WPTC 0603
I ional Agricultural Trade and Policy Center
WORKING PAPER SERIES
'V
UNIVERSITY OF
SFLORIDA
Institute of Food and Agricultural Sciences
THE FUTURE OF GLOBAL WARMING: WILL IT BE
POLLUTION ELIMINATING TECHNOLOGY OR
ENVIRONMENTAL CATASTROPHE?
By
Ram Ranjan
WPTC 0603 March 2006
INTERNATIONAL AGRICULTURAL TRADE AND POLICY CENTER
THE INTERNATIONAL AGRICULTURAL TRADE AND POLICY CENTER
(IATPC)
The International Agricultural Trade and Policy Center (IATPC) was established in 1990
in the Institute of Food and Agriculture Sciences (IFAS) at the University of Florida
(UF). The mission of the Center is to conduct a multidisciplinary research, education and
outreach program with a major focus on issues that influence competitiveness of specialty
crop agriculture in support of consumers, industry, resource owners and policy makers.
The Center facilitates collaborative research, education and outreach programs across
colleges of the university, with other universities and with state, national and
international organizations. The Center's objectives are to:
* Serve as the Universitywide focal point for research on international trade,
domestic and foreign legal and policy issues influencing specialty crop agriculture.
* Support initiatives that enable a better understanding of state, U.S. and international
policy issues impacting the competitiveness of specialty crops locally, nationally,
and internationally.
* Serve as a nationwide resource for research on public policy issues concerning
specialty crops.
* Disseminate research results to, and interact with, policymakers; research, business,
industry, and resource groups; and state, federal, and international agencies to
facilitate the policy debate on specialty crop issues.
The Future of Global Warming: Will it be Pollution Eliminating
Technology or Environmental Catastrophe?
Ram Ranjan
Postdoctoral Associate
International Agricultural and Trade Policy Center
Department of Food and Resource Economics, University of Florida
Email: rranian(@),ifas.ufl.edu, Ph: (352) 392 188326; Fax: (352) 392 9898
Selected Paper to be presented at the 3d World Congress ofEnvironmental and
Resource Economics
Abstract
This paper looks at the role of risk perceptions in influencing public policy related to
global warming. In a global optimization framework, the choice between investment in
pollutioneliminating technology and other carbon mitigating polices is shown to be
influenced by the degree of probability weighing of the participating nations. Two main
issues are considered. First, the extent of mitigation efforts such as investment in research
related to technological breakthrough, abatement and emissions is derived for a globally
optimizing manager. Second, the same question is addressed in light of the possibility
that nations may differ in their risk perception over the extent of damages from global
warming and therefore would take differential steps for mitigation. In case of a globally
optimizing manager, relative risk perceptions over catastrophe and technological
breakthroughs lead to substantial difference in policy choices as compared to a case when
risks are not weighed at all. For instance, if the manager fears a high risk of catastrophic
event in future following which later technological breakthroughs may become useless,
his optimal action is to minimize investment in such research efforts. Differential risk
perception is also shown as a possible cause for failure of coordination between rich and
poor nations.
Introduction
Global warming is becoming a cause of greater concern for policy makers with an
increase in the frequency of natural hazards such as hurricanes and melting of polar ice
caps that are supposedly correlated with it. While scientific evidence related to the
causes of global warming increasingly points towards man made emissions, the question
of how and how much effort to put towards mitigation and technology development still
needs to be tackled. One crucial question is over the timing and extent of damages from
global warming. Catastrophic outcomes such as a rise in the sea level, breakdown of the
thermohaline circulation belts, etc. have been predicted to happen with a crossing over of
the thresholds for accumulated atmospheric carbon. Yet, it may be possible to avoid such
disastrous consequences through a combination of available mitigative and adaptive
tools. Pollution eliminating technology is one possibility that holds tremendous potential
for reducing the hazards of global warming significantly. However, most of such efforts
would come at a cost to the society in terms of forgone consumption, plus the uncertainty
associated with the investment in discovery of new technology could be daunting.
