A Long-term Permit Program for Long-term Climate Change Mitigation

Transcription

1 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg A Long-term Permit Program for Long-term Climate Change Mitigation Stephen C. Peck and Thomas J. Teisberg 12/05/97 revised starting 01/07/02 ending 05/22/02 The Greenhouse Issue Science Greenhouse gases such as carbon dioxide accumulate in the atmosphere as they are emitted from human activities such as combustion and are removed only slowly by natural processes. These gases, by trapping heat in the lower atmosphere, can cause global warming with uncertain and possibly undesirable consequences. The most significant man-made greenhouse gas is carbon dioxide. The preindustrial concentration of carbon dioxide is generally believed to have been 280 parts per million by volume (PPMV) and it is expected that it will reach 400 PPMV by The rate of removal of carbon dioxide from the atmosphere is just under 1% per annum i. The Problem It would be to the advantage of the world to decide on some reduction of carbon dioxide emissions relative to trend in order to mitigate changes in climate. Yet if only some nations act alone to reduce emissions, they pay all the costs and receive only some of the benefits. In order to receive benefits greater than their costs, all (major) emitting nations should participate in the agreement ii. Currently the European Union is pursuing the Kyoto Protocol for the period ; the EU nations intend to introduce and trade carbon permits to reduce the cost. The United States is relying on a policy of reducing its energy intensity and reviewing the success of the program and the outcome of the science of climate change in We suggest here an approach to the second and subsequent stages of the Kyoto agreement, which we believe to have several attractive features. We assume as the most likely result, that the OECD, the FSU and the US participate from 2013 onward and over the course of two decades, they persuade the developing nations to join - particularly the three most populous. Having only partial participation in 2013 should not be too expensive because in the program we suggest - a least cost program of emissions reductions designed to stay below a given concentration level, the market determined prices of emission permits do not become high until several decades have elapsed. Technical and behavioral mitigation options Carbon dioxide emissions worldwide are currently about 6 billion tons (by carbon mass) and are expected to grow over time to billion tons by 2100 due to world income growth and the larger amount of combustion which accompanies that growth. There are technical options available now and probably more in the future to reduce emissions. For instance, fuels with more hydrogen produce less

2 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg carbon dioxide on combustion, nuclear power plants produce no carbon dioxide from combustion, and it is possible, at a cost, to drive smaller cars, use more efficient light bulbs and so on. Current focus of the Policy Debate The UN Framework Convention on Climate Change has set an objective of stabilizing greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system. The Convention addresses the issue of keeping the concentration of the greenhouse gases below some critical level, recognizing the gases long atmospheric lifetimes, yet the policy debate around the first Kyoto commitment period has mostly been conducted in terms of the emissions flows that would be allowed by each nation over a period lasting five years and centered in 2010 iii. Such an approach can be very expensive, partly because the chosen emissions reductions may be arbitrary and second because the five year period is too short relative to the payoff periods of the capital investments needed to reduce emissions of carbon dioxide. Attention is now turning to the second Kyoto commitment period and it may be time to re-focus attention on the ceiling concentrations of the greenhouse gases. Principles underlying policy Principles are now stated for a world-wide agency to control emissions at least cost so as to not exceed a given carbon dioxide concentration in the atmosphere iv. The least-cost plan would call for a time trajectory of marginal costs for curtailing emissions of carbon dioxide over time, and households, companies and others would reduce their emissions so that the marginal cost of emissions reduction equals that trajectory. Later we describe a practical scheme which can be decentralized, but nevertheless captures most of the advantages of such an ideal conceptual scheme. Concentration maximum While not fully justified in cost benefit terms, a pragmatic policy is to keep below a given concentration target for carbon dioxide v. As a concrete but purely illustrative example, assume that a joint decision were reached by the signatory nations of the UN Framework Convention on Climate Change not to exceed a carbon dioxide concentration of 550 parts per million (PPMV), about twice the preindustrial concentration of 280 PPMV. Making assumptions about world income growth, interest rates, the cost and availability of mitigation options and the removal of carbon dioxide from the atmosphere, we and others have estimated that this ceiling will be reached in 2070 and that the marginal cost of emissions reduction (i.e. the cost of the most expensive reduction of carbon dioxide) will rise from $6.50 per ton of carbon in 2010 to $226 per ton in 2070 vi. This transition to a marginal cost of $226 is sufficient to cause a switch of the world economy over six decades from one based primarily on fuels that produce carbon dioxide when combusted, to one based primarily on energy producing methods which do not release free carbon dioxide to the atmosphere. Annual

