JOURNAL OF ENVIRONMENTAL ECONOMICS AND MANAGEMENT ARTICLE NO.
31, 1]18 Ž1996.
0028
Learning and Stock Effects in Environmental Regulation: The Case of Greenhouse Gas Emissions CHARLES D. KOLSTADU Department of Economics, Uni¨ ersity of California, Santa Barbara, California 93106-9210 Received March 24, 1994; revised April 14, 1995 This paper concerns the optimal regulation of greenhouse gases that lead to global climate change. In particular, we focus on uncertainty and learning Žwhich, over time, resolves uncertainty .. We present an empirical stochastic model of climate]economy interactions and present results on the tension between postponing control until more is known vs acting now before irreversible climate change takes place. Uncertainty in our model is in the damage caused by global warming. The results suggest that a temporary carbon tax may dominate a permanent one because a temporary tax may induce increased flexibility. Q 1996 Academic Press, Inc.
I. INTRODUCTION Uncertainty is a dominant characteristic of environmental externalities, including the accumulation of greenhouse gases leading to climate change. We understand well neither the effects of climate change nor the costs of controlling greenhouse gases. This is one reason considerable sums are expended in trying to better understand this problem. An additional factor frequently comes into play having to do with the cumulative or stock effects of greenhouse gases. It is not the emissions of greenhouse gases that directly cause adverse effects; rather it is the stock of these gases that may lead to climate change and these stocks change slowly with a great deal of inertia. The process of emitting is not readily reversible. These two aspects of the problem}stock effects and uncertainty }lead to a tension between instituting control and delaying control.1 Some in society will desire control of greenhouse gases before climate change is well understood, citing the irreversible nature of additions to the global stock of greenhouse gases. Others in society may urge delaying control until the problem is clearly delineated. If, ex post, the problem turns out to be less severe than expected then those urging delay will have been proved correct Ž ex post .. If on the other hand, the problem turns out to be more severe than expected, then delay can be very costly indeed. Irreversibilities in climate change have many facets; stock effects are one type of irreversibility. At the simplest level, some investments in controls on greenhouse * Research supported in part by a grant from the Research Board of the University of Illinois, by NSF Grant SBR-94-96303, by USAID cooperative agreement DHR-5555-A-00-1086, and by DOE Grant DE-FG03-94ER61944. Comments and suggestions by John Braden, Lars Mathiesen, Michael Schlesinger, two anonymous referees and an associate editor have been appreciated. I am grateful to Prof. William Nordhaus for providing an early version of the ‘DICE’ model. 1 There are other examples with these basic characteristics: hazardous wastes and groundwater, acid rain, species extinction, pesticide accumulation, and the list could go on. 1 0095-0696r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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CHARLES D. KOLSTAD
gas emissions are irreversible}one cannot uninvest in a CO 2 scrubber. Environmental irreversibilities are more multidimensional. On the one hand, the fact that CO 2 emissions contribute to the long-lived CO 2 stock and cannot be ‘‘unemitted’’ is an irreversibility analogous to the capital irreversibility. However, there may be environmental effects triggered by a buildup of CO 2 that are irreversible: an ice age, a shift in the Gulf Stream, or the extinction of a species. The focus of this paper is on stock-type irreversibilities. Uncertainty gives rise to two quite different issues, often confused. One relates to risk aversion. It would appear that the disutility from low-probability severe climate damage may not be equivalent to the disutility from the certain wasting of a corresponding amount of control capital; thus a risk averse decision maker might find it optimal to ‘‘bias’’ control decisions toward over-control, relative to the deterministic case. In this case, the fact that one cannot uninvest control capital or actively remove carbon from the atmosphere is irrelevant to the optimal regulatory strategy. A second and quite distinct issue relates to uncertainty where that uncertainty is being resolved over time; i.e., information is being acquired over time. Since control decisions are not made at one time for all future times, but rather sequentially over time, today’s decision may well be influenced by learning. In this case, stock effects, either environmental or in control capital investment, may significantly affect the optimal level of control today. The literature on irreversibilities tells us that with learning, we should avoid decisions that restrict future options. This paper concerns the latter issue, one of the most fundamental questions in the climate changergreenhouse gas control policy arena: what delay or acceleration of the generation of greenhouse gases should be pursued when uncertainty exists and learning is taking place? Thus this paper seeks to determine how the fact that we are learning about climate change influences our actions today to control greenhouse gases. In our application, there is uncertainty on the damage from climate change. While there has been some work related to this question w29, 34, 35, 37, 38, 41, 42x, explicit treatment of the learning process has yet to appear in the empirical literature on climate change.2 Our approach to the problem is to adapt a simple optimal growth economy-climate model w35, 37x to include uncertainty and learning. There are two major results of this paper. One is that we are unable to find any significant stock effect associated with greenhouse gas accumulation, where by stock effect we mean an effect associated with an inability to reduce the stock of greenhouse gas. The rate of change of the climate is just too slow. In retrospect, should we turn out to be emitting slightly too much today, then that can be corrected in the future by emitting slightly too little. Thus if greenhouse gas emission control decisions are perfectly reversible, then there should be no bias in greenhouse gas control, upward or downward, relative to the deterministic case. In other words, uncertainty and learning are second order effects. A second result is that there is a modest stock effect associated with sunk control capital. In other words, when control capital investment is not reversible, then the capital stock effect appears to be stronger than the environmental stock effect. 2 Hammitt et al. w16x and Manne and Richels w28, 29x have examined the costs of delaying action until more is known.