Consequently, risks of failed investments and of catastrophes play a crucial role in
decision making. However, significantly important than the actual risks is the perception
of such risks borne out by the ambiguity associated with the science of global warming.
A lot of work has been done on comparing the costs and benefits of global
warming mitigation efforts, optimal risk reduction policies and predicting the risks of
damages (Keller et al. 2005, Kelly and Kolstad 1990, and the references therein). Some
approaches have also involved exploring international cooperation to jointly mitigate
global warming through carbon abatement and sequestration (Nordhaus and Yang 1996,
Yang 2003, etc.). Reis (2001) has looked at the possibility of pollutioneliminating
technological breakthrough in an endogenous growth model. Gjerde and Kverndokk
(1999) look at the role played by possibility of catastrophe in shaping optimal abatement
policies. Simultaneous consideration of possibility of both events has not been
considered in the literature so far. Further, work related to the impact of public
perception of risks on public policies to combat global warming has been devoted little
attention. Yet, it is the public opinion formed through risk perception that is going to
have major say in public policies, as governments would seldom against the interests of
the public they represent, if the public itself does not perceive the risks to be a great
threat. To the extent that public perceptions are reflected in governmental policies, the
strategies implemented to mitigate global warming would differ from normative
approaches that suggest decision making based upon objective evaluation of expected
costs and benefits. While there has been some interest in researching the public
perception of risks related to global warming, this research is still being confined to the
field of behavioral psychology or at best at qualitative levels in mainstream
environmental economics.
Bleda and Shackley (2005) argue that businesses would not change their
perceptions towards climate change until affirmative signals are received consistently for
a long period of time. They further argue that reality is perceived by businesses after
being filtered through a reference frame and is not perceived objectively. Consequently,
'experienced reality' may differ from 'actual reality' due to perceptions which are based
upon their interests, expectations, etc. The interpretations of the signals or experiences
are governed by the frame of reference of the receiver and could be resilient to objective
revisions (Daft and Weick, 1984). It is further argued that direct signals of climate
change may be subject to misinterpretation as isolated weather related signals and thus
could be discarded if reinterpretation of these signals requires significant organizational
changes (Berhout et al 2004). Risk perception could also vary based upon gender, race,
education, sex, political affiliation, nationality, etc. Kahan et al. (2005) mention that
whites are more likely to be less risk averse as compared to minorities to certain types of
risks involving abortion, guns, global warming, etc. This phenomenon often termed as
the 'white male effect' is explained by their more individualistic and antiegalitarian
lifestyles. For instance, in case of global warming, individuals who see the cutback on
carbon emissions as a threat to their individualistic life styles are more likely to discount
the available information related to global warming. Whereas, individuals who are
egalitarian in nature would be more willing to accept the possibility of global warming as
they would be more concerned about the intergenerational equity.
While it is not clear how risk perception at the local level amongst these various
groups transforms into a collective societal risk perception over global warming, the
impact of such perceptions on public policies cannot be denied and therefore ignored.
Such risk perception could have significant impact on the success or failure of
international collaboration to mitigate global warming through investment in research
related to technology invention or direct carbon abatement and sequestration polices.
The approach in this paper involves modeling the risk of catastrophe and chance
of technological breakthroughs as a Poisson process whose hazard rates are influenced by
the accumulated stock of atmospheric carbon and stock of research in pollution
eliminating technology. In a dynamic programming framework, the total intertemporal
benefits following a catastrophe or a technological breakthrough are solved for and then
integrated back into the main objective function that seeks to maximize the long term
expected benefits from being in all the possible states of the world. The choice variables
include level of emissions, abatement efforts and the level if investment in research. The
weighing of probabilities is modeled as an inverted sshaped function following the
literature on risk weighing in behavioral economics.