3 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg carbon emissions in 2100 switch from about 20 billion tons to between 5 and 6 billion tons. Flexibility in space and time To achieve the least cost emission trajectory for staying below a concentration maximum, two conditions must apply, one relating to flexibility in space and the other flexibility in time. Flexibility in space is easy to understand. If the marginal cost of reducing a ton of carbon in the US is $10 and in France is $12, then $2 can be saved by making the reduction in the US. In an optimal program, we should be indifferent between reducing emissions in the US or France. So, to achieve flexibility in space, the marginal cost of emissions reductions at a point in time must be the same anywhere in the world. Flexibility in time is harder to comprehend vii. A reduction of 1.01 tons in 2010 and a reduction of 1 ton in 2011 are equivalent in terms of the effect on carbon dioxide concentration in the year 2070 because about 1% of the carbon dioxide is removed annually by natural processes. If the real interest rate is 5% per year, and the marginal cost of reducing a ton is $10 in 2010, then we will be indifferent, as of 2010, between saving 1.01 tons in 2010 and saving a 1 ton in 2011 only if the marginal cost in 2011 is given by the equation MC (2011) * 1 / 1.05 = 10 * 1.01 tons viii Multiplying both sides of the equation by 1.05: MC (2011) = 10 *1.01 *1.05= $ It follows that to achieve flexibility in time, emissions reductions should be timed so the marginal cost rises at 6 % annually: the sum of the real interest rate (5%) and the atmospheric depreciation rate of C0 2 (1%) Extrapolating back at 6% from the marginal cost of $226 per ton in 2070 to 2010 implies that the marginal cost of emissions reductions in 2010 would be about $6.50 per ton. This would raise, in 2010, the price of crude oil by about 80 cents per barrel, natural gas by 10 cents per thousand cubic feet and coal by $3.60 a short ton. Permit program to achieve a 550 PPMV concentration maximum A permit based program is now described which provides a significant role for the markets ix to stabilize carbon dioxide concentrations, and does so at the lowest cost. Permit specification A permit is invented that would allow the emission of one ton of carbon in the form of carbon dioxide in the year 2070 or the equivalent emission in any other year. For instance in 2069, one permit would allow the emission of 1.01 tons of

4 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg carbon, since.01 tons would be removed from the atmosphere by natural processes between 2069 and In general one permit would allow the emission of 1.01 raised to the power n where n is the number of years elapsing between the year of the emission and the year Thus one permit would allow the emission of 1.71 tons in 2010 ( ). Government s roles are 1) to distribute the permits, either by auction or gift, 2) to ensure that when emissions of carbon dioxide occur, the appropriate number of permits is surrendered to government by the emitter, and 3) when atmospheric carbon dioxide is captured, the appropriate amount of permits is provided by government to the sequesteror. Permit price and total value of all permits No more than 435 billion permits could be issued in order to stabilize atmospheric concentrations at 550 PPMV by 2070 x. Each permit would be a capital asset which could be sold or held. In order to induce the permit to be held from one year to the next, it would have to appreciate in value at the real interest rate. We have calculated that given the cost of mitigation options, and assuming a real interest rate of 5 % per annum, the permit price in 2010 would be $ As explained above, this would allow the emission of 1.71 tons of carbon and thus the marginal cost of emissions reduction would be $11.25 divided by 1.71 or $6.5 per ton in 2010, just the same as in the least cost program. Thus from 2010 to 2070, the permit price grows from $11.25 to $226 at 5% per annum while the marginal cost grows from $6.50 to $226 at a rate of 6% per annum. Note that although we have calculated the permit price in this illustration, the price, in reality, would be determined in the market. If a stock of permits adequate for 20 years is maintained, about 100 billion permits would be issued in With a permit market price $11.25, the market value of all permits issued in 2010 would be over $1 trillion. The maximum value of outstanding permits, over $13 trillion, would be achieved in The market value in 2070 should be close to zero because virtually all permits should have been used. After 2070, the world s emissions should match the removal of carbon from the atmosphere, between 5 and 6 billion tons a year - about 1% of the excess mass of carbon in the atmosphere (580 billion tons). Each country would reduce emissions such that the marginal cost ($226/ton) is equalized across space. Unwilling partners would be provided incentives to conform by foreign aid, reductions in tariff barriers etc. Reduction of the theoretical approach to practice. Suppose, for consideration of the greenhouse issue, we divide the world into four blocs; 1) the United States (US), 2) the other members of the OECD (OOECD), 3) the nations of the Former Soviet Union (FSU) and 4) the rest of the world (ROW) particularly including the countries with large and rapidly growing populations, Brazil, China and India.