THE OPTIMAL REGULATION OF GREENHOUSE GASES
3
Thus we should bias emission control downward, relative to the deterministic case. This might suggest that we should do little to control greenhouse gases. While this is one interpretation, another interpretation is that greenhouse policies that result in re¨ ersible actions are to be preferred Žall other things being equal. to those involving capital investments that are long-lived and not easily reversed. For example, a permanent carbon tax may be dominated by a temporary carbon tax. The next section of the paper reviews some important contributions to the theoretical literature on learning, as well as existing empirical analyses of climate change and learning. The subsequent section presents our model of optimal regulation. We then examine the case of uncertainty in the disutility of pollution and, finally, the results. II. BACKGROUND The basic issues here are Ža. how stock effects determine current period emissions, Žb. how optimal emission policies for greenhouse gas can be computed, and Žc. how learning can be represented in a model of pollution accumulation in an economy. The first two of these issues are considered in this section. We defer learning to the subsequent section. A. Irre¨ ersibilities and Stock Externalities A major literature has developed in the area of investment under uncertainty in the presence of externalities. Arrow and Fisher w1x initiated much of the work in this area by focusing on a two period model with uncertainty about the benefits of an environmental asset that is to be exploited Že.g., a canyon flooded to make electricity .. With some uncertainty resolved between the two periods and the impossibility of undoing development of the environmental asset, it turns out to be optimal to bias development in favor of preservation of the environmental asset. Henry w17x published similar results. In essence, taking an irreversible action has a cost in terms of reducing the value of information. Arrow and Fisher w1x introduced the notion of quasi-option value, the value of the information gained by waiting before exploiting the environmental asset. Since then, there has been considerable literature on irreversibilities and on quasi-option value Že.g., see w3, 10, 11, 13, 31x.. Of course there is also a large literature in finance on option value. In particular, a number of recent papers concern the optimal timing of capital investments Že.g., oil field development. when learning is taking place Že.g., oil field exploration.; see Paddock et al. w40x. Another related literature, primarily from the early 1970s, concerns optimal growth in the presence of environmental externalities, particularly stock externalities. This was a natural extension of the optimal growth models that were popular in the 1960s and early 1970s Žand earlier}see w46x.. An important and characteristic paper in this genre is that of Keeler et al. w19x. In that paper a simple optimal growth model is posited where utility is a function of consumption and a stock of pollution. Optimal paths for accumulation of capital and pollution are developed for several different types of pollution control. Other papers of this type include w6, 12, 44, 45, 48x. Cropper w4x also considers such a model of optimal growth but focuses on catastrophic environmental effects}the ultimate in irreversibilities. See also Viscusi and Zeckhauser w51x.
