40]). Why could we have
expected an effect in the model MIND? As shortly
discussed in Section 1, optimal first period decisions
change, if the derivative of the second period, ex post
value function V2(x1, π y
s)=maxx2Psπy
su2,s(x1,x2)
with respect to the first-period decision x1is non-linear
in the vector of posterior probabilities πy
s(Epstein [9]),
α∂/∂x1V2(x1, πi
s)+(1−α)∂/∂x1V2(x1, π j
s)6= ∂/∂x1V2
Table 1 The EVOI measured in %CBGE of the no-learning case and the EVOA/EVOI ratio for different scenarios: perfect learning
about CS and damages separately as well as jointly and for early, intermediate, and late learning
CS Damages CS and Damages
tlp EVOI (%) EVOA/EVOI (%) EVOI (%) EVOA/EVOI (%) EVOI (%) EVOA/EVOI (%)
2030 0.006 0.004 0.09 0.29 0.45 0.022
2050 0.004 0.15 0.06 0.53 0.33 0.112
2070 0.002 0.20 0.03 1.77 0.22 0.287
CS climate sensitivity
76 Anticipating Climate Threshold Damages
Anticipating Climate Threshold Damages
1.5 2 2.5 3 3.5
0
0.05
0.1
0.15
0.2
0.25
0.3
T0
∆ CBGE [%]
EVPI(Dx,T0)
Dx=20%
Dx=10%
Dx=5%
Tc
Te
T1
T2
Fig. 2 The EVPI for different values of Dxand T0. The EVPI
is measured in %CBGE of the no-learning case. Tcdenotes the
temperature the decision maker is already committed to cross.
For T0>T1(Dx), avoiding the threshold is optimal for perfect
information that Dthresh =Dx. For T0>T2(Dx), avoiding the
threshold is optimal even in the no-learning case. Teis never
reached for any information setup
(x1, απi
s+(1−α)π j
s). Obviously, a necessary precon-
dition for this is that the optimal second period utility
V2actually depends on the first period decision x1
and the derivative is non-zero. MIND includes several
such cross-period interactions that are not present in
other integrated assessment models. More specifically,
it features multiple capital stocks, a knowledge stock,
and learning by doing in technologies. However, the
numerical results above clearly show that the effect of
anticipation is negligible in this setting.
3.2 Learning About Threshold Damages
3.2.1 The Expected Value of Perfect Information
We start by considering two extreme cases: Either the
decision maker has perfect information, i.e., learning
occurs before any decision is to be taken, or she does
not learn at all. Figure 2shows the associated expected
value of perfect information (EVPI)4for different val-
ues of the threshold specific damages Dxoccurring
4The EVPI is defined as the difference in welfare between the
case of perfect information and the no-learning case. It is mea-
sured in %CBGE of the no-learning case.
with mean-adjusted probability p(Dx)(see Section 2.3)
and different threshold temperatures T0. Also shown
is the critical temperature T2(Dx)that divides the pa-
rameter space into two regimes: (A) For all threshold
temperatures T0<T2, it is optimal without learning
to cross the threshold, and (B) for all T0≧T2, it is
optimal without learning to stay below the threshold. A
further separation occurs within regime A: For thresh-
old temperatures T0<T1(Dx), it is optimal to cross
the threshold even in case of perfect information as
the mitigation costs more than outweigh the threshold
damages.
The EVPI is zero for high values of T0>Tebecause
information about a threshold that is not crossed for
the optimal policy without threshold damages is useless.
However, the same is not true for very low values
of T0<Tc, when the decision maker is committed to
cross the threshold. The information about the received
threshold damages is still valuable as it is used to adjust
the savings rate. At a certain T0, the EVPI reaches a
maximum. For lower T0, the emissions reductions that
are necessary to avoid the threshold are too costly.
For higher T0, the avoided threshold damages decrease
because higher T0are only reached later in time, and
thus, the corresponding damages are discounted.
Since the EVOA is bounded from above by the
EVOI and the EVOI is bounded from above by the
EVPI, the potential benefits from anticipation are
larger in regime A than in regime B. We also note
from Fig. 2that the EVPI is increasing in Dx, although
expected damages are held constant by reducing the
probability of the threshold when increasing Dx. This
is due to the risk aversion of the decision maker, which
makes her prefer a low Dxwith a higher probability to
a higher Dxwith a low probability.
3.2.2 The Value of Anticipation
Now we investigate the dependence of the EVOI and
the EVOA on the time of learning tlp. Figure 3shows
the EVOA and EVOI for learning points between the
year tlp =2010 and tlp =2080 in steps of 5years. It also
shows the cumulative anticipatory changes in emissions
(1E) before learning relative to the no-learning case.
The EVOI decreases from the EVPI obtained in 2010
to zero for tlp =2200. The latter is essentially the no-
learning case. The EVOA has to be zero for tlp =2010
because there are simply no pre-learning decisions to
be made. It is also zero for tlp =2200 because the
discounted utility after this time is too small to justify
anticipation.
