metadata: authors: 0: author: Baranzini, Andrea
1: author: Chesney, Marc
2: author: Morisset, Jacque
docna: 0: docna: Uncertainty and global warming : an option -
            pricing approach to policy
docty: Policy Research Working Paper
subtopic: Science of Climate Change,Climate Change and Health,Climate Change and Environment,Environmental Disasters & Degradation,Food Security,Energy Demand,Energy and Environment,Energy and Mining,Global Environment
teratopic: Public Sector Development
abstracts: cdata!: Uncertainty is inherent in the analysis
            of global warming issues.  Not only is there considerable
            scientific uncertainty about the magnitude of global
            warming, but even if that problem were resolved, there is
            uncertainty about what monetary value to assign to the costs
            and benefits of various policies to reduce global warming.
            And yet the influence of uncertainty in policymaker's
            decisions is ignored in most studies of the issue.  The
            authors try to explicitly incorporate the effect of
            uncertainty in the choice of global warming abatement
            policies.  The approach they develop draws on the emerging
            literature on investment under uncertainty - in particular,
            that on the option-valuation approach.  Their numerical
            applications focus on the Cline's (1992) analysis of
            global warming, but it may be applied to a range of global
            warming analyses.  First, they assess whether it is optimal
            to implement Cline's strategy of limiting global
            warming today, or whether it should be postponed, and for
            how long.  Then, they identify the optimal policy to be
            implemented today for different levels of uncertainty about
            the costs and benefits of policies to reduce global warming.
repnme: repnme: Uncertainty and global warming : an option
            - pricing approach to policy
display_title: Uncertainty and global warming : an
            option - pricing approach to policy
listing_relative_url: /research/1995/02/697691/uncertainty-global-warming-option-pricing-approach-policy
topicv3: Agriculture and Food Security m1326205 435,Energy and Extractives m1326595 507,Climate Change m1326829 567,Environment, Natural Resources and Blue
            Economy m1327391 566,Deforestation m1503714 3386
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Public Sector Development
origu: Development Economics and Chief Economist
no_of_pages: 38
model_extractions: POLICY RESEARCH WORKING PAPER 1417

# Summary findings

Uncertainty is inherent in the analysis of global warming issues. Not only is there considerable scientific unccrtainty about the magnitude of global warming, but even if that problem were resolved, there is uncertainty about what monetary value to assign to the costs and benefits of various policies to reduce global warming. And yet the influence of uncertainty in policymakers' decisions is ignored in most studies of the issue.

Baranzini, Chesney, and Morisset try to explicitly incorporate the effect of uncertainty on the choice of global warming abatement policies. The approach they develop draws on the emerging literature on investment

- undcr unccrtainty in particular, that on the optionvaluation approach.

Their numerical applications focus on Cline's (1992) analysis of global warming, but it may be applied to a range of global warming analyses.

First, they assess whether it is optimal to implement Cline's strategy of limiting global warming today, or whether it should be postponed, and for how long Then, they identify the optimal policy to be implemented today for different levels of uncertainty about the costs and benefits of policies to reduce global warming.

- This paper a product of the Country Operations Division, Latin America and the Caribbean, Country Department I-is part of a larger effort in the region to understand the role of uncertainty in environmental policy issues. Copies of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Celinda Dell, room Q7-106, extension 85148 (30 pages). February 1995.

_Th e Workiog Policy Resarch Paper_ P _Sncr dinata the fin&gs of_ **_uk_** X **_prog_** _to_ **_erag the c of i*ea abwt_** _dtvelopmext issulesAn ob*tiueof the series is toge_ **_t_** _hermdingsoutqukkciy, cuen if thc presmtiom are kss tbanfilypolWshd. ne |pa pers amry the names of the authos and should be used axd cied acrwrdngl. The frtigs, bapdaim od mdsso are the aud;=x' owx and sdm not be au&c d to the World Bank, its ExcYtive Board of Dirers, or axy of its mknbr aoitries._

Produced by the Policy Rescarch Dissemination Center
# Uncertainty and Global lWarming