Following the above approach, two main issues are addressed. First, the extent of
mitigation efforts such as investment in research related to technological breakthrough,
abatement and emissions is derived for a globally optimizing manager whose risk
perceptions reflect those of the collective perceptions of the society. Second, the same
question is addressed in light of the possibility that nations may differ in their risk
perception over the extent of damages from global warming and therefore would take
differential steps to mitigate it. In case of a globally optimizing manager, relative risk
perceptions over catastrophe and technological breakthroughs lead to substantial
difference in policy choices as compared to a case when risks are not weighed at all. For
instance, if the manager fears a high risk of catastrophic event in future following which
later technological breakthroughs may become useless, his optimal action is to minimize
investment in such research efforts. This is because later technological breakthroughs are
of no use if a catastrophe happens before them. This highlights the role of risk
perception. A globally coordinated effort when participants tend to weigh the probability
of catastrophes differentially may differ significantly from one where uniformity in risk
perception is assumed. Yet, it is an issue of great practical concern in view of difficulties
faced in coming to a common consensus over the extent of carbon mitigation and other
steps required to expedite pollution eliminating technological breakthroughs. In a global
optimization framework, the choice between investment in pollutioneliminating technology
and other carbon mitigating polices is shown to be influenced by the degree of probability
weighing of the participating nations. Differential risk perception is also shown as a possible
cause for failure of coordination between rich and poor nations to come to terms over an
agreeable carbon mitigation plan.
Model
The modeling approach involves incorporation of risks of global warming related
catastrophes as being endogenous to the stock of carbon accumulated in the atmosphere
and then exploring optimal combination of abatement, research and emissions paths to
influence this stock of carbon. The probabilities of catastrophe and that of discovery in
technology is modeled as a Poisson process. The stock of carbon in the atmosphere
evolves as:
(1) s =ea&s
Where e is the emissions per period and a is the abatement of carbon. Stock of carbon
has a natural rate of decay which is governed by the rate at which it gets assimilated into
the deep oceans. Society can also make investment i in research to discover pollution
eliminating technology which would lead to no further stock accumulations. This
investment is considered to be worthwhile, given the possibility of catastrophic changes
in the earth's climate caused by high concentrations of atmospheric carbon. The
possibility of a pollutioneliminating technological breakthrough is modeled as a Poisson
process, characterized by its hazard function:
(2) =a log(i)
where p is the instantaneous probability of a technological breakthrough given that it has
not occurred in the past. The hazard rate is a function of the level of investment in
technological research (i). The methodology used for modeling the probability of
breakthrough and catastrophe in this system is that of a Poisson process, based upon the
work of Clarke and Reed (1994), Reed (1988) and Reed and Heras (1994). Further,
Knowler and Barbier (2005) have applied this approach to model invasive species
management under uncertainty. The probability density function of this technological
progress is given as:
(3) f(p) = pexp(p(t))
The possibility of a catastrophic change in earth's climate is modeled as a Poisson
process too, characterized by its hazard function :
(4) q = plog(s)
In the above equation c is the instantaneous probability of a catastrophe given that it has
not occurred in the past. The hazard rate is function of the stock of carbon already
accumulated into the atmosphere. The probability density function of this possibility is
given by:
(5) f(q) = exp(q(t))
In the event of a catastrophe, society suffers damages, d(s) per period for all times to
come. Society's problem is to maximize the expected sum of values in the states
characterized by neither invention nor catastrophe, invention (with the possibility of a
catastrophe) and catastrophe as:
(6) E(v0) + E(v) + E(v2)
where vO is the discounted sum of profits received by society before either a
technological change or a catastrophe happens. vl is the discounted sum of value