Within regime A, where the threshold is crossed
in the case of no learning, three different regimes of
Results 77
A. Lorenz et al.
2010 2020 2030 2040 2050 2060 2070 2080
0
0.07
0.14
0.21
0.28
I II III
Learningpoint tlp
∆CBGE [%]
−10
−5
−2
0
2
5
10
∆E [%]
∆E
EVOI
EVOA
Fig. 3 Expected value of information (EVOI), expected value
of anticipation (EVOA), and relative changes in cumulative pre-
learning emissions in anticipation of learning (1E) are shown
depending on the time of learning tlp. The dashed lines mark
three distinct regimes of anticipation (I–III). The two black points
in 2040 and 2045 mark local optima that are only slightly worse
compared to the shown “optimal” path
anticipative behavior can be identified. They are indi-
cated in Fig. 3: (I) For early learning, it is possible to
avoid the threshold easily by adjusting the post-learning
decisions. Doing so in case Dthresh =Dxis learned leads
to a substantial EVOI without the need for down-
ward anticipation. Not having to anticipate downward
benefits the case where Dthresh =0is learned. There is
even some upward anticipation to come closer toward
the solution that would be optimal for perfect informa-
tion about Dthresh =0.
(II) For increasing tlp, there is less time between
learning and crossing the threshold (without adjust-
ments). Since mitigation costs are convex, this in-
creases the costs of avoiding the threshold in the
“bad case” (Dthresh =Dx)by post-learning adjustments
alone. Therefore, in regime II, the DM lowers pre-
learning emissions compared to the no-learning case.
The benefits of doing so experienced in the “bad”
case outweigh its costs in the “good” case. For further
increasing tlp, avoiding the threshold with post-learn
adjustments alone becomes physically infeasible. The
motive for anticipation is then to keep the option open
to avoid the threshold in the bad case in the first place.
The associated costs increase with tlp.
(III) At the border between regimes II and III, these
costs reach a point, at which the decision maker is
indifferent between keeping the option open and not
keeping the option open, i.e., crossing the threshold
also in the “bad” case. This leads to local optima with
identical expected utility. Two of them are indicated by
black dots in the upper panel of Fig. 3. Although the
threshold is crossed for both learning paths in regime
III, learning about the damages has a value, as wit-
nessed by the significant EVOI for tlp >2040 in Fig. 3.
The reason is that learning still enables the DM to
adjust her savings rate to damages and thus to perform
consumption smoothing. More specifically, savings are
decreased after crossing the threshold if the threshold is
“bad”. Finally, regime III shows a positive anticipation
effect in emissions. However, the benefits from this
anticipation are negligible.
In conclusion, downward anticipation for being able
to avoid the threshold at all, or at low costs, in the bad
case is the dominant effect. Anticipation of learning
about threshold damages leads to a significant welfare
gain only if the learning occurs within a specific time
window t1<tlp <t2. This “anticipation window” is nar-
row, and it spans at most one decade. Due to the 5-
year time steps in MIND, it is not possible to determine
its exact extent. The fact that the anticipation window
is narrow is explained by the relatively high flexibility
of the model in increasing or decreasing emissions. We
will discuss this further in Section 3.2.4.
3.2.3 Availability of Renewable Energy
We investigate the origin of the anticipation window
by focusing on the anticipation effect in the decision
variables. These are investments in renewable energy,
fossil energy, RnD aimed at improving labor or energy
efficiency, and investments in the aggregate macro-
economic capital stock. The cumulative anticipatory
changes of the decision variables relative to the case
without learning are shown in the left panel of Fig. 4.
The right panel shows the cumulative post-learning
adjustments up to 2200 separately for Dthresh =0and
Dthresh =Dx. The resulting EVOI and EVOA are
shown in Fig. 5.
The main option for reducing emissions used by
the model is substituting fossil energy by renewable
energy. Renewables are used to avoid the threshold
after learning in regime I and for anticipatory emission
reductions in regime II. The latter can be seen by
comparing the “all options” case in Fig. 5with the case,
where the usage of renewables is restricted to be lower
than in business-as-usual (“no renewables”), which is
not zero but very little. The EVOA vanishes in the lat-
ter case. Apparently, anticipatory emissions reductions
via reductions in energy demand or increased energy
efficiency would be too costly. Hence, the existence of
the anticipation window rests on the availability of a
sufficiently cheap and flexible, carbon free, substitute
78 Anticipating Climate Threshold Damages
Anticipating Climate Threshold Damages
2010 2020 2030 2040 2050 2060 2070 2080
−.2%
−.1%
0%
+.1%
+.2%
−10%
−5%
0%
+5%
+10%
−1%
−.5%
0%
+.5%
+1%
−10%
−5%
0%
+5%
+10%
−20%
−10%
0%
+10%
+20%
CO2 Emissions
Investments in agg. Capital
R&D Energy Efficiency
Renewable Energy
R&D Labor Efficiency
pre- learn changes
time
all options
no renewables
2010 2020 2030 2040 2050 2060 2070 2080
−2%
−1%
0%
+1%
+2%
−50%
−25%
0%
+25%
+50%
−20%
−10%
0%
+10%
+20%
−10%
−5%
0%
+5%
+10%
−20%
−10%
0%
+10%
+20%
CO2 Emissions
Investments in agg. Capital
R&D Energy Efficiency
Renewable Energy
R&D Labor Efficiency
post- learn changes
time
all options
no renewables
"good" learning case
"bad" learning case
Fig. 4 The anticipation effect (left) and post-learning decisions (right) both in cumulative decision variables and with and without the
availability of renewable energy
for fossil energy. However, too much flexibility would
again diminish the EVOA because adjustments could
2010 2020 2030 2040 2050 2060 2070 2080
0
0.05
0.1
0.15
0.2
0.25
∆E
EVOI
EVOA
Learningpoint tlp
∆CBGE [%]
−10
−5
−2
0
2
5
10
∆E [%]
all options
no renew
only fossils
Fig. 5 Expected value of information (EVOI), expected value
of anticipation (EVOA), and relative changes in cumulative pre-
learning emissions in anticipation of learning (1E) are shown
depending on the time of learning tlp. Shown are three scenarios
differing in the availability of mitigation options. In the “no
renew” case, the usage of renewable energy is restricted to be
lower than in the business-as-usual case where renewables are
only used in the twenty-second century to counter the scarcity
of fossil energy. In the “only-fossils” case, other options, like
investments into “R&D” in energy and labor efficiency, are also
not available
be made entirely after learning. This suggest that an
intermediate flexibility generates anticipation.