## An Option-Pricing Approach to Policy'

_Andrea Baranzini' Marc Chesneyb Jacques Morisset'_

- _a. University of Geneva, Department of Economics, 102 bd Carl-Vogt, CH-1211 Genstve 4, Switzerland_

- _b. Department of Finance, HEC-ISA Centre, F-78350Jouy-en-Josas, France_

- _c. World Bank, 1818 H Street NW, Washington, DC_ 20433, _USA_

1. We would like to thank Antonio Estache and Danny Leipziger for helpful comments and William Cline for providing us data on the costs and benefits of global warming.
# UNCERTAiNTY AND GLOBAL WARMUNG

# TABLE OF CONTENTS

# Summary

|I.|<br>Introduction...................................|.........<br>1|
|---|---|---|
|2.|The Option-Pricing Model.....|2|
|3.|An Application<br>..|7|
||3.1.<br>TheBaseline Scenario....|8|
||3.2<br>Parameters for the Option-Pricing Model...........|..........<br>14|
|4.|Concluding Remarks..............................|........<br>18|
||Appendix...................................|21|
||Bibliogrphy.<br>................................|........--<br>23|
||Tables........................-|25|
# SUMMARY

Uncertainty is an inherent phenomena in global warmning issues. First, there is considerable scientific uncertainty around the magnitude of global warming, and second, even if this problem was resolved, it remains difficult to measure in monetary tenns the benefits and the costs associated with the policies aimed at reducing global warming. Although uncertainty affects the policy-makers' decision to reduce (or not) global warming, such an influence is ignored in most existing studics.

The aim of this paper is to explicitly incorporate the effect of uncertainty on the choice of global warming abatement policies. The approach developed in this paper is related to the emerging literature on investment under uncertainty, and in particular to the option-valuation approach. Our numerical applications will focus on Cline's (1992) analysis of global wanning, but our approach may be applied to a wide range of global warming analyses presented in the literature. We are able to obtain two kind of results. First, we assess whether it is optimal to implement a given strategy of limiting global wanning today or whether it should be postponed, and for how long. Second, we identify the optimal policy to be implemented today, for different levels of uncertainty around the costs and benefits of limiting global warming.
# 1 Introduction

Global wanning has received considerable attention in the last few years, yet few concrete actions have taken place. Recently, Cline (1992) and Nordhaus (1993) have shown that, from an economic perspective, the resulting recommendations greatly depend on some key issues. First, it is necessary to evaluate the costs and benefits of limiting global warming. Second, these benefits and costs should be evaluated in a reference period and discounted accordingly. Third, one should consider uncertainties about evolving scientific knowledge and economic environment.

This paper focuses on the issue of uncertainty. In the context of global warming uncertainty affects the decision-maldng through two kinds of irreversibilities, that work in opposite directions. First, uncertainty, as already pointed in the seminal work by Weisbrod (1964), Arrow and Fisher (1974) and Henry (1974), delays the decision on an irreversible action if the passage of time is likely to bring significant new information. This is particularly true for global wanming, in which the political and economic repercussions of abandoning policy actions once they are well under way are so high as to make abandonment impracticable. Second, uncertainty biases the traditional cost-benefit analysis against policy adoption. It may be desirable to adopt a policy now, even though the traditional analysis declares it uneconomical, because greenhouse gases possess long lifetime, increasing the level of irreversible damages compared to that if the action was taken from the start.

The approach developed in this paper will be related to the emerging literature on investment decision under uncertainty, in particular to the option-valuation approach. In this paper, we will obtain two kdnds of results. First, we will assess whether it is optimal to implement a given strategy of limiting global warming today or whether it should be postponed, and for how long. Second, we will identify the optimal policy to be implemented today, for different levels of uncertainty around I
# the costs and benefits of limiting global warming.