received by society in the event of a technological breakthrough and v2 is the discounted
sum of value received in the event of a catastrophic collapse of the earth's atmosphere
before a breakthrough in technology. The three terms can be further defined as:
(7) vO= (b(e) c(a) c(i) d(e))exp(rt)dt
In the simplest case when technological change is of the nature that it also eliminates the
possibility of future catastrophe besides eliminating pollution, the value function vl, after
the technological change can be defined as:
(8) vl = jb()exp(rt)dt, where e is the level of emissions that maximizes societal
0
benefits. Note that, emissions however would not accumulate into the atmosphere, thus,
obviating the need for any abatement. The above formulation is for a case where the
technological change happens at the beginning of time period. The value function after
the occurrence of a catastrophe is defined as:
(9) v2= d(s)exp(rt)dt, where s is the level of carbon accumulated into the
0
atmosphere. Now, let t be the time when a technological change occurs. Then the value
to the society before such an event occurs is given by:
(10) vO = p exp(p(t))exp(q(t))(b(e) c(a) d(e) c(i))exp(rt)dt ,
o o
where p exp(p(t)) exp(q(t)) is the probability that the technological change happens at
time t and the environmental catastrophe happens after time t. After integration by parts,
this can be further written as:
(11) vO = exp(p(t) q(t)) (b(e) c(a) d(e) c(i)) exp(rt)dt
0 P(t) + q(t)
Similarly, if t be the time when a catastrophe occurs, and a technological change happens
after time t, then the value to the society before such an event is given by:
(12) vO = j exp(q(t))exp(p()) ((e) c(a) d(e) c(i))exp(rt)dt where
Lo o Jo
q exp(q(t))exp(p(t))is the probability that the environmental catastrophe happens at
time t, and the technological change happens after time t. After integration by parts, this
can be further written as:
(13) vO = f exp(p(t) q(t)) (b(e) c(a) d(e) c(i)) exp(rt)dt
0 P(t) + 4(t)
Therefore, the expected value to the society, before any of the two events occur is given
by:
vO = p exp(t q(t)) (b(e) c(a) d(e) c(i)) exp(rt)dt +
(14) exp( (t) + (t)
exp( p(t) q(t)) (b(e) c(a) d(e) c(i)) exp(rt)dt
{0 p(t) + c(t)
After simplification (14) can be further rewritten as:
(15) vO = exp(p(t) q(t))(b(e) c(a) d(e) c(i))exp(rt)dt
Similarly, vl and v2 can be defined as:
(16) vli= pexp(p(t))exp(rt)dt
0 r
The equation of motion for atmospheric carbon accumulation after the technological
innovation is given by:
(17) s = s
(18) v2 = d(s)cexp(q(t))exp(rt)dt
0
= s(t)2 q exp(q(t)) exp(rt)dt
0
= (t)2 exp(rt)dt = s(T)2 exp(t)2 exp(rt)dt = s(T)
o o r+23
where T is the time of collapse
Society's problem is to maximize the sum of vO, vl & v2 given the constraints posed by
the equations of motion for carbon stock and the hazard functions for technological
discovery and the environmental catastrophe. The current value Hamiltonian is given by:
b(e)
exp(p(t) q(t))(b(e) c(a) d(e) c(i)) + )p exp(p) d(s)c exp(q) +
(19) r
S(e a &s) + Aalog(i) + AJ3 log(s)
Where 1, A2,and A3 are the shadow prices of stock of carbon, stock of cumulative
probability of invention and stock of cumulative risk of catastrophe. One would expect
the shadow prices, A, and A3 to be negative as they are 'bads'. First order conditions
w.r.t.:
(20) e => b'(e) d'(e) exp(p q) + A1 0
(21) a=> c'(a)exp(pq) A =0
(22) i=> c'(i)exp(p q) + + vl exp(p) 0
i i
The first order conditions require dedication of emissions, abatement and investment
efforts to a point where there marginal utility equals their shadow prices. This ensures
long term optimization of the societal benefits.
More interpretation of the shadow prices themselves could be had from observing their
equations of motion over time. No arbitrage conditions are given by:
(23) =r ( d'(s) exp(q) d'(s) exp(q) + 3)
r r s s
(24) = rA, + exp(p(t))(b(e) c(a) d(e) c(i))vlp exp(p)
(25) A = rA3 + exp(q(t))(b(e) c(a) d(s) exp(q)
r
Rate of growth of shadow price of stock of carbon is positively related to the stock of
damages as given by equation (23). It is also negatively influenced by the shadow price
of stock of cumulative hazard rate. This should be obvious as a higher A3 would imply a
higher risk of catastrophe which would require 4 to be lowered for adequate
optimization. Shadow price of invention chances, as given by (24) above grow positively
with an increase in vl as an increase in value after invention would make postponing it
costlier. Finally, (25) presents the growth path for shadow price of cumulative risk of
catastrophe as increasing in stock of carbon, which is again obvious.