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CHARLES D. KOLSTAD
The general problem we are examining is one where a regulator must make a sequence of decisions, acquiring information between each decision. This is not a new problem. In fact, one of the first papers illuminating this question is Simon w47x, who showed that if the objective is quadratic, uncertainty is Gaussian and the problem is unconstrained, then uncertainty andror learning are irrelevant to today’s decision. Malinvaud w25x generalized this somewhat, showing that if one allows any well-behaved objective but requires uncertainty to be small, then the same result emerges. In both of these cases, irreversibilities in terms of today’s actions constraining tomorrow’s opportunities are not allowed. Freixas and Laffont w14x showed that under very special conditions, if today’s actions constrain tomorrow’s opportunities, then there should be a bias in today’s decisions when learning is occurring, relative to the no-learning case. The most general theory was developed by Epstein w9x. His result was that curvature of benefits and costs determine the sign of the bias and that learning can induce a bias in the absence of an irreversibility. His result has been applied in the climate change context by Ulph and Ulph w50x and Kolstad w21x. B. Climate]Economy Models and Learning Economic models have played a critical role in the formulation of environmental policy in the United States over the past three decades. The main function of these models has been to simulate the economy’s response to particular environmental regulations. In the arena of climate change policy, a number of economic models have been developed, particularly over the last 5 years. We should acknowledge that one of the first economics papers on the subject of global warming was Nordhaus w32x and the first models were developed by Nordhaus and Yohe w33x and Edmonds and Reilly w8x. We refer the reader to the survey by Weyant w52x for a comprehensive discussion of the applied economic models for examining climate policy.3 We focus here on several papers that have addressed the specific question of the effect of uncertainty and learning on climate change policy. Peck et al. w42x was one of the first papers to explicitly examine learning in the context of climate change. However, analysis of this issue within the context of an empirical model had to wait for the development of the Peck and Teisberg w41, 43x model of climate and the economy. In w42x, the model is used to compute the value of information on climate uncertainty. They find particularly valuable information about the warming rate and damages, although that value is not much diminished if the information is acquired as late as 50 years from the present. In other words, there is no rush to resolve uncertainty. Hammitt et al. w16, 24x have conducted analyses of the optimal sequential decision strategy when there is uncertainty about climate sensitivity and the target for an equilibrium temperature rise, uncertainty which is assumed to be resolved in the next decade. Their model is not an optimal growth model but rather a simple Because the field is changing so rapidly, even the Weyant w52x survey misses some work in this area. In particular, Nordhaus w34, 37, 38x has produced several models of the climate and economy, evolving into the DICE model}Dynamic Integrated Climate]Economy Model. Furthermore, the Global 2100 model of Manne and Richels w29x has recently metamorphosed into a model Ž‘‘MERGE’’. with more explicit accounting of damages w27x. 3
THE OPTIMAL REGULATION OF GREENHOUSE GASES
5
decision analytic framework. In w16x, the authors conclude that moderate emission controls should be pursued unless climate sensitivity is high while the target for emission increases is low. In w24x, the model is modified to include abrupt climate changes; the conclusions remain qualitatively valid. Manne and Richels w29x explicitly examine how an optimal decision Žwithin an optimal growth framework. will be changed by the acquisition of information. They focus on the resolution of uncertainty about climate damage. Although the primary contribution of their work is to establish a framework for analyzing this issue, in that their results are highly dependent on specific assumptions, they do demonstrate that there can be a large payoff from improved information. Furthermore, their results suggest that information acquisition is preferred to precautionary emissions reductions. III. A STOCHASTIC MODEL WITH LEARNING In this section we will introduce a stochastic, discrete time optimal growth model in the spirit of Ramsey w46x. What makes the problem more complex is that learning is a stochastic process whereby as every decade passes, more is learned, although we do not know ex ante what that will be. Thus the state of knowledge evolves as a stochastic process. We first introduce the optimal growth model and then incorporate a specific type of learning into the model. A. A Stochastic Growth Model In this section we present a general model of the dynamic evolution of an economy, incorporating emission control, pollution accumulation, and pollution damage. To a large extent it is a standard optimal growth model, although some aspects having to do with the climate are nonstandard. It is based and draws heavily on the climate]economy model of Professor William Nordhaus w35]38x. His model is deterministic however, and our model is stochastic. The model is not regionally differentiated and involves the maximization of the worldwide net present value of expected per capita utility Žfor a representative consumer., summed over the population. Utility is enhanced by consumption and depressed by pollution damage. Output can be channeled to consumption, emission control, or investment. Knowledge is represented by the probability vector on states of the world and that knowledge follows a random walk through time, associated with learning. In other words, the probability vector evolves over time depending on the uncertain outcome of learning. There are two ways to represent a problem such as this. One is to use dynamic programming, solving the Bellman equation w49x. The other is to formulate a stochastic programming problem, which is our approach. Essentially, one must look at all possible trajectories learning might take in the future and condition all variables on the learning that has taken place up to the point in time where the variable’s value is determined. It is standard w7x to write the variables as functions of histories of the stochastic process; i.e., the realization of the random variable up to a point in time. Thus assume that there are several states of the world Ž s . with knowledge represented by a probability vector on those states. The variable h t is the history of
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CHARLES D. KOLSTAD
learning and we will define precisely how knowledge evolves with learning. The model follows with the variables and parameters defined in Table I. Maximize the expected net present value of utility: max I, E
Ý Ž 1 q r . yt Ý m Ž h t , t . Ý ps Ž h t , t . u c Ž h t , t . , d Ž h t , t . , s L Ž t . t
Ž 1.
s
ht
s.t.