3.2.4 Sensitivity of the “Anticipation Window”
Now we investigate the sensitivity of the anticipation
window with respect to T0,Dxand the flexibility of
the decision maker to change emissions over time. The
results are shown in Fig. 6a–c.
Dependence on Threshold Position T0With rising
threshold specific temperature T0, the maximum of the
EVOI decreases because the threshold is crossed later
in time and less mitigation efforts are needed to stay
below the threshold. For the same reason, the antici-
pation window is pushed toward later learning points.
As already discussed above, for T1<T0<T2, which is
the case for T0∈ [2,2.3]°C, anticipation occurs to stay
below the threshold in the high-damage case. Now we
compare this result with one for T0>T2, where the
threshold is avoided even in the no-learning case. In the
latter case, there is no incentive for downward antici-
pation, but the before mentioned incentive for upward
anticipation in order to optimize the good learning case
occurs. This leads to an EVOA that slowly increases
with tlp up to a maximum beyond which a higher pre-
learning deviation from the optimal no-learning path
Results 79
A. Lorenz et al.
2010 2020 2030 2040 2050 2060 2070 2080
0
0.05
0.1
0.15
0.2
0.25
a
∆E
EVOI
EVOA
Learningpoint tlp
∆CBGE [%]
−10
−5
−2
0
2
5
10
∆E [%]
T0=2°C
T0=2.3°C
T0=2.75°C
2010 2020 2030 2040 2050 2060 2070 2080
0
0.05
0.1
0.15
0.2
0.25
b
∆E
EVOI
EVOA
Learningpoint tlp
∆CBGE [%]
−10
−5
−2
0
2
5
10
∆E [%]
Dx=20%
Dx=10%
Dx=5%
2010 2020 2030 2040 2050 2060 2070 2080
0
0.05
0.1
0.15
0.2
0.25
c
∆E
EVOI
EVOA
Learningpoint tlp
∆CBGE [%]
- 10
- 5
- 2
0
2
5
10
∆E [%]
∆E≤
≤
50%/yr
∆E 5%/yr
∆E≤2%/yr
Fig. 6 Sensitivity of the anticipation effect: EVOI, EVOA, and
the anticipatory relative changes in cumulative pre-learning CO2
emissions are shown as a function learning time tlp.aThe depen-
dence on different threshold temperatures T0,bthe dependence
on the damage amplitude Dx,cthe dependence on different ex-
ogenous inflexibilities 1Emax of the decision maker in reducing
or increasing emissions
leads to too high costs in the bad case. Although the
absolute values of the EVOA and EVOI are smaller
for high T0, anticipation remains important in relative
terms (EVOA/EVOI ratio).
Dependence on Mean Threshold Damages DxFig. 6
shows that both the EVOI and the EVOA are in-
creasing in the threshold damage Dx. The anticipation
window is slightly shifted toward earlier learning points
for small threshold damages. This is due to the fact that
the equilibrium between the mitigation costs to keep
the threshold open in the bad case and the threshold
damages is shifted toward lower values by decreasing
Dx. The relative importance of anticipation remains
large.
Dependence on an Artificial Emissions Flexibility
1Emax The limited maximum emission flexibility
1Emax is assumed to originate in processes that are not
represented in the model, such as political and socioe-
conomic inertia. The first effect of limited flexibility is
to move the curve toward lower values of tlp. Since the
ability to react to new information is now limited, antic-
ipation becomes necessary for earlier learning times. In
the limit of very low flexibility (1E<1%/year) (not
shown), the EVOA vanishes and even the EVOI for
perfect learning in 2010 decreases as the decision maker
cannot avoid crossing the threshold. In this case of low
flexibility, the information can only be used to postpone
the crossing of the threshold to later times by reducing
emissions, but not to avoid the threshold.
80 Anticipating Climate Threshold Damages
Anticipating Climate Threshold Damages
4 Conclusions
We first introduced and clarified some terminology that
can be used to assess the importance of anticipation
of future learning. In particular, we introduced the
concept of an expected value of anticipation.
We then investigated future learning about two key
parameters of the climate problem, climate sensitivity
and climate damages. We used the integrated assess-
ment model MIND to calculate the welfare benefits
from learning and the implications of anticipation of
future learning for optimal near-term climate policy
in terms of changes in the cumulative pre-learning
emissions. The welfare benefits from learning were
significant but benefits due to anticipation of this learn-
ing were not. This confirmed previous results in the
literature.
We then investigated anticipated learning about un-
certain threshold damages. The anticipation of learning
leads to both higher and lower pre-learning emissions
depending on the severity and position of the threshold.
The welfare gains from this anticipation were in general
considerably higher for downward anticipation (lower
pre-learning emissions) than for upward anticipation
(higher pre-learning emissions).
However, anticipation was only important if learn-
ing occurred within a specific, narrow time window,
which depended on the flexibility of the decision maker
to reduce and increase emissions. Inside this window,
the welfare benefits due to anticipation can contribute
almost the entire value of information (≈95%). The
strongest anticipation effect on pre-learning emissions
did in general not lead to the strongest welfare gain.