Alternate scenarios:
While the above model assumed simplification in terms of postbreakthrough or
catastrophe scenario, it is possible for catastrophe to take place even after a technological
breakthrough or a breakthrough to take place after a catastrophe. Consider a case when
technological change leads to elimination of pollution but does not eliminate the risk of
catastrophe. vO and v2 remain the same as before however vl is now redefined as:
o s(T)72 1
(26) vl = p exp( p) b(e) exp(q(t)) exp(rt)dt + f exp(q(t)) exp(rt)dt
Sr+2
Another situation is where catastrophe does not eliminate the possibility of technological
change later on. Under this scenario, vO remains the same as before, vl is given by the
above equation and v2 is now redefined as:
(27)
v2= { exp(q) {d(s)exp(p(t))exp(rt)dt + ) exp(p(t))exp(rt)dt
r
We will consider these possibilities for detailed exposition in the numerical simulation
section. So far we have not incorporated the probability weighing aspect in the
optimization problem. In the next section we look at how collective weighing of risks
could be incorporated to be reflected in the risk weighing by the global planner or a
country level planner.
Subjective Weighing of Risks
We now add the behavioral economics element into the model. Based on
accumulated evidence in economics and psychology literature (see summary in Hurley
and Shogren, 2005), assume the planner assigns higher weights to low probabilities of an
event and lower weights to high probabilities of the same event (also see Starmer, 2000).
We add these weights to the hazard rate p(x), which represents the probability of the
event happening at time t, given that it did not happen previously. Let the weighing
function follow an inverse Sshape. Following Prelec (1998), we use a twoparameter
weighting function
(28) w(p) = e (1np)
where 0 is the parameter that primarily determines elevation, and y is the parameter that
primarily determines curvature. Elevation reflects the inflection (reference) point at
which a person switches from overestimating low probability events to underestimating
high probability events, i.e., the degree of over and underestimation; curvature captures
the idea that people become less sensitive to changes in probability the further they are
from the inflection point (Tversky and Kahneman, 1992; Gonzales and Wu, 1999).
Depending on the values of 0 and y the path of the hazard rate can differ from the
traditional expected utility model. By adding a subjective weighing that discounts high
probabilities, a planner cares less about high risks of loss. The inflexion point of the
inverted sshaped curve is critical and can only be determined empirically. Weighing of
hazard rates implies that predominance is given to the probability of the event happening
at time t, given they would survive until that time. In an exponential distribution, this
hazard rate is constant (but could be endogenous). Figure below shows the subjective
weighing of hazard rates.
INSERT FIGURE 1 HERE
Numerical Simulation:
We perform numerical simulations with hypothetical parameters to study the evolution of
optimal control policies. Choice of parameters reflects the nonlinearity involved with
various costs and benefits functions. Figure 2 shows the value functions before
breakthrough or catastrophe (vO), after breakthrough (vl) and after catastrophe (v2).
Note that the value function after technological breakthrough (vl) is lower than one
before either of the events (vO) as the former is weighed by the probability of a
technological breakthrough. Also, the value after a catastrophe has negative value all
throughout due to damages from accumulated stock of carbon in the atmosphere.
INSERT FIGURE 2 HERE
Figure 3 shows the survival probabilities (or the probability of surviving until a given
time) for environmental catastrophe. The base case scenario predicts the lowest survival
probability for any given time period. Survival probability increases with an increase in
perceived chance of technological breakthrough. Due to a high chance of technological
breakthrough, emissions actually increase substantially initially (as shown in Figure 4).