5 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg Figure 1 (left hand panels) provides a summary of the carbon dioxide emissions of the OECD and the Rest of the World (ROW) under two conditions, first with no emissions stabilization, and then with stabilization at 550 PPMV. The right hand panels show the reduction in the number of permits (about 44 % of the 435 billion permits) in moving from the No Stabilization (625 billion permits) to the 550 PPMV (435 billion permits) case xi. How would this work? Let us suppose first that all four blocs agree to the second phase of the Kyoto agreement. The US, for instance would announce in 2003 that, while in the absence of controls it would use 125 billion permits by 2070, it will cut its cumulative emissions and use only 75 billion permits. Hopefully the OOECD, with its strong commitment to climate change mitigation, will make a similar commitment starting in 2010 xii.similarly the FSU having been willing to abide by the Kyoto Agreement should be willing to agree to this less burdensome agreement. And the major countries of the developing world also agree to restrict their cumulative emissions. Any bloc agreeing to the program is expected to enforce on their citizens the rule that if carbon dioxide is emitted the appropriate quantity of permits should be surrendered to the government. By 2010, approximately 100 billion permits, enough to last for about 20 years will be in private hands. The governments of the US, OOECD, FSU and ROW may achieve this by grandfathering some, and by auctioning other permits. As each year passes, one more year s worth of permits is transferred to private hands, so that there is always about 20 years worth of permits outstanding. This prevents any national government from changing the outstanding stock of permits without long and careful debate and agreement. The amount of permits allocated to the four blocs should be such that it is unlikely that any interregional trading will be needed. As a safety valve, if, during the 20 years, the price of US permits is consistently expected to be higher than the price of FSU permits, for instance, then a government to government agreement could be made to transfer some FSU permits to the US at a price intermediate between each bloc s prices. The FSU governments would have to agree to reduce emissions by that number of permits. Thus flexibility in space could be achieved approximately. Since this proposition of limited private international permit trade is so inconsistent with the Kyoto approach, it is worth spending some time on it in a more familiar context. Suppose that there are two nations, each with an electricity industry. One nation has 34,000 MW of coal capacity and the other has 32,000 MW of natural gas capacity; natural gas is assumed to be cheap, so the operating cost of coal and natural gas plants are the same. There is also unlimited geothermal capacity which produces no carbon dioxide but it is slightly more expensive than coal or gas generation. Since coal emits about twice as much carbon as gas, the output of carbon dioxide, measured as carbon, is 100 units (34 * 2 for coal plus 32 * 1 for gas). Suppose that the governments of the