I Ž t , h t . , E Ž t , h t . G 0,
Ž 2a .
given the consumption identity c Ž t , h t . s Y Ž t , h t . y I Ž t , h t . rL Ž t . ;
Ž 2b .
the production function Y Ž t , ht . s f K Ž t , ht . , LŽ t . , EŽ t , ht . , t ;
Ž 2c .
damage from climate change d Ž t , h t . s g T Ž t , h t . , Y Ž t , h t . rL Ž t . ;
Ž 2d.
sunk nature of abatement
Ž 1 y dE . s Ž t . Y Ž h t , t . y E Ž h t , t . F s Ž t q 1 . Y Ž h tq1 , t q 1 . y E Ž h tq1 , t q 1 . ;
Ž 2e .
TABLE I Model Variables and Parameters I E K M T O c d Y r h dK dM dE L s t s b ps m l
Investment Žtrillion $ per decade. Emissions of greenhouse gases Žgigatons per decade. Capital stock Žtrillion $. Stock of greenhouse gases Žgigatons. Mean atmosphere temperature Ž8C relative to base. Mean deep ocean temperature Ž8C relative to base. Per capita consumption Žthousands of $. Per capita climate damage Žthousands of $. Gross output of goods and services Žtrillion $ per decade. Pure social rate of time preference Ž0.01 or 0.03 per annum. History of learning Capital depreciation rate Ž0.35 per decade. Greenhouse gas stock decay rate Ž0.0833 per decade. Emission control depreciation rate Žparameter. populationrlabor supply Žbillions, L ) 0. Greenhouse gas emissions]output ratio, uncontrolled Ž s˙ - 0. Timertechnology Ždecades. State of the world Greenhouse gas emission factor Ž0.64. Probability of state s given learning history h t : Ýps s 1 Probability of learning history h t Rate of learning Ž0 s no learning; 1 s complete learning in one period..
Žcontrol. Žcontrol. Žstate. Žstate. Žstate. Žstate.
THE OPTIMAL REGULATION OF GREENHOUSE GASES
7
capital accumulation K Ž t q 1, h tq1 . s Ž 1 y d K . K Ž t , h t . q I Ž t , h t . ;
Ž 2f .
greenhouse gas accumulation M Ž t q 1, h tq1 . s Ž 1 y d M . M Ž t , h t . q b E Ž t , h t . ;
Ž 3a .
atmospheric temperature evolution T Ž t q 1, h tq1 . s s T Ž t , h t . , M Ž t , h t . , O Ž t , h t . ;
Ž 3b .
ocean temperature evolution O Ž t q 1, h tq1 . s r T Ž t , h t . , O Ž t , h t . .
Ž 3c .