There was even one point in time such that learning
at this point leads to two equally preferred solutions
whereof one avoids the threshold and the other one
does not.
The existence of a significant anticipation effect
rested on the assumption of highly nonlinear dam-
ages and the availability of a flexible, scalable, and
relatively cheap substitute for fossil energy. However,
the anticipation effect was increased if the flexibility
of adjusting emissions was reduced by other means
than the availability of renewable energy. We showed
this by introducing exogenous constraints on emissions
changes motivated as political constraints or processes
not represented in the model.
The analysis we have performed is only semi-
quantitative and conclusions come with some caveats.
The known limitations of all integrated assessment
models with their highly simplified representation of
the socioeconomic and physical processes apply. The
representation of the threshold, the resulting dam-
ages, flexibility, uncertainty, and the learning process
(as one-time perfect learning) could certainly be im-
proved. More complex learning processes could be
studied by changing toward a dynamic programming
framework. Studying multiple and partly reversible
thresholds occurring at uncertain temperatures could
lead to more complex pattern of anticipation. All this,
of course, would make the numerical solution more
difficult.
Beside these limitations, a clear implication for real
world climate policy can be drawn from our study:
Although we are actually uncertain about both the po-
sition of potential thresholds as well as about their eco-
nomic impacts, anticipating uncertain thresholds can
be an important argument for lower emissions but not
higher emissions.
Acknowledgements We are grateful for the helpful comments
of two anonymous reviewers. A.L. acknowledges support by the
German National Science Foundation and M.G.W.S. acknowl-
edges funding by the BMBF project PROGRESS (03IS2191B).
Appendix 1: Climate Sensitivity
The climate module of MIND calculates the tempera-
ture response to anthropogenic forcing induced by CO2
and SO2(which are coupled to CO2emissions) and
exogenous forcing from other greenhouse gases:
˙
T=µ¡ln ¡C/Cpi¢+fSO2+fOGHG¢−αT, (13)
where Cis current and Cpi pre-industrial atmospheric
CO2concentration, Tdenotes global mean tempera-
ture anomaly, and µthe radiative forcing for a doubling
of pre-industrial atmospheric CO2content divided by
the heat capacity of the ocean (dominating the inertia
of the climate system) and ln 2. The parameter αis
the response rate of the climate to changes in radiative
forcing. It is linked to climate sensitivity CS via:
CS =µ
αln 2 . (14)
Actually, both µand αin the temperature equa-
tion are uncertain and correlated via the global mean
temperature record of the last two centuries (e.g.,
see Forest et al. [10]; Frame et al. [11]). For simplic-
ity, we assume a perfect correlation and 1
µ=1
¯µ−10 ·
exp(−0.5CS). The acceptability of this assumption can
be assessed in Fig. 7.
The temperature response is now fully determined
by CS. As prior information about CS we take a
log-normal distribution from Wigley and Raper [41]:
¯π(CS) =LN (0.973,0.4748).
Conclusions 81
A. Lorenz et al.
0 2 4 6 8
10−2
10−1
100
101
CS [°C]
α [1/yr]
Sample from Frame et al.
Fit for µ(CS)
Fig. 7 Correlation of αand CS from the temperature record of
the last two centuries (see Frame et al. [11]) [green dots]. By
assuming a strict relationship between 1
µand CS as 1
µ=1
¯µ−10 ·
exp(−0.5CS)the correlation narrows to the [red curve]
Appendix 2: Climate Damages
The uncertain parameters aand bin the exponential
damage function DF(T)=1
1+a Tbare determined from
an expert-based assessment done by Roughgarden and
Schneider [33]. They provide a joint probability distri-
bution for both parameters. We use their methodology
to derive the damage functions that are representative
for the quantiles described by the sampling probability
0 1 2 3 4 5 6
−10
0
10
20
30
40
50
60
70
80
90
100
∆Tpreind [°C]
Damages [% of net output]
q < 1%
63% < q < 99%
q > 99%
1% < q < 21%
21% < q < 63%
Fig. 8 Samples taken according to a descriptive sampling scheme
from a joint probability distribution of the damage function pa-
rameters aand bfrom Roughgarden and Schneider [33]. Shown
are the damage functions representative for the quantiles qwith
probability weights ωi= [1,20,60,18,1]% that have been used
within the experiments
weights ωi. Figure 8shows the damage functions that
represent the quantiles chosen for our experimental
setup: ωi= [1,20,60,18,1]%.
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84 Anticipating Climate Threshold Damages
85
Chapter 5
Synthesis and Outlook
The focus of this thesis is the assessment of the role of accounting for uncertainty and learning
in the integrated assessment of optimal global climate change mitigation policy. Within the inte-
grated assessment of climate change, the climate problem is framed as a decision problem for a
representative agent (decision maker; DM) who aims at maximizing global social welfare by opti-
mally chosing a level of greenhouse gase mitigation that balances the economic costs of mitigation
and the climate change induced damages. In this framework the decision maker is facing substan-
tial uncertainties in all parts of the underlying cause-effect chain. The importance of accounting
for uncertainty can be assessed by comparing a situation in which the decision maker (DM) faces
multiple possible outcomes of his decisions, each with a known probability of occurrence, to the
situation in which the DM is certain to face the expected outcome. Some of the DM’s uncer-
tainty about the socio-economic and the climate system will be resolved by future observations
and scientific progress. The importance of anticipating these future learning possibilities about
uncertainty can be assessed by comparing two situations: The case of uncertainty is compared to
a situation in which the DM initially is uncertain about the outcomes, but over time receives new
information about the probabilities of the outcomes and thus can adjust his or her decision towards
the new situation. In this context, the three main research questions are: how important is uncer-
tainty for optimal climate mitigation policy? How important is the prospect of future learning for
optimal climate mitigation policy? And which decision frameworks are applicable to analyzing
optimal climate policy under uncertainty and learning? This thesis interprets the questions for the
importance of accounting for uncertainty and learning along the line that uncertainty and learning
are important for the analysis if their inclusion significantly changes both, the optimal emission
policy itself and the net welfare gain from adopting an optimal climate policy instead of following
the business as usual approach. As the answers to the first two questions strongly depend on the
decision framework the climate problem is stated in, the third question underlies both of the oth-
ers.