6 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg two nations decide to cut back emissions of carbon (from electricity production) by 17 % to 83 units. To achieve this without trading, the nation with the gas plants should receive 32 permits and the nation with the coal plants should receive 51 permits. The industry composition after the permits have been allocated is about 25,000 MW of coal capacity and 9,000 MW of geothermal capacity in one nation and 32,000 MW of gas capacity in the other nation. Note that this allocation has nothing to do with equity but only with efficiency. If persuasion is needed for the nation producing electricity from coal, then the other nation could make a gift of grain for instance. The fundamental point is that permits need not be used for equity. Indeed if the international spatial market for permits is likely to be inefficient, as David Victor has argued, privately traded permits should not be used for accomplishing emissions reductions xiii. In contrast, within countries or blocs (e.g. US or EU) where the permit trading takes place under a unified legal system, using permits to achieve flexibility in space is to be encouraged. Next suppose that in 2010, only the US, OOECD and FSU commit to the Kyoto agreement for the second commitment period. They would state that they expect the ROW to commit to an efficient share of permits, to enter the climate stabilization program between 2020 and 2030, and to enforce on its countries citizens the obligation to surrender permits for carbon dioxide emissions. If the ROW does not look likely to conform, then the three participating blocs could offer inducements in the form of foreign aid, investment support etc. If the ROW gives no indication of joining the agreement by 2030, then the participants can ultimately pull out of the agreement, paradoxically by issuing more permits in each year so that, for the OECD, for instance, 250 billion permits are issued to cover the period between 2010 and In this case the permit price would be approximately zero.the decision of how hard to work to induce the nonparticpants to join and the number of permits wil be dependent on the scientific knowledge base. Suppose that the ROW does join the agreement in 2030 xiv. Then the governments of the ROW should issue approximately enough permits to last for 20 years and augment them annually. On a continuing basis there will be international review of the agreement as to how well it is working. If it appears that many nations are allowing emissions without CO2 permits, then the other members can decide to increase the rate of increase of their own permits or to increase the incentives for participation. After 2070, each bloc is issued the number of permits needed for efficient maintenance of the atmospheric concentration at 550 PPMV of carbon dioxide. Since an attempt is being made to stabilize emissions, it would be appropriate to allow industry and the public to maintain a constant stock of about five years worth of permits - 25 billion permits. The market value would be about $6 trillion. Again inter-bloc government exchange is kept as a safety valve in case the initial allocations were not the most efficient.

7 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg In the description above, we have assumed that each bloc implements the carbon dioxide cumulative constraint in the same way. But what is most important, of course, in this parallel but independent strategy is the commitment by each bloc not to exceed the use of a given cumulative number of permits over 60 years. The United States has strong institutions and a preference for market solutions and should, we believe, adopt the market based approach as described. The OOECD also has strong institutions, nascent markets for carbon permits but may be more inclined to command and control ; we believe this is likely to be overly expensive but if the OOECD governments and their citizens prefer that approach, it is their decision to make. The same applies to the FSU and ROW. So long as they meet their cumulative commitment, it is a matter of secondary importance to the US how they do so. In making their decision, they will presumably consider the fact that both command and control and the market approach require strong institutions. Extensions to other greenhouse gases and carbon sequestration activities The effect of delayed commitment to the climate agreement can be investigated xv. More complex models of the carbon cycle can be employed xvi. The approach can be used to achieve a balance between benefits and costs over time xvii. Other important greenhouse gases including methane and nitrous oxide can be accommodated in a similar framework in which the goal is to stay below a given global temperature elevation relative to pre-industrial levels in contrast to a given concentration level for each gas xviii.carbon sequestration activities can be accommodated. Providing incentives for R&D and for capital intensive technologies An attractive feature of our market based scheme is that the marginal cost of emissions reductions rises at 6% annually, thus doubling every 12 years. Such a rise is sure to attract significant R&D interest. One other key point is that for now, technologies which emit little C0 2 are generally capital intensive and some like nuclear power plants, can take close to a decade to complete. The financial risks associated with these capital intensive technologies are high and not easily diversifiable. If the price of oil and gas is low and stays low after the construction of these technologies, then the shareholders of companies that built them are likely to suffer a substantial capital loss. This fact obviously acts as a disincentive to the construction of long lived capital intensive technologies. This capital side irreversibility can be managed by setting aside some funds from the permit auctions to pay for capital losses incurred because the price of oil and gas remains low after the completion of construction of the capital intensive plants. It should be emphasized that these funds should not be used to repay the costs of construction programs that over-ran. Conclusion By the invention of an asset, the carbon (dioxide) permit, it has been shown how to craft a solution of the climate program that combines, in the most appropriate