When risk perception is high over catastrophe or over both catastrophe and technological
breakthrough, survival probability from catastrophe is pushed even farther up. This is
primarily achieved through increased abatement and reduced emissions compared to the
earlier cases. When the risk of future catastrophe remains even after a technological
breakthrough (as specified in equation 26), the survival probability is the highest which is
achieved through very high level of abatement compared to the rest of the cases.
INSERT FIGURES 3 ,4 and 5 HERE
Investment in technological research is highest in the base case, and lowest in the cases
when the perceived risk of a breakthrough is higher. When the perceived risk of a
catastrophe is higher, the investment is lower than the base case, which seems
counterintuitive. However, this anomaly is due the comparative costs and their
effectiveness of various control options that have been assumed. Note that investment in
research falls drastically with an increase in the risk of catastrophe as a catastrophe would
negate all the investment towards technological breakthrough if technological
breakthrough happens after the catastrophe. This highlights a very important challenge
to policy formulation related to investment; a very high perceived risk of catastrophe may
discourage investment in discovery of technology that could become ineffective after the
catastrophe. This of course is related to the nature of catastrophe that follows a crossing
of the environmental threshold for atmospheric carbon. Sharp shifts in the environmental
balance, (such as a rapid rise of the sea level) may render pollution eliminating
technology useless as the focus may shift towards adaptive measures. When the
technological breakthrough is of the nature that sucks out accumulated carbon from the
atmosphere, thus helping revert back the catastrophic consequences of a threshold
breaching, investment in such research would be encouraged. However, it is unlikely
that a rise in the sea level or a break down of the thermohaline circulation belt could be
reverted back. Consequently, such possibilities are negligible.
INSERT Figure 6 HERE
Figure 7 shows the possibility of a technological breakthrough which is highest in the
base case and increases as the risk of catastrophe increases. In case when the risk
perception is high for the both the events of breakthrough and catastrophe, the risk
perception over catastrophe dominates.
INSERT FIGURE 7 HERE
Case of Two Countries with Differential Risk Perception
Consider now that there are two countries (developing and developed) that differ in their
risk perception over the catastrophe and also over the probability that a pollution
eliminating technology could be discovered in the near future. The developing country
has lower benefits from emission, higher damages from catastrophe, higher costs of
abatement and investment in technology.
Figure 8 shows the emission paths of the two countries when their welfare is
jointly managed by a global optimizer who assigns equal weights to both of them.
Emissions of the developed country are higher than those of the developing country due
to their higher benefits from emissions. This maintains even as the risk perception of a
catastrophe is high. However, abatement by the developed country too is higher than the
developing country as determined by its low relatively lower costs of abatement. This is
shown in figure 9 below.
INSERT 8 and 9 HERE
The developed country also makes higher investment in research for technological
breakthrough as shown in figure 10. However, this is not always the case as it turns out
that risk perception can alter the roles of the two sides. When the risk perception over
catastrophes for the developing world is higher and the globally optimizing planner
assigns a slightly higher weight to its welfare (w=.6) as compared to the developed world,
the investment patterns change dramatically. It is now the developing country that is
taking more investments in research all throughout.
INSERT FIGURE 10 HERE
Conclusion
Quite a few key results emerge from the above analysis. While the tradeoff between
abatement efforts, reduction of emissions and investment in research is obviously linked
to their relative costs and benefits, some results are more highlighted than others.
Consider the influence of high perceived chance of technological breakthrough, which
leads to both higher emissions and abatement as compared to the base case. Whereas,
when catastrophic risks are hard to eliminate (continued risks case), the situation
obviously warrants highest levels of abatement. Similarly, if risk perception over
catastrophe is very high, it might discourage investment in pollution eliminating research
if breakthroughs after catastrophe are deemed to be useless. International cooperation is
heavily influenced by how welfare of countries is weighted in the global scheme and how
different countries perceive the risks of catastrophe.