Contributions to the answers of all three questions have been made within this thesis. They are
summarized in the following. The final section concludes with a general outlook and future re-
search requirements.
86 Synthesis and Outlook
5.1 Formulating the Climate Problem under Uncertainty and
Learning
The most commonly used framework within the integrated assessment modeling of climate
change under uncertainty is the expected utility (EU) maximization. This is due to the fact, that
the von Neumann-Morgenstern axioms underlying the EU framework represent widely accepted
norms of rational decision making under uncertainty (Machina, 1987). The straight forward
implementation of the EU framework is given by the so-called cost-benefit analysis (CBA), that
incorporates the costs of mitigation as well as a monetized representation of climate induced
damages and thus the benefits of mitigation. Two of the main criticisms of CBA are the pure
inability to monetize certain non-market damages, and more broadly, the incommensurability
of different kinds of damages, e.g. the comparison of losses of biodiversity to damages to
infrastructure.
Another EU implementation, the also often used cost-efficiency analysis (CEA) refrains from
monetizing climate induced damage directly and thereby avoids the criticism. The trade-off
between mitigation and benfits of avoiding certain levels of climate change is conducted implicitly
by choosing a certain limit (or target) in a variable describing the amplitude of climate change
which then constrains the EU maximization.
In the more recent past, many studies have calculated the implications of a certain temperature
target, e.g. the 2◦C target adopted by the UN in Cancun (UNFCCC, 2010). Chapter 2 confirms
findings from the older literature in decision theory that this cost-efficiency approach runs into
axiomatic problems if uncertainty and learning are included in the analysis. As argued by Held et
al. (2009), for applying a climate target under uncertainty the information about the maximum
temperature to reach is not sufficient, but it has to be accompanied by a measure of probability
with which the target is to be kept, as it is in principle not possible to keep the “whole distribution”
of temperatures below a given threshold. This method, of constraining not temperature but rather
probability for crossing a temperature, is called chance constraint programming. Chapter 2
shows that this chance constraint programming framework can lead to normatively unappealing
consequences when future learning is included. These consequences are the possibility of
negative expected value of information and the possibility of infeasibility of the decision problem
due to learning, both are derived from an axiomatic analysis and are demonstrated within the
context of an intuitive example. The possibility of negative value of future information stems
from the fact that the framework does not include an explicit tradeoff between the risk of crossing
the temperature threshold and the associated costs. Keeping the target fixed irrespective of what
is learned is more costly in a convex, nonlinear way if “extreme” messages are received. Thus
a decision maker complying with this framework would actually pay for not having to receive
new information, which is not desirable from the common normative perspective. The potential
infeasibility of the decision problem due to uncertainty results from the fact that in some future
learning scenarios it is not only more costly, but impossible to maintain the given probabilistic
climate target. The point here is that the decision criterion is incomplete as it does not give advice
on how to proceed in this situation.
Formulating the Climate Problem under Uncertainty and Learning 87
Chapter 2 continues by proposing an alternative decision criterion that does not use a fixed proba-
bilistic climate target but allows for, and requires, a trade-off between mitigation costs and the risk
of exceeding the target. This so-called cost-risk analysis (CRA) pays tribute to the fact that some
kind of trade-off between the “bad” and “good” consequences of a decision needs to be included
for each state of the world, to make the EU maximization feasible under uncertainty and learning.
However, this trade-off does not necessarily include a complete monetization of climate damage
but can be done on a more aggregated level. It may also still include the notion of a climate target.
The information requirements for assessing the DM’s preferences are equally high in a CBA and
a CRA. The question of which formulation is more practical, in that real decision makers really
behave as if following one of the problem formulations and the factual estimation of the necessary
preference parameters is left for further studies.
There are other critiques of the standard EU framework, two of which are the inability to han-
dle so-called structural, or deep, uncertainty, and the representation of different kinds of aversion
against large differences in consumption (across states of the world, time, people, etc.) with only
one parameter. The former is a feature in observed decision-making of people being averse to not
knowing the probabilities of the outcomes they have to decide about. Several frameworks have
been proposed to handle this aspect of deep uncertainty from which the model of ambiguity due
to Klibanoff et al. (2009) seems to be the most promising one. However, due to computational
complexity, up to now, the model still awaits its application in the more complex integrated as-
sessment models of climate change. The latter problem of disentangling the parameters of relative
risk aversion from the parameter of inter-temporal substitution elasticity in consumption has been
on the agenda at least since the 1970s. The necessity to do so arises from the observation that real
world decision makers reveal different levels of aversion to both effects. Thus, a consistent disen-
tangling of both parameters could deliver an explanation of several observed paradoxes. Kreps &
Porteus (1978) first proposed such a framework. It allows the desired disentanglement but it in-
troduces an intrinsic preference of the DM for early or late resolution of uncertainty. This is again
normatively undesirable as it means that the DM would even pay money for useless information,
i.e. if she could not adjust her actions according to the new information. This deficiency has cur-
rently been overcome by a framework that simultaneously solves both problems: disentanglement
of aversion parameters, in combination with neutrality towards the timing of uncertainty resolu-
tion (see Traeger, 2009, and references therein). As with the Klibanoff model for ambiguity, the
widespread application of the Traeger model, and especially the combination of both approaches,
proposes very exciting challenges for further research.