8 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg way, the necessary functions of government - to persuade other nations to join and to police emissions - with self interested economic interests in the emissions and capital markets. This approach may be extended in a self evident manner to the other five greenhouse gases and to greenhouse gas capture activities as well as emission reduction. Authors Associations and Acknowledgements Peck, President Flèche, 3720 Redwood Circle, Palo Alto, Ca, (Peck_Stephen@msn.com) Teisberg, Teisberg Assoc, 1475 Ingleside Drive, Charlottesville, Va, 22901(TJTeisberg@compuserve.com) While the central concept of this paper, the evaporating permit is our original invention xix, we are grateful to many colleagues with whom we have shared ideas over the decade that we have worked on global climate change issues. We particularly acknowledge: Richard Richels and Hung-Po Chao for many insights they have shared with us over the years: Steve Wan, who helped some years ago with the Optimal Control approach: David Victor who has argued convincingly, in our opinion, that an international market in emissions permits will not work and who commented helpfully on this paper: Jesse Ausubel who has worked on these issues for many years xx : Kurt Yeager who suggested some expositional improvements: Dale Heydlauff who made us more aware of some political sensitivities: Larry Williams who has helped specifically with the carbon cycle: Bill Thompson and David Hone who made us aware of some of the European political constraints and sensitivities: Jae Edmonds, Jim Sweeney and John Weyant who have been consistently supportive and Darius Gaskins for his continued skepticism on permits. Finally our work would not have been possible without the continuing pioneering contributions, both conceptual and modeling of Alan Manne, Bill Nordhaus and Thomas Schelling. We take, as usual, complete responsibility for all errors and opinions.

9 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg Figure 1 Annual Emissions Rates and Permits by Region Billions tons/year Carbon Emissions - OECD No Control 550 PPM Decade Beginning Billions tons/year Carbon Emissions - ROW No Control 550 PPM Decade Beginning Billion Billion OECD Permits ROW Permits Unrestricted 550 PPM "Minus 5%" Unrestricted 550 PPM "Plus 5%" i Data for three key gases in 1995: [preindustrial concentration], {current concentration}, (annual rise), <e-folding rate>, /committed temperature increase per doubling of preindustrial concentration/ For C0 2 [280 PPMV], {358}, (1.5), <0.9%>, /3 0 C/. For N 2 0 [275 PPBV], {312}, (0.8), <0.7%>, / C/. For CH 4 [700 PPBV], {1720}, (10), <10%>, /0.3 0 C/. The best source for background information on climate change issues is the series of reports by the Intergovernmental Panel on Climate Change, e.g. Climate Change 1995: the Science of Climate Change, W. Houghton et al. eds. Cambridge University Press For purposes of expositional convenience, we use an e-folding rate of 1% for C0 2. ii Thomas C. Schelling, Some Economics of Global Warming, American Economic Review, Vol. 82, No. 1 pp. 1-14, March iii In the Kyoto Protocol, a number of developed nations agreed to reduce emissions of a basket of six greenhouse gases 5% below 1990 levels between

10 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg 2008 and Such a reduction is equivalent to a fall of approximately 30% relative to 2010 emissions of the gases. iv Wigley, T. M. L., R. G. Richels and J. A. Edmonds, Economic and Environmental Choices in the Stabilization of Atmospheric C02 Concentrations, Nature, Vol. 379, No. 18, January v Such an approach was first described in Kosobud R.F., T.A. Daly, D.W. South and K.G. Quinn, Tradable Cumulative CO 2 Permits and Global Warming Control, The Energy Journal, Vol. 15, No. 2, 1994, pp vi The computational results in this note were derived from Peck S.C. and T. J. Teisberg, A Property Rights Approach to Climate Change Mitigation, Working Paper, EPRI, November This study exercised our Carbon Emissions Trajectory Assessment Model (CETA).Its first publication was Peck S.C. and T. J. Teisberg, CETA: A Model for Carbon Emissions Trajectory Assessment, The Energy Journal, Vol. 13, No. 1, 1992, pp Studies of the Stanford Energy Modeling Forum, which have applied many models to the climate issue, give similar results; see for instance D. W. Gaskins and J. Weyant, Model Comparisons of the Costs of Reducing C0 2 Emissions, American Economic Review, Vol. 83, No. 2, May vii Nordhaus, William D., How Fast Shall We Graze the Global Commons? American Economic Review, 5, 1982 pp was an early analysis of this issue. viii The units in the equation work as follows: {2011$/ton}*{tons}/ {2011$ / 2010$} = {2010 $/ton}*{tons} ix We refer to private markets because actions of households and firms respond to the prices of carbon permits, and actions of private individuals in capital markets set the price of permits. x In 2010, the atmospheric concentration of carbon is about 400 PPMV. This is 120 PPMV higher than the preindustrial level. This excess will have depreciated to 70 PPMV by Thus in 2070, we have room for ( ) or 200 PPMV. This is equivalent to 425 billion tons in 2070 or 425 billion permits. The difference between this figure and that in the text of 435 is due to some second order considerations in the model. xi Also shown is a further illustrative transfer of 5% of the outstanding permits that might be needed to induce the ROW to join the climate stabilization agreement. We discourage the use of permits as inducements because an international market to monetize these permits is unlikely to work well.