The option pricing approach developed in this paper can be applied to a wide range of global wanning projections presented in the literature, but our numerical applications will focus on the recent aggressive policy proposed by Cline (1992). The impact of changes in uncertainty on the optimal date of intervention of this proposal will be closely examined throughou **t** he paper. In the face of uncertainty, the optimal strategy could be to wait, or, eventually, to proceed with a less aggressive policy. Cline's proposal is to limit the level of CO2 emission to 4 GtC annually, but a ceiling of 5.5-6.5 GtC --about the existing level in 1990-- seems more appropriate for a level of uncertainty ranging around 8-10 percent annually. This results, we believe, have high policy content and are in line with European Union's recent proposals.

The paper proceeds as follows. In section 2 we review the basic theory of the option pricing model. Section 3 introduces Cline's cost-benefit analysis and integrates the option pricing model in this approach. Section 4 concludes and presents some qualifications.

# 2 The Option-Pricing Model

The economically optimal decision to invest depends on the benefits and costs associated with a given project or policy proposal. Typically, the discounted benefits (V) and costs (P) of reducing global warming are expressed as follows: 2
where r is the appropriate discount rate, and T the time horizon. The discounted costs are the monetary costs of abatement policies, while discounted benefits are the level of damage avoidance --the difference between the cost of global warming in the absence of intervention and the costs of global warming which can not be avoided because greenhouse gases possess many years of life once they are emitted.

In conventional models, a policy will be implemented when discounted benefits are greater than discounted costs (V/F> 1). Although this approach is typically applied in the global warming context, it suffers from two major shortcomings. First, it does not account for the uncertainty surrounding the costs and benefits of limiting global warming. One way around has been to calculate different scenarios and assign probabilities, but only a few number of outcomes can be considered. Second, it does not consider the possibility of waiting to take advantage of better information. As discussed in the introduction, the decision to invest can be considered as irreversible because it requires initial investments on the order of hundreds of billions of dollars a year (see Cline (1992) for some detailed figures). Therefore, the policy-makers have strong incentives to wait in order to acquire additional information and thus the decision to wait has a value. The question is how to derive the value (or the price) to wait and to detemnnine the optimal time of intervention.

The financial literature provides useful tools to calculate the price of waiting and the critical threshold ratio of benefits to costs which renders the policy efficient.

3
Indeed, an irreversible investment is similar to a financial call option where in exercising the decision to invest, the policy-maker forgoes the potential gains of postponing the decision. We use the model developed by Samuelson (1965) and McDonald and Siegel (1986) and known as the perpetual option model, also called: the option to wait to invest.

The benefit to cost ratio is defined as Y = V/F. The critical level at which it becomes optimal to implement the policy is Y', which is greater than 1 because of the value to wait to invest. For simplicity, we assume that Y follows a geometric Brownian process: where u is the drift of the process, **a** its volatility, and z a Brownian motion.

We assume that the drift and the volatility are constant over time. In our numerical simulations, the first variable will be determined by Cline's analysis, while the second one will be defined exogenously. One major caveat is that disasters cannot be analyzed because this stochastic process assumes that the benefit/cost ratio is continuous (on this topic, see Drepper and Mansson (1993)).

Assuming a perpetual and american option, McDonald and Siegel (1986), following Samuelson (1965), have demonstrated that the option value (W) at time u can be written as;:

**2. W is the discounted expected pay-off in a risk-adjusted economy. In equation (4), (Y*-l)F is equivalent to the** pay-off **of** early exercise. **(VY/F)'** is **an expected discount factor.**

4
with where rv = (r - Gv ) and rF = (r _-GF_ ) denote the effective discount rates, _(GV_ the growth rate of benefits, GF the growth rate of costs, cFv and aF the standard deviations of benefits and costs respectively, a: the total variance associated with the variable Y. and _OeV_ the correlation coefficiene.

The investment decision is optimal when the benefit to cost ratio is greater than the critical ratio (Y > Y) as the value to invest in the future is lower or equal than that of investing today (W < F - V). Yet the value of the option to wait to invest could be large enough to invalidate the usual decision rule, to invest when benefits exceed costs. In effect, the correct decision rule under such circumstances shouild be to invest when benefits exceed the costs by an amount at least equal to the value of the lost (foregone) option.

The influence of (exogenous) uncertainty is explicitly taken into account in the decision maldng process. Overall, an increase in uncertainty (o2) augments the value 3. Here the risk premium is assumed to be zemr.