But even if the correct formulation of the decision problem for the integrated assessment of cli-
mate change has been found, and one could argue that with the models mentioned above we are
close to getting there, the challenge remains of finding, or estimating, or choosing, the normative
parameters. The outcomes of the integrated assessment and the role of uncertainty and learning
rest upon the choice of those parameters. The philosophical debate is still open as to whether
one should measure those parameters in the markets, as done by Nordhaus (2008b), or set them
normatively as done by Stern (2007).
88 Synthesis and Outlook
5.2 Importance of Uncertainty for Global Climate Analysis
There is a widespread intuition, at least amongst scientists, that uncertainty surrounding climate
change and its potential impacts is a crucial element of the problem of climate change. Hence,
intuitively, the explicit inclusion of uncertainty about, say, climate sensitivity and climate damage
amplitude into an integrated assessment model should have a strong impact on both the optimal
global mitigation policy and the resulting net welfare benefits of acting upon climate change.
However, from the standard IAMs, which formally implement a CBA, this result is not supported
(see e.g. Nordhaus & Popp, 1997; Saphores, 2004). The standard solution of a low carbon price
that moderately increases over time, the so-called policy ramp, does not change significantly when
uncertainty is included.
Several changes to the model formulation have been proposed that lead to considerable effects
from uncertainty, such as the consideration of fat-tailed probability distributions by Weitzman
(2010), or including heterogeneous damage, as in Schmidt et al. (2011a). However, only a few
other studies have investigated the influence of including uncertainty (and perfect learning) on the
welfare gain from acting upon climate change, e.g. Pizer (1997). Chapter 3 investigates the origin
of the negligible welfare effect in a “standard” IAM and proposes several changes to the structure
of the IAM itself that would lead to considerably higher welfare effects from the inclusion of un-
certainty.
The overall net welfare benefit of acting optimally with respect to climate change under uncer-
tainty is decomposed into three components: The benefit of optimally acting on climate change
under certainty, the change of the benefit of this action due to the inclusion of uncertainty, and the
benefit of adjusting the action from the optimal action under certainty to the optimal action under
uncertainty. It is proposed to use the last term, the welfare benefit of adjusting the mitigation ac-
tion to uncertainty, relative to the overall welfare benefit of acting on climate change as a metric
for the importance of including uncertainty into the integrated assessment of climate change.
Furthermore, the IAM MIND is projected onto an a-temporal marginal cost-benefit picture of cu-
mulative emission reductions that lead to marginal mitigation costs and marginal benefits from
reducing climate damages. This picture allows linking the elementary components of the benefit
of climate policy (BCP) to the functional structure (temperature response, consumption losses due
to damages and mitigation effort, and welfare effects of the former, all depending on the level of
cumulative emission reductions) of the climate cause and effect chain within MIND.
The key finding is that the welfare gain from explicitly including uncertainty is only significant if
it leads to nonlinear shifts in the marginal functions within the cause-effect chain, e.g. a nonlinear
increase in marginal damages (marginal w.r.t. cumulative emission reductions). Such a change in
turn leads to a difference in the expected marginal net welfare benefits of mitigation under uncer-
tainty compared to certainty, the so-called marginal risk premium, that is convex increasing. In this
situation the additional benefit from adjusting the mitigation level to the situation of uncertainty
can take up a significant part of the overall net benefit of mitigation, thus uncertainty is important.
These necessary shifts in the marginal functions within the climate cause-effect chain originate
from the nonlinear damage function and the nonlinear utility function. The function of maximum
Importance of Anticipating Future Learning 89
temperature increase in cumulated mitigation efforts is only weakly nonlinear, and uncertainty
does not change the form of the marginal function, only shifts it slightly upward. This weak non-
linearity explains why uncertainty about climate sensitivity, however prominently discussed, has
only a very small influence on optimal policy decisions.
Another important finding is that for the importance of uncertainty the curvature of the marginal
functions in the climate cause-effect chain matters. Here the curvature of the maximum temper-
ature function in cumulative mitigation efforts and the curvature of the corresponding damage
function compensate each other to some extent and thereby weaken the overall impact of uncer-
tainty.
Generalising this result, it shows how the structure of one part of the cause-effect chain can in-
fluence the importance of uncertainty within another part. The effect of including uncertainty in
climate sensitivity on the choice of optimal climate policy and the resulting increase in welfare is
small. However, if changing the damage function from a quadratic to a more nonlinear shape, the
importance of CS uncertainty increases.
The utility function and thus the normative parameters of pure time preference and of constant
relative risk aversion also have a very strong impact on the importance of uncertainty, as well as
on the overall net welfare benefit from acting upon climate change. This also corroborates the
findings in the literature, e.g. by Pizer (1997), that uncertainty about normative parameters would
by far dominate uncertainty about the climate system.