11 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg xii We split the OECD total in half because in 1990 the US annual emissions of carbon dioxide approximately equaled those of OOECD. To determine the efficient split (the appropriate criterion) would take further analysis. xiii David G. Victor, The Collapse of the Kyoto Protocol and the Struggle to Slow Global Warming, A Council on Foreign Relations Book, Princeton University Press, 2001 xiv By , with a large enough R&D funding commitment, it might conceivably be possible to measure fluxes of the major greenhouse gases from space based satellites in order to detect if a party to the climate convention is emitting more than its quantity of surrendered permits. The next four footnotes are not central to the exposition and are added for those with some mathematical interest. All results were derived by inspection of the first order conditions of the Hamiltonian function of Optimal Control Theory. xv In this case: C 1 {X 1 (t), t} = M (0)*exp{(r + d)*t} for 0 < t < T e. This equation means that the marginal cost C 1 {X 1 (t), t} of the participating region, which depends on the emissions reduction X 1 (t) and time t, increases from the initial level of marginal cost M (0) at an exponential rate equal to the sum of the real rate of interest r and the rate of depreciation of carbon from the atmosphere d, until atmospheric equilibrium at the ceiling level has been established at time T e. M (0) is endogenously determined by the constraint that the atmospheric concentration should not exceed some ceiling level, 550 PPMV in our case. The higher the concentration ceiling, the lower will be M (0). For the region that initially does not participate until time T i, the two equations below mean that the marginal cost, dependent on its emissions reduction X 2 (t) = 0 and time t, stays at 0 until time T i, then jumps to the same level as for region 1. C 2 {0, t} = 0 for 0 < t < T i C 2 {X 2 (t),t} = M(0)*exp{(r + d)*t} for T i < t < T e The lack of institutions that prevents region 2 from participating until T i clearly raises the cost of keeping under the ceiling, since at year T i -1, region 2 could reduce emissions by (at least) one unit at cost zero and increase emissions in year T i by exp (-d) for a present value savings as of year T i -1 of M (0)*exp {(r + d)*(t i -1)}

12 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg xvi In this case: C{X(t),t} = w*m 1 (0)*{exp{(r + d 1 )*t} + (1 - w)*m 2 (0)*{exp{(r + d 2 )*t} for 0 < t< T e When a ton of carbon dioxide enters the atmosphere, a fraction, w, enters pool 1 with a rate of depreciation d 1 and the remainder, (1 w) enters pool 2 with a rate of depreciation d 2. The equation above implies that marginal cost of emissions reduction C{X(t),t} is simply equal to the weighted sum of the marginal cost of the two pools at time 0 growing at different rates, each equal to the real rate of interest and the relevant rate of depreciation. The marginal costs at time zero, M 1 (0) and M 2 (0) may be determined by the fact that the amount of carbon in the atmosphere should not exceed some level. Two permits would be created, one associated with each pool. The first permit would evaporate at a rate d 1 and the second at a rate d 2. When a ton of carbon was emitted, a surrender would be made of w of permit 1 and (1- w) of permit 2. xvii To understand the workings of an intertemporal program that balances the costs and benefits of control, it is best first to derive the intermediate result that the rate of permit price appreciation equals the rate of marginal cost appreciation less the rate of depreciation. The relationship between the marginal cost and the permit price is (T t) M(t) = P(t)/(1+d) So, ln (M(t)} = ln{p(t)} - (T t) * ln(1+ d), where ln represents log to the base e Differentiating with respect to time yields; {1/M(t)} d{m(t)/dt} = {1/P(t)} d{p(t)/dt} + d, since, ln (1 + d) = d for small d. This, on moving d to the other side of the equation is the desired result. Now it may be shown that in an intertemporal program that balances benefits and costs, the real rate of return is made up of three components, as in the equation immediately below. The first is the rate of appreciation of marginal cost, the second is (minus) the rate of depreciation of the carbon permit (the sum of these two is the rate of appreciation of the permit as shown just above) and the third (positive) may be interpreted as a dividend yield to be paid by the issuing government to the permit holder. This dividend is made up of the marginal damage due to the carbon dioxide stock in the atmosphere (above preindustrial levels) divided by the marginal cost. Assume that the damage is a power function