5
to invest in the future as compared with that of investing today; (dW/do<sup>2</sup> > 0).' If uncertainty is higher, the value of waiting to receive more infornation is indeed higher and the required flexibility premium should be higher too. In the numerical application, we will see that the investment decision is very sensitive to the estimate or perception of the underlying uncertainty. One caveat is in order at this point. The uncerainty is assumed to be exogenous, but it can be influenced by the damages from global warming and the degree of policy intervention --uncertainty may increase (or alternatively decrease) with the level of cumulative emission from the use of fossil fuel., As discussed in the introduction, not only is the optimal investment timing influenced by the possibility of waiting for better information, but also by the evolution of irreversible damages during the waiting period. Two basic assumptions can be tested in the model. First, it can be assumed that irreversible damages would remain constant, whatever is the starting date of intervention. The advantage is that the expected time when the investment will take place can be directly deduced from the option-pricing approach:' Second, it is certainly more realistic to assume that the level of irreversible damages is correlated positively with the length of the waiting period. Postponing the intervention will therefore translate into a higher level of CO2 emission which, in turn, 4. Note that this is not always true for the individual standards deviations, since the total effective standard deviation is a quadratic function of the two standard deviations and the correlation, see the Appendix.

5. On this issue, see the recent paper by Chichilnisky and Heal (1993).

6. For a prof see World Bank (1991).

6
will incrcase the temperature and irreversible damages, shortening the waiting period in comparison to that suggested by equation (8). In that case, we will determine the optimal timing by simulating the Cline's proposal with different dates of intervention, starting from 1990 (as proposed by Cline). The optimal timing will be determined by the furst date when the benefit to cost ratio is greater tnan the critical ratio (Y > Y*). This exercise will be done in the next section.

In short, the model can thus be used (i) to determine if governments should intervene to reduce global warming; (ii) to examine the influencc of uncertainty on the process; (iii) to determine the optimal date of intervention; and (iv) to identify the optimal level of CO2 cutback to be implemented today.

# 3 An Application

The objective of this section is to apply the option-pricing model developed in the preceding section. In order to proceed with application we need two types of data:

* The estimated costs and benefits associated to the abatement in global warming, as well as the relevant discount rate. This data will be extracted from Cline's analysis.

* Variables relevan **t** o the option model, such as quantitative measures for the underlying uncertainty (on costs and benefits), specifically the sandard deviation of uncertain variables and the correlation between these variables.

7
# 3.1 The Basdine Scenario

The analysis presented by Cline (1992) is certainly the most thorough study of climate change and, therefore, it will be used as a reference for our baseline scenario in the absence of uncertainty. However, it is worth underscoring that two aspects from the original model have been modified: First, the expected increase in temperature from global warning will be explicitly linked to the CO2 emission rather than to be determined by a linear approximation between the long-term global warming and the current level of tempeature. Second, the ratio of unavoidable damages to total damages in the absence of intervention will not be fixed during the entire period, but it will vary over time in response to the variations in the stock of CO 2 .

## Expected Costs

In the recent literature, the cost of abaten"'nt polices are generally detennined by: (i) afforestation or diminishing deforestation; (ii) energy substitution -non-fossil fuel for fossil fuel energies; (ffl) non-energy inputs substitution --capital and labor for energy; (iv) change in product mix; (v) adaptive measures such as population migrations; (vi) 'climatic engineering' such as ocean fertilizaon.