Due to the limitations of the model employed within our analysis, Chapter 3 can provide no gen-
eral answer to the question of the importance of uncertainty for the integrated assessment, but what
it does offer is a comprehensive set of measures to assess the importance of the different effects of
uncertainty and learning separately within a fairly complex integrated assessment model. Future
research will investigate the applicability of the approach to even more complex models which
comprise more realistic representations of the climate system. The study is also to be expanded
to the impact of other uncertainties, as it already has been shown that a combination of two single
parameter uncertainties is in no way additive in the aggregated outcome. A similar model decom-
position could be tested for the assessment of anticipative learning (as partly done in Chapter 4)
and for the inclusion of other changes to the model, like the introduction of regional inequity, new
industry sectors, changes in the climate representation, etc.
5.3 Importance of Anticipating Future Learning
As already demonstrated with respect to the importance of uncertainty, this thesis is not only
concerned with the question of the change of near-term decisions due to anticipation of future
learning, but also with the resulting changes in welfare gains from climate policy. To investigate
these changes, the notion of the so-called expected value of anticipation is introduced by decom-
posing the overall expected value of future information into the welfare gain from pre-learning
and post-learning decision adjustments.
This decomposition is then applied to the study of future learning about climate sensitivity and
90 Synthesis and Outlook
amplitude of climate induced damage within a simple exogenous one time learning framework
with the integrated assessment model MIND. Thereby we confirm the findings in the literature
that although future learning about those quantities has a significant overall expected value of
information the value of anticipating this future learning by changing near-term decisions is neg-
ligible. Chapter 4 continues by introducing uncertainty and future learning about the amplitude
of additional damage stemming from the crossing of a tipping point like threshold in temperature,
representing e.g. a breakdown of the north Atlantic thermohaline circulation as introduced by
Keller et al. (2004), or the partial destabilization of the west Antarctic ice sheet. We show that an-
ticipatory changes in near-term emissions towards more mitigation can become crucial to harvest
the value of learning. This is the case, if the value of new information cannot be used after the
learning occurred, due to the fact, that the DM would already be committed to cross the threshold
by pre-learning actions. Thus the expectation of future learning alone, without knowing what will
be learned, leads to stronger pre-learning mitigation action, thereby keeping the option to mitigate
the threshold in case the threshold damages would be severe. In this case almost the complete
value of information is due to anticipation. However, this significant value of anticipation only
occurs if the learning takes place in a specific “anticipation window”. If learning happens earlier,
the whole value of information can be simply harvested by post-learning adjustments. If learning
happens later, anticipatory changes necessary to maintain the option to avoid the threshold become
too expensive and thus the DM already commits herself to crossing the threshold irrespectively of
what is learned. This significantly lowers the overall value of information and thus also the value
of anticipation. Within the standard model setting of MIND the “anticipation window” is quite nar-
row, spanning approximately one decade. This is due to the high flexibility of the model to change
emissions on short notice to moderate costs. The location and width of the window is strongly
sensitive to this flexibility. When introducing an additional inflexibility into the model, whether it
represents political or social barriers to fast emission reductions, the anticipation window moves
towards the present and becomes broader. The maximum value of anticipation decreases, as the
inflexible emission trajectories are necessarily suboptimal compared to the flexible ones.
A straight forward extension of the work of Chapter 4 would be the inclusion of multiple thresh-
olds that might depend on different climate variables and not only on global mean temperature.
The estimates of damages arising from the crossing of the thresholds should be revised in the light
of new research on climate change impacts. However, the main result that future learning only in-
fluences near-term decisions if strong nonlinearities, e.g. from irreversibility, come into play, will
remain. Another extension would be to replace the simple one-time perfect learning by a more
advanced representation of the learning process, whether it is learning at multiple points in time
or even endogenous learning. The latter in particular might lead to stronger dependencies between
future learning possibilities and short-term decisions as the expected marginal welfare after learn-
ing is then potentially influenced by both, potential path dependencies (stock effects, thresholds,
irreversibilities) from the first period, and changes in the post-learning probability distribution due
to first period actions. However, the formal requirements in terms of computational needs (like
efficiently implementing dynamic programming for large models) and informational needs (for-
malizing endogenous learning functions for different technologies and sectors) to represent the
Importance of Anticipating Future Learning 91
uncertainties and learning processes are huge; at this point of time they seem prohibitively so.
One very interesting extension of the work would be its application to the so-called Dismal Theo-
rem by Weitzman (2009). He proposed that under certain conditions, that include fat-tailed proba-
bility distributions for uncertain climate responses, exponential damage from climate change and
sufficiently high risk aversion of the DM, the marginal utility of mitigation can be unbounded.
This means that any single ton of mitigated carbon would be worth nearly the whole economic
output, or putted the other way round, the DM would spend nearly the whole economic output
to avoid a very unlikely but catastrophic tail event. Several critics of the applicability of Weitz-
man’s argument have been put forward, e.g. by Nordhaus (2009). Another crucial assumption
that has received less attention, is the fact that Weitzman’s DM acts in an a-temporal framework,
i.e. she cannot change her decisions in the mid-term. Thus an argument against the Dismal The-
orem put forward (not formally) by Myles Allen and David Frame (communication at the 2011
Tanner lecture, on 2011-05-21 in Oxford, UK) refers to the possibility of mid-term corrections
once the DM can foresee that the unlikely tail events are becoming reality. From the perspective
of Chapter 4 this argument refers to the flexibility of the decision maker to harvest the (really
large) expected value of future information from learning about the (potentially catastrophic) cli-
mate damage nearly completely by post-learning decisions. From what we have learned from
Chapter 4, however, this is only true when either the climate damage are reversible or the learning
leading to mid-term corrections takes place early enough to be outside the anticipation window.