Equations Ž1. ] Ž2. constitute the basic economic model and Eqs. Ž3. describe the evolution of the climate. Equation Ž2e. captures the extent to which emission control decisions are irreversible. Uncontrolled emissions are s Y. We assume investment in emission control capital is proportional to controlled emissions, s Y y E. Control capital in period t q 1 is bounded from below by the previous period’s control capital Žrepresented by controlled emissions., depreciated by 1 y d E . Note that if d E is zero, Eq. Ž2e. requires that next period’s controlled emissions must be at least as great as this period’s controlled emissions; in other words, you cannot reverse a decision to control emissions. If d E is equal to 1, Eq. Ž2e. simply states that controlled emissions cannot be negative, a rather innocuous assumption. The intermediate territory with d E between zero and one corresponds to partial reversibility of emission control decisions. Analogous to d E is d M , found in Eq. Ž3a.. This variable controls the persistence of the stock of CO 2 : a small value of d M corresponds to a very pronounced stock effect. A value of d M close to 1 implies that the pollutant is not a stock pollutant but rather a flow that does not accumulate. The links between the economic model and the climate are E and T. Emissions in Eq. Ž2c. are good in that they allow increased output. However, emissions Ž E . increase CO 2 levels Ž M . which increase temperature ŽT . which causes damage Ž d . which yields disutility ŽEq. Ž2d... The goal is to choose the investment path and emission path that maximize expected utility. The specific functional forms of the functions f, g, s, and r in Eqs. Ž2c., Ž2d., Ž3b., and Ž3c. are described in the Appendix. We note here that environment damage is quadratic in the temperature change. Damage is of course reversible in the stock of pollution but much less so with respect to the flow of pollution because of the difficulty in quickly changing the stock. B. Learning There are three basic types of learning which are potentially applicable to global warming. One is active learning whereby observations on the state of the economyrclimate convey information about uncertainty. Thus by perturbing emissions, one can obtain information about uncertain parameters Že.g., see w2x or, for an application to climate change, w5x.. A second type of learning is purchased learning whereby knowledge is purchased and the amount of knowledge purchased ŽR & D
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CHARLES D. KOLSTAD
expenditures. depends on its cost and benefits Žsee w15x.. A third type of learning can be called autonomous learning where the mere passage of time reduces uncertainty. It is this third type of learning that we examine in this paper. 1. Information structures. The typical approach to including autonomous learning in models of irreversibility is to posit a two or three period model where uncertainty changes from one period to the next. Miller and Lad w31x use a two period model with an ex ante probability distribution on period i benefits Ž bi . of f Ž b1 , b 2 .. After observing period one benefits, the ex post marginal distribution is obtained: f Ž b1 , b 2 < b1 .. While this is clearly learning, we need a way to parameterize the rate of learning so that the effects of the rate of learning can be deduced. Jones and Ostroy w18x, Olson w39x, and Marshak and Miyasawa w30x provide such a framework through the concept of an ordering on information structures. Start with a set of possible states of nature and a probability vector associated with those states of nature being realized. Add to this the receipt of an informative message, and a vector of probabilities of receiving specific messages. The information in the message is a conditional probability on states of nature. An information structure consists of a prior on states of nature, a vector of probabilities of receiving specific messages, and, for each message, and ex post probability of states of nature. Of two information structures with the same prior on states of nature, the one that has the greater variability in terms of possible posteriors is viewed as being ‘‘more informative.’’ This is equivalent to the more informative structure yielding a higher attainable expected utility when the consumption bundle depends on the state of nature Žmore flexibility can only be advantageous w18x.. Thus if two learning processes are associated with two comparable information structures, then the structure that is more informative corresponds to greater learning. To quantify this concept of learning further, suppose there is a set of possible states of nature, indexed by s s 1, . . . , S. Furthermore, suppose there is a finite set, Y, of possible ‘‘messages’’ containing information on the state of nature. Suppose the prior on receiving particular messages is q Ždimension equal to the size of Y . and the conditional probability on states of nature Žafter the message ye Y has been received. is p Ž y .. We use the term ‘‘prior’’ to refer to a probability distribution on states, before the message is received and ‘‘posterior’’ to refer to distributions on states of nature after a message has been received. Let P be a matrix with columns consisting of p Ž y . with a different column for each y. Thus P has S rows and the same number of columns as members of Y. Ž P, q . is an information structure. A first goal is to develop an economically relevant ordering on information structures. A standard definition of the comparative value of information is provided by w18x Žsee also w23x.. 2. A special parameterization of learning. We consider a special restriction on the set of comparable information structures. In particular, if there are S possible states of nature, we assume a message consists of a noisy signal as to the true state of nature and thus there are S possible noisy signals. Let le w0, 1x reflect the level of information in the signal with 0 being no information and 1 being perfect information. Thus given a prior p we define the star-shaped information structure
THE OPTIMAL REGULATION OF GREENHOUSE GASES
9
FIG. 1. Star-shaped spreading of beliefs from p .
Ž P, q . where q ' p and the sth column of P is
p S s Ž 1 y l. p q l es ,
Ž 4.