Accordingly, Cline considers that the costs of abatement policy basically arise from the reduction in fossil fuel emissions due to output reduction (Q), the need tO set aside land for affoation (FA), and the need to curtail frontier agricultural land use and thereby carbon release from deforestation (FD). These costs are expanded by the proportion (w) to take into account likely costs associated with curtailing all greenhouses gases in a way commensurate with reducing carbon dioxide emissions. This cost is further increased by considering the portion of the cost that would have 8
gone into investment (x). Therefore, the costs of abatement polices at time t (in percent of World GDP) can be estimated by: _CD ' (Q FA, + D,) (I + w)x_ (9) _rGDP,_

# where GDP, is world GDP at time t.

Clearly, the cost associated with Q varies with the level of abatement as shown by Cline. This basic equation is defined as follows:<sup>7</sup> _(E,_ - _K_ - _'RF')_ - _Z[ 1[a_ f+ .y(t - 35)1 (10) _(E, - E4o_ with: Zo = percentage carbon reduction at zero cost (set a 22 percent). E'o = Base year deforestaion emissions (set a 1 billion ton of Carbon (GtC). o = Abatement cost parameter (set at 0.0678) y = Abatement cost parameter (set at -0.00039) K = target ceiling emission (set at 4 GtC) At= Level of global carbon emission at time t (baseline)

**RFA,** = Emission reduction from afforestation (set at 1.6 GtC per year, 1991-2020) RvPt = Emission reduction from lower deforestation (set at 0.7 GtC per year, 19912275)

FImally, the parameters w and x are defmed to be equal to 0.2 and 1.12 of total 7. Now that equation (10) is slightly different over the period _1990-2025_ (see Cline (1992), p. _282_ for _detils)._ 9
# costs respectively, foUowing the arguments proposed by Cline.

For the aggressive policy of 4 GtC of CO2 emissions annually proposed by Cline, the estimated costs for the period 1990-2275 are depicted in Figure I and in Table 1.' The estimates assume that a reduction of up to 22 percent of emissions can be achieved at zero cost, on the basis of the body of engineering estimates. The overall pattern that emerge is one in which there is a phase of initially low cost carbon reductions, followed by a period when these costs rise to a peak of 3.5 percent of World GDP, a level that then tapers off to some _2.5_ percent of GDP as the passage of time should permit the development of a wider range of technological alternatives.

It is worth underscoring that these projections are calculated on the basis of the costs measured by Cline (1992), but a considerable debate is taldng place currently in the literdture. For example, important additional damage estimates have been made by Titus (1992) and Fankhauser (1992), the fit author author arguing that potential damages in forest loss are much more important than estimated by Cline, while the second one have extended Cline's analysis by including a more extensive set of countries. The model can be easily extended to include alternative measures of damages costs.

## Expected Benefits

# Expected benefits are defined as the damages that can be avoided by a policy

### 8. The parmeter x is defined by Cline as follows:

x = (1 - **a1 )** + P k **_Tk_** with _-l_ the portion of the cost that would have gone into investment, and P*k a shadow price of capital to convert investment in consumption-equivalents.

9. The time horizon of 2275 is fixed by Cline, because at that time, it is believed that reources will be exhauted under a scemunio of high fossil fuel consumption.

10
or, in other terms, as the differences between the damages in the absence of intervention and the unavoidable damages. The furst requirement is therefore to detemine the damages of global warming in the absence of intervention. Damages from global warming are determined by the estimated increase in temperature; which largely depends on the so-called climate sensitivity parameter.

In expression (11) is specified the linkage between the increase in temperatue from global wanning and the level of CO2 emission. Two scenarios are successively examined (the central and the high cases) which correspond to an increase in temperature in the long term of about 10 C° and 18 C° respectively.'<sup>0</sup> The increase in the level of temperature is assumed to depend on (i) the degree of radiative force above industial level, which in tum is influenced by the level of CO2 concentration at time t, (ii) the relationship between the degree of radiative force and the increase in temperature, and (iii) the feed-back effect caused by water vapour, snow and ice albedo. In short, global warming is defined as follows: where _ZO_ is the initial atmospheric concentfation of carbon dioxide in 1990, E, the C02 emission at time t, X warming per unit of radiative forcing before utking account the feedback effect (set a 0.3), and B is the feedback multiplier (set at 1.9 in the central case and 3.4 in the high case). The values of these parameters are those used by Cline (1992).

The increase in temperature will produce world damages that are esfimated

> 10. The 'central casoe is in line with the IPCC estimates, while the 'high case' is relatively pesimistic.