13 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg (power > 1) of the atmospheric C0 2. Initially the marginal damage will probably be small so that the marginal cost of control appreciates at the sum of the rate of depreciation and the real rate of interest. But as the stock of carbon dioxide in the atmosphere becomes large, the marginal damage term will be larger and the appreciation rate of the marginal cost will become smaller. This is in contrast to the concentration stabilization examples where marginal costs continue to appreciate at the sum of the real rate of interest and the rate of depreciation of carbon dioxide from the atmosphere. The effect of adding the dividend is to raise the initial marginal cost relative to a concentration stabilization case. We have noted this behavior in exercises with CETA where a cost benefit analysis will cause the atmosphere to reach some steady state in concentration. We then exercise the model to find the least cost approach to reach that concentration level. Generally the marginal cost path for the benefit/cost case lies above that for the least cost concentration approach for the first portion of the program. If benefit/cost balancing were to be done in practice, the government would compute the marginal damage every half decade, say, and change its dividend payment accordingly. Proceeds from auctioning some of the permits could be used to pay the dividends. r = {1/M(t)}*{d{M(t)}/dt} - d + [df{g(t), t}/d{g(t)}]/ M(t) xviii In this case we consider two greenhouse gases: gas 1, carbon dioxide and gas 2, nitrous oxide, and we must expand our criterion from keeping below a ceiling concentration of one gas to keeping below a given rise in global temperature relative to pre-industrial levels. Three more parameters are needed. The first two, a i, i=1, 2 represent the effect of an increase in concentration of each gas on the committed temperature rise. The third parameter is f. It represents the rate at which actual global temperature catches up with committed temperature. (It is usually assumed that 2% of the gap is made up each year, hence f=0.02) Three permits are needed therefore, one for carbon dioxide, one for nitrous oxide and one for global temperature. The behavior of the marginal cost of control of each gas is: C i {X i (t),t} = [M i (0) {a i * (f / (d i f))}*m(0)] *{exp{(r + d i )*t} + {a i * (f / (d i f))}* m(0)*{exp{(r + f)*t} for 0 < t < T e, i = 1,2 Suppose this equation represented the marginal cost for N 2 0. Then the marginal cost of control would equal an initial expression representing the marginal cost of the gas stock. This term increases at the sum of the real interest rate (5%) and the rate of atmospheric depreciation (0.7% annually for N 2 0). This rate of marginal cost increase (5.7%) can be achieved by issuing permits that evaporate by 0.7% annually. The second term has to do with the stock of temperature. It

14 Printed 5/27/ Stephen C. Peck and Thomas J. Teisberg appreciates at the sum of the real rate of interest (5%) and the lag of actual temperature behind committed temperature (2%). Such a rate of marginal cost increase (7%) can be achieved by issuing permits whose purchasing power evaporates at 2% annually. Note that C i {X i (0), 0} = M i (0). The values M 1 (0), M 2 (0), and m(0) are endogenous and can he solved by consideration of the temperature limit and certain other conditions. If the equation above were to refer to methane, the first exponential term would be exp {( )*t}. When t = 10, this equals 4.5 and when t = 20, it equals This means that initially the first expression in the methane equation will be small. For a numerical optimization approach to this problem, see: A.S. Manne and R. G. Richels, An Alternative Approach to Estimating Tradeoffs among Greenhouse Gases, Nature, Vol. 4. No. 10, 4/5/01. xix There is an interesting analogy between the evaporating permit and work done by Chao and Peck on electricity transmission pricing. See Chao, Hung-Po and Stephen Peck, A Market Mechanism for Electric Power Transmission. Journal of Regulatory Economics, Vol. 10, No. 1, 1996, pp In that work, the market would produce optimal results only if the model s power flow rules closely replicated the behavior of the power system. This is true here also where the evaporating permit replicates the gas s behavior. xx Jesse H. Ausubel, Does Climate Still Matter? Nature, 25 April 1991, 350,