It would be highly desirable to understand the risk perceptions of different
countries and their hopes over future pollutioneliminating technological advances. It is
also possible that weighing of probabilities over these key events is used as a strategic
tool to while negotiating deals over emissions reductions between nations. For instance,
by pretending that a country does not actually believe in the risks of catastrophe, it could
extract out a much milder emissions plan as compared to other nations.
References
1. Clarke, R.H, and W.J. Reed, 1994. Consumption/Pollution Tradeoffs in an
Environment Vulnerable to Pollution Related Catastrophic Collapse, Journal of
Economic Dynamics and Control 18, 9911010.
2. Gjerde, J.G., and S. Kverndokk (1999): "Optimal Climate Policy under the
Possibility of a Catastrophe", Resource and Energy Economics, 21(34), 289317.
3. Knowler, D., and E. Barbier, 2005. Importing Exotic Plants and the Risk of
Invasion: Are MarketBased Instruments Adequate? Ecological Economics 25:
341354.
4. Keller, K., M. Hall, SR. Kim, D.F. Bradford, and M. Oppenheimer (2005).
"Avoiding Dangerous Anthropogenic Interference with Climate System",
Climatic Change, Vol. 73 (3)
5. Nordhaus, W.D. and Z.L. Yang (1996). "A regional Dynamic General
Equilibrium Model of Alternative ClimateChange Strategies", American
Economic Review, 86, 741765.
6. Kelly, D.L., and C.D. Kolstad (1999). "Bayesian Learning, Growth and
Pollution", Journal of Economic Dynamics and Control, 23: 491518.
7. Reis, A.B. (2001). "Endogenous Growth and the Possibility of Eliminating
Pollution", Journal of Environmental Economics and Management, Volume 42,
Issue 3, Pages 360373.
8. Reed, W.J., 1998. Optimal Harvesting of a Fishery Subject to Random
Catastrophic Collapse. IMA Journal of M htllem, tii Applied in Medicine and
Biology, 5: 215:235.
9. Reed, W.J. and H.E. Heras, 1992. The Conservation and Exploitation of
Vulnerable Resources, Bulletin of Mathematical Biology, 54 (2/3): 185207
10. Tol, R.S.J. (2003). "Is the Uncertainty about Climate Change too Large for
Expected CostBenefit Analysis?", Climatic Change 56: 265289.
11. Yang, Z.L. (2003). "Reevaluation and Renegotiation of Climate Change
CoalitionsA Sequential ClosedLoop Game Approach", Journal of Economic
Dynamics and Control, 27: 15631594.
12. Gonzalez, R. and G. Wu. "On the Shape of Probability Weighting Function",
Cognitive Psychology 38, 129166 (1999).
13. Prelec, D. "The Probability Weighting Function", Econometrica, 66(3) 497527
(1998).
14. Starmer, C. "Developments in the NonExpected Utility Theory: The Hunt for a
Descriptive Theory of Choice under Risk", Journal of Economic Literature, 331
382 (2000).
15. Tversky, A., and D. Kahneman. "Advances in Prospect Theory: Cumulative
Representation of Uncertainty", Journal of Risk and Uncertainty, 5, 297323,
(1992).
Appendix
Functional Forms and Parameter Values for the Global Planner Case:
Benefits from emission b0e + ble2 bO = 5, bl= .5
Cost of abatement cOa + cla2 cO =.2,cl =.5
Cost of innovation cOi + cli2 cO = 1, cl = .5
Damages from emissions dOe + dle2 dO =.5,dl =.01
Damages from stock s2 (t) s2 (t)
Hazard function for c = p log(s(t)) = .05
catastrophe
Hazard function for p = a log(2.71828 + i(t)) a = .05
technological breakthrough
Weighing of Catastrophic risks 02(log(alog(2.71,2+, 2 02= .5 y2 = .5
(high risk perception of
catastrophe case)
Weighing of technological e1(log(alog(2.7121+ .1 1111 yl = .5 01 = .5
possibilities
(high risk perception of tech.