11
on the basis of studies on the U.S. economy." The central estimate for economic damage from global warming is set by Cline at I percent of world GDP at benchmark of doubling CO2 concentration. Finally, the function relating damage to warming is assumed to be geometric with an exponent of 1.3. Therefore, non-discounted damages (in percent of world GDP) from global warming in the absence of intervention are given by: where do is the benchmark economic damage for carbon-dioxide-equivalent doubling (set at 1 and calibrated for a climate sensitivity of 2.5 C°), W, the projected temperatre at time t as defined by equation (I1), and y the exponent in the geometrical function (set a 1.3).

Equations (11) and (12) can be used to determine the costs of global warming in the absence of intervention --the accumulation of carbon dioxide leads to an increase in temperature which, in turn, produce damages to the world economy.

In Cline's approach, the benefits from an aggressive abatement policy are fixed at 80 percent of the costs of global wanning in the absence of intervention during the entire period. This fraction, equivalent to the unavoidable dLmages in the long term, overestirnates the benefits from intervention in the beginning of the period since the level of greenhouse gases is higher in 1990 (6.7 GtC) than the ceiling proposed by Cline (4 GtC). A more precise approach is here followed since the benefits are It. The general approach has been to analyze the different economic sectors affected by global warming. Some problems arise for agriculture, with the controversial so-called ferdlization effect, according to which CO% concentration may, up to a certain level, improve photosynthesis. Other major problems concem the monetary value of non-maket damages, such as helth effects, changing amenities, species extinction and social costs of migrtions due to sea-level rise. The indirect damages, due to greenhouse gases and conventional air pollutants should also be included in the estimates (Ayres and Walter, 1991).

12
measured as the difference between the costs of global warming in the absence of intervention Bij) and those if the global CO2 emission is limited at 4 GtC (B'). Thus: The benefits from intervention are further expanded to take into consideration the fact that some of these gains accrue to production going into investment (the parameter iq is set a 1.06 following Cline's analysis) and the benefit from reduction of the excess tax burden (T).

The evolution of the expected benefits of an aggressive abatement in global warming is described in Figure 1 and Table l. It is worth underscoring that the fraction of avoidable damages is not constant over time as originally assumed by Cline. As depicted in Figure 2, this ftaction is only 45-50 percent in the first decades and gradually increase up to 82 percent in 2275. Benefits of limiting global warming would be considerably higher in the long-term than in the short-term because of the importance of the inreversible accumulation ef CO2 and thus unavoidable damages in the first decades.

# Cost-Benefit Analysis

As discussed by Cline, the results of the cost-benefit analysis are greatly influenced by the discount rate (r) because of the long time horizon (up to 285 years). Although the is an ongoing debate on tfis issue, we remain attached to Cline's analysis by using a relatively low discount rate of 1.5 percent.<sup>12</sup> The rtio of 12. By incorporating the influence of the portion of resources diverted from capitl investment by applying a shadow **prie** on capital and **converting thes reurces to consmption equivalents. the overall** discount **te is close to** 2 percent. For an **extensive** discussion on **the issue** of **the** discount **rate, see for example,** Birddl **and** Steer (1993).

13
acualized benefits to costs is: <!-- Start of picture text --> b,<br>C+, (14)<br>(1 + r)'<br><!-- End of picture text -->

# with C, and b, defined in equations (9) and (13) respectively.

The results that emerge from the traditional cost-benefit analysis are that the aggressive policy should be rejected in the central case because the ratio Y is lower than 1 (Y=0.94). In contmst, in the high case, the aggressive abatement policy should be implemented since the discounted benefits are largely greater than the discounted costs of intervention (Y = 1.94).' Therefore, the results of the traditional cost-benefit analysis do not permit to support or to reject aggressive abatement policies. However, Cline concludes that these empirical results support his policy proposal to the extent that policy-makers are risk averse and apply a higher weight to the high case scenario than the central case scenario.