If the DM finds herself inside the anticipation window, the learning about the potentially catas-
trophic damages is only valuable if she reacts in the short-term, thus the fat tails again have a
paramount influence on current decision making. As the processes that may lead to catastrophic
climate damage are arguably all highly nonlinear and include significant hysteresis behavior, only
a situation outside the anticipation window could be used as an argument against Weitzman. Thus
the open question again is an empirical one, for the position of the thresholds in the system and
for the possibility of our future ability to cope with irreversibility by means of geo-engineering or
climate control options.
Another interesting extension of this work, especially in the context of potentially catastrophic
climate change is the combination of the framework from Chapter 4 with the preference structure
by Traeger (2009), mentioned above. In his framework discounting due to pure impatience is no
longer allowed, instead the DM discounts the future due to increasing uncertainty. The interesting
part of this extension is that now different future scenarios are discounted differently accord-
ing to the confidence in this specific projection. Thus the application of Traeger’s framework
to catastrophic climate change would probably deliver a solution for the problem of dominating
catastrophic fat tails of climate damages. The framework established in Chapter 4 could be used
to check the validity of such a proposition, especially when including mid-term corrections and
endogenous learning as above.
92 Synthesis and Outlook
5.4 General Outlook and further Research Questions
The results of this thesis rest on the general assumptions of the welfare economic framework em-
ployed. Thus a general direction for further research is to release or extend some of the underlying
assumptions.
One possible expansion is the recognition of heterogeneity within different parts of the model
setup. Schmidt et al. (2011a) investigate the influence of heterogeneous climate damage on the
certainty premium a decision maker would be willing to pay for ruling out uncertainty. They
find that if the climate damage is only imposed on a minor fraction of the population the risk
premium can increase substantially. This can partly be countered by implementing efficient insur-
ance markets or allowing for measures of self-insurance. In expansion one could be interested in
additionally considering an initially heterogeneously wealthy population. This would surely again
increase the effect of damage uncertainty.
The same is true when considering other sources of inequity between different actors and parts of
the total population, like inequity between regions (e.g. see Anthoff et al. 2009).
An even stronger impact of uncertainty in terms of changes in welfare gains from different policies
would be expected when adopting a framework put forward by Sterner and Persson (2008), who
argued that climate impacts would be more severe when not hitting one aggregated global output
providing sector but would hinder the production of very specific single sectors, like an envisaged
“environmental good production”. If the goods from those separate sectors where only limited in
their possibility to being substituted for each other, damage to one sector could not only lead to
limited growth or even recession of this single sector but due to the low elasticity of substitution,
the overall welfare impact from uncertainty about damage would be highly increased.
The assumption of infinitely ongoing exponential aggregated economic growth itself is a very
strong one. Besides the normative question of whether such a growth regime is a “good” thing,
in that it optimally provides happiness, it is also questionable whether such an exponential tra-
jectory is an adequate description of realistic assumptions about future economic growth. As the
theoretical foundation of long-term economic growth is still not satisfying, a first step for further
research about systematic uncertainty would be a sensitivity study with respect to the assumption
of exponential economic growth.
Gerst et al. (2010) go in this direction, by modelling economic growth as an exponential process
with a stochastic yearly growth rate. However, one could argue, as done e.g. by Ayres & Warr
(2009), that the processes driving economic growth are not stochastic in nature but result from
several processes that are partly continuous and partly discontinuous in nature. These are popula-
tion change, expansion and interconnection of markets, and technological, scientific, and societal
innovation. To include those processes in a satisfying way into a general framework of long-
term economic growth proposes an enormous challenge to the economic profession. Once this is
achieved, the climate problem, or better, the ecological constraints of our planet, can be understood
as (temporary) boundaries that induce technological change. So it might even be the other way
round, that the climate challenge and the planetary constraints work as drivers for the economy
and overcoming them, whether by behavioral change, by decarbonizing the economy, introduction
General Outlook and further Research Questions 93
of climate management or even the expansion of markets beyond the planetary boundaries through
access to cheap space travel, will be what ensures ongoing exponential growth in the future. To
investigate the role of scientific uncertainty about the system boundaries of the earth system in
such an integrated growth framework is an exciting challenge for future research.
Summarizing, this thesis has contributed to evaluating the importance of including uncertainty and
learning into the integrated assessment of climate change mitigation policy. This importance turns
out to be low under standard assumptions about the cause-effect chain of climate change. How-
ever, several plausible structural changes in the representation of the climate cause-effect chain
have been positively tested with respect to their potential to significantly increase the importance
of uncertainty and learning. This thesis has also provided the means to conduct more of these tests
with regard to changes proposed in this final chapter. Without waiting for the results of those tests,
a, in this sense somewhat hasty, personal conclusion can be drawn from this thesis: that I person-
ally tend to lean towards our initial intuition that uncertainty and learning are dominant aspects
of climate change analysis and that our formal framework is far from being able to represent this
important feature properly. Some important first steps have already been made to improve this
situation. But another bold effort has to be undertaken, maybe even more so than the original idea
to put together economic and physical models, to overcome the shortcomings of model represen-
tation of uncertainty and its impacts. Only than can the integrated assessment of climate change
lead to policy implications that real world decision makers could base their real decisions upon.
94 Synthesis and Outlook
95
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