breakthrough case)
Discount rate r .1
Initial value of hazard rate pO .1
Initial value of hazard rate qO .1
Initial value of stock of carbon sO 30
Decay rate of stock of carbon 6 .001
Parameters for the two country case:
Parameters Country Developed CountryDeveloping
Benefits from emission bO = 7, bl =.9 bO = 5, bl =.5
Cost of abatement ca0 =.2, cal =.2 ca0 = .5,cal = .5
Cost of innovation ciO =. 1, cil =. 5 ciO = .2, cil =.7
Damages from emissions dO= 5,dl =.01 dO= 7,dl= .05
Damages from stock s2 (t) s2 (t)
Hazard function for c = log(s(t)) = .05
catastrophe
Hazard function for p = alog(2.71828 + i(t)) a = .05
technological breakthrough
Weighing of Catastrophic risks 82(log(alog(2.71828+i(t)))2 02 = 1 y2 = 1
(base case)
Weighing of technological 81(log(alog(2.71828+i(t))))Y1 y = 1 O1 = 1
possibilities
(base case)
Weighing of technological yl = .4 01 = .4 yl = .4 01 = .4
possibilities and catastrophe
(high chance oftech. 02 = .99 72 = .99 02 = .99 72 = .99
breakthrough)
Weighing of technological yl = .9 01 = .9 yl = .9 01 = .9
possibilities and catastrophe
(w=.6 and high perception of 02 = .99 /2 = .99 02 = .4 y2 =.4
catastrophe by the developing
country)
Figure 1: Inverted SShaped Weighing of Probability of an Event
Weighted Probability
1\
Probability
0.2 0.4 0.6 0.8
Note: Parameters in the function: w(p) = e ( lnp) 0 .5, y
0.O
Figure 2: Value Functions under Base Case
6 *v0(t)value before either event takes place
mv1(t)value after a tech. breakthrough
5 ^ v2(t)value after a catastrophe
4
3
2
1T
1
2
3
Time
Figure 3: Survival Probability of Catastrophe
1 11 21
Time
Figure 4: Time Path of Emissions
e(t)base case
 e(t)high risk perception of catastrophe
e(t)high perception of chance of breakthrough
e(t)high risk perception over both events
Time
1
Figure 5: Time Paths of Abatement Efforts
25
20
15
10 
5
a(t)base case
*a(t)high risk perception of catastrophe
a(t)high perception of chance of breakthrough
a(t)high chance of both events
 continuedriskunweighted
1 2 3 4 5 6 7 8
Time
Figure 6: Time Path of Research Investment
35
i(t)base case
3 li(t)high risk perception of catastrophe
i(t)high perception of chance of breakthrough
2 5 i(t)high chance of both events
2 
15
05
0
1 21 41
Time
Figure 7: Probability of Technological Breakthrough over Time
0.9
*exp(p(t))high perception of chance of technological breakthrough
0.8 Uexp(p(t))high perception of risk of catastrophe
exp(p(t))base case
0.7 exp(p(t))high risk perception over both events
0.6
0.5
0.4 
0.3
0.2
0.1
1 21
Time
Figure 8: EmissionsTwo Countries
0.95
0.85
0.75
0.65
*developing country (base case)
udeveloped country (base case)
developing country (high risk perception of tech. breakthrough)
developed country (high risk perception of tech. breakthrough)
5
Time
0.55
0.45
0.35
0.25
0.15
0.05
0.05 1
9
Figure 9: AbatementTwo Countries
*developing country (base case)
c *developed country (base case)
developing country (high risk perception of tech breakthrough)
developed country (high risk perception of tech breakthrough)
i i ^ ^ < ^ ^ ^ ^ ^ ^ ^ ^ ^ < ^ >
Time
Figure 10: Investment: Two Countries
*developing country (base case)
*developed country (base case)
10 developing country (high perception of tech breakthrough)
developed country (high perception of tech breakthrough)
 developing country (w=.6,high risk perception of catastrophe for developing world)
8 developed country (w=.6, high riskperception of catastrophe for the developing world)
1 21 41 61
Time