# 3.2 Parameters for the Option Pricing Model

As explained in the preceding sections, our objective is to introduce uncertainty in the analysis of global waming. The principal questions to be answered are: should we invest now in the aggressive abatement policy advocated by Cline or should we 13. These ratios slightly differ from those found by Cline because the tempeoates and so the benefits from intervention are not defined in the same ways. If we use the same approach than Cline, the benefit-cost ratios ae 0.77 and 1.59 for the central and high cases respectively, close to the results obtained by Cline (chapter 7, scearios I and 9).

14
wait, and if yes how long? Finally, what would be the appropriate policy to be implemented today?

There appears to be much more uncertainty about the benefits of global warming abatemen **t** han about costs in the literature (as reflected in TAbles 2 and 3). The major source of uncertainty regarding the benefits lies in the uncertainties and impondemble impacts of climate change (scientific uncertainty), but major doubts also remain on the magnitude of the damages, and their conversion in monetary values. The uncertainty about the costs of limiting global warming principally lies in the choice of instruments to be implemented (e.g. carbon tax, regulations).<sup>4</sup> To illustrate the degree of uncertainty, Tables 2 and 3 report the estimates of recent studies on the expected costs and benefits. The estimated costs of a 40 percent abatement policy from baseline in 2025 range from only 0.26 percent of World GDP for Edmonds-Bums to 3.52 percent for Whally-Wigle, while the estimatod benefits (of 1 5 eliminating the damages of CO2 doubling) vary between 0 and 5.5 percent of GDP. 1<sup>6</sup> The overall uncertainty of costs and benefits in 2025 is 74 percent and 107 percent respectively, equivalent to a standard deviation of about 3.2 percent and 3.8 percent annually.

The correlation coefficient between discounted benefits and costs of global warming also influences greatly the optimal timing of intervention --the critical ratio 14. See Nordhaus (1993). for a good summary of the uncertainties of limifing global warming.

15. Which is expected to occur as early as 2025 (IPCC, 1990).

16. Although Nordhaus (1993) reports that most studies give quite similar results and find damages rnging between _1.0-1.5%_ US 1988 GDP for a CO2 doubling and a survey of scientific and economic experts by Nordhaus (1993) shows that a 3 C increase of average temperature in 2090 would cost on average 1.8% of GDP, an order of value close to the preceding ones, the great dispersion of the answers, ranging from 0 to 5.5% of GDP, illuswrates the uncertainty surrounding these estimations.

15
(Y*) is a decreasing function of this coefficient. If benefits and costs are poorly correlated, the effect of uncertainty would increase since there will always be the possibility of having simultaneously higher than expected costs and lower than expected benefits. The correlation coefficient, calculated by the standard formula, equals 0.065 (see Table 4).

From equations (9) and (13), we infer that the annual average growth rate of benefits (G.) is 0.9 percent in the high case and 0.8 percent in the central case, while the annual growth rate of costs (GF) is 0.02 percent.17 Tbe discount rte (r) remains the one applied by Cline (1.5 percent).

All parameters are summarized in Table 4. The option-pricing model is successively applied to the centml and high cases. The introduction of uncertinty does not delay the investment decision in the high case scenario, but accentuated the non-profitability of policy intervention in the centrl case scenario. Below are some details.

High Case: In the face of uncertainty, the decision to proceed now remains optimal. However, as discussed below in detail, this result is quite sensitive to the volatility associated with the costs and benefits.

Central Case: Policy intervention is even less attractive, when analyzing it in the face of uncertainty. The optimal time of investment would be in about 133 years from now. This result may appear redundant, knowing the deterministic result, but it reveals option prices (i.e. the variable W in Table 4).

17. Defind as geometric average growth raztes per annum for the period 1990-2275.

16
# Sensitivity Analysis

The decision to invest is sensitive to the degree of uncertainty as depicted in Table S for the high and most pessimistic case of global warming. When the volatility of benefits and costs is lower than 6 percent annually, it remains optimal to invest now in aggressive policies against global warming. In contrast, these aggressive poLicie **s** hould be deferred for about 35 years with a volatility of 6 percent, 72 years with a volatility of 8 percent or 115 years with a volatility of 10 percent.

As explained earlier, the optimal date of intervention will depend on the evolution of irreversible damages during the waiting period. Following the assumption that the level of irreversible damages would remain the same as if the action was taken from the start, the optimal waiting period is defined by equation (8). The results from this assumption are summarized In Table _5_ (denoted by E(m)). However, if the irreversible damages increase over time, the waiting period will be shorter. To illustrate, assuming a volatility of 10 percent per year, the optimal delay would be about 115 years compared to 152 years with constant irreversible damages.

# Optimal Policy

In the face of uncertainty, the aggressive policy proposed by Cline may appear sub-optimal. The response could be to wait, as examined earlier, or, eventually, to proceed with a less aggressive policy. The second option is actualy recommended by Nordhaus (1993) who, based on the Dynamic Integrated Climate-Economy (DICE) model, proposes an initial abatement policy of 10-15 percent rather than 40 percent as proposed by Cline.<sup>18</sup> 18. Recently, Clinie (1993) has however shown that the DICE model can reproduce his results if the discount rte is appropriately choen.Aithough tde selection of the discount rate is fundamental for a period of time over 300 years, our objective remais to examme the impact of uncerainty.

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Using the approach developed in this paper, it is relatively easy to identify the optimal poLicy to be implemented today for a given degree of uncertainty. Notice that, for simplicity, we assume that the degree of uncertainty on costs and benefits are equivalent, so that the resulting impact on the option value is unambiguously positive as demonstrated in Appendix. The results of this exercise are presented in the last line of Table _5_ in tenns of CO2 ceiling. For example, the optimal ceiling would be only 6.4 GtC for a volatility of 10 percent, while it would decline to 4.8 GtC for a volatility of 6 percent. As expected, higher is the uncertainty around the cost and benefits, lower should the CO2 cutback.

Figure 3 depicts the optimal path of CO2 emission over the period 1990-2275 for different degree of uncertainty.'9 The optimal path is sensitive to the degree of uncertainty, specifically in the beginning of the period. The optimal path, when the uncertainty is around 8 percent annually, would cut emissions by 15-20 percent in the first decades from baseline, rising gradually to 80 percent in the next century. However, when the uncertainty rises to 10 percent, the CO2 cutback would be only 5-10 percent in the furst decades. These results can be compared with the 40 percent cutback recommended by Cline in the next few decades, and are close to the level proposed by Nordhaus and those recommended recently by the European Union.

# 4 Concluding Remarks

Uncertainty is an inherent phenomena in global warming issues, and thus it must be explicitly taken into account in the evaluation of policies. Although the major concern remains the uncertainty around the scientific evidence of climate changes, the economic analysis provides some guidance whether governments should intervene in

> **19.** The abatement **ratio** is defined as **the** CO2 **cutback** from **baseline.**

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the foreseen future.

On the basis of the option-valuation approach, this paper has examined the impact of uncertainty on the costs-benefits analysis. The major conclusions are the following:

* The aggressive proposal presented by Cline (1992) appears to be optimal for a relatively low degree of uncertainty around the costs and benefits of limiting global warming. This result is valid in the case where the increase in temperature in the long term is about 18 C° (high case scenario), and the discount rate is equal to 1.5.

* However, the action should be delayed if the uncertainty is higher than 6 percent per annum. The optimal date of intervention calculated in the paper, ranging from 35 years to 126 years from now, accounts for the possibility to accumulate future information and for the increase in irreversible damages during the waiting period compared to those which would have prevailed if the action was taken from the start.

* In case of relative high uncertainty, it may be optimal to implement today a less aggressive policy. If the Cline's proposal is to limit the level of CO2 emission to 4 GtC annually, a ceiling of 5.5-6.5 GtC --about the existing level in 1990- seems more appropriate for a level of uncertainty ranging around 8- 10 percent annually.

Finally, we would like to conclude that the approach followed in this paper could be imprved in numerous ways. As discussed earlier, uncertainty around cost and benefits is assumed to be exogenous and constant over time. It would be certainly more realistic to consider uncertainty as endogenous, varying, for example, with the increase in global warming or the magnitude of the policies to limit global carbon 19
emissions. Clearly, additional work is required in this area in buth an analytical and empirical perspectives.

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