where e s consists of all 0’s except for a 1 in the sth position. Clearly E y wp Ž y .x s P q s p . Furthermore if l s 0, each column of P is p and if l s 1, P s I. As an example, suppose you can receive one of three messages indicating whether the state of nature is 1, 2, or 3. We are now assuming that the number of possible messages equals the number of possible states of nature. A message that conveys the maximum amount of information would resolve all uncertainty on the state of nature. If the message is too noisy to contain any information, then the posterior on states of nature is the same as the prior. This is illustrated in Fig. 1 where the simplex of probabilities on states of nature is shown. The prior is p . The set of posteriors associated with a star-shaped spreading of beliefs, spread all the way out to the vertices, is shown by the three lines radiating out from p . Perfect learning would move you to one of the three vertices following receipt of the message. Less learning would move you to one of the three points marked with circles. Even less learning would move you to one of the three points marked with x’s after receiving the message. The advantage of representing learning by this star-shaped spreading of beliefs is that the process can be parameterized by the l in Eq. Ž4.. The disadvantage is that we have eliminated perfectly legitimate and orderable learning processes Žemanating from p in Fig. 1.. C. Learning in the Optimal Growth Model Introducing learning into our stochastic growth model involves introducing a second set of states corresponding to different paths that learning might take; i.e., sequences of messages that might be received. Each message yields a different outcome of the learning process where an outcome is a new probability vector on states of nature, p Ž t .. As before, let Yt be the set of possible single period
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CHARLES D. KOLSTAD
outcomes Žmessages. of the learning process at time t. At any point in time, to know the current state of knowledge it is essential to know the learning that has preceded t.4 We call this the history of learning, Ht s Ž y 0 , . . . , yt . < yi « Yi ;0 F i F t 4 . Notationally, Ht contains the learning that has occurred by time period t Žbefore decisions are taken in time period t .. An element of Ht is a particular history, h t . For instance, consider a 10 period world in which learning can proceed in three directions y1, 0, 14 at any point in time. One possible history would be Ž0, 1, y1, y1, y1, 1, 0, 1, 1, 0.. If we partition any h t as Ž h ty1 , yt ., then we define the ‘‘predecessor’’ and ‘‘most recent’’ functions, w : Ht ª Hty1 and c : Ht ª Tt as w Ž h t . s h ty1 , c Ž h t . s yt . The function c indicates the most recent learning whereas w indicates learning that occurred earlier. This allows one to functionally represent the learning path and to compute the probability vector on states of the world, p Ž h t , t .. Define the transition matrix P t Ž h t . such that each column is a posterior probability vector corresponding to a different element of the message space, Ytq1. Thus p Ž h t , t . is the column of P ty1Ž w Ž h t .. corresponding to c Ž h t .. Furthermore if qt Ž h t . is the probability vector associated with different elements of Ytq1 , then
p Ž h t , t . s P t Ž h t . qt Ž h t . .
Ž 5.
w P t Ž h t ., qt Ž h t .x is an information structure as described earlier. In our model, we define qt Ž h t . ' p Ž h t , t .. Thus the probability of receiving a message that reinforces state i is the same as the prior on state i. It is ‘‘easy’’ to modify the stochastic growth model Ž1. ] Ž3. to incorporate this learning. Two things remain to be specified: how the probability vector Žp . evolves and the probability Ž m . of a specific history Žlearning path. occurring. Following Eq. Ž4.,
p Ž h tq1 , t q 1 . s
½
Ž 1 y l. p Ž h t , t . q l e s Ž 1 y l. p Ž h t , t .
if s s ˆ s otherwise
Ž 6a .
m Ž h tq1 , t q 1 . s m Ž h t , t . pˆs Ž h t , t . ,
Ž 6b .
ˆs s c Ž h tq1 . ,
Ž 7a .
h t s w Ž h tq1 . ,
Ž 7b .
where
where e s is a vector of 0’s and 1’s with a 1 in the sth position and 0 in the rest of the positions, and l is the rate of learning. As indicated above, we assume the message space is the same as the space of possible states of the world. Thus messages are noisy indications of the true state of the world. Thus for each possible learning path, we know the probability vector on states of nature and the probability of that learning path actually occurring. Since Eqs. Ž6. are defined recursively, initial values need to be defined for p and m. p is of course the prior 4
Of course the current state of knowledge can be characterized by the probability on states of nature. However, this is an infinite set whereas the possible learning histories are finite. Finiteness is more amenable to stochastic programming which is the solution approach used here.
THE OPTIMAL REGULATION OF GREENHOUSE GASES
11
on states of nature and the initial m corresponds to q in the previous discussion, which is also equal to the prior on states of nature. There are many ways uncertainty can enter a model such as this. As indicated above, we assume uncertainty in the damage from global warming.5 Specifically, we write utility as u c Ž t . , d Ž t . , s s log c Ž t . y D s d Ž t . .
Ž 8.
We further assume there are two states of the world, B and L, corresponding to global warming being a big problem ŽB. vs global warming being a little problem ŽL.. Learning can proceed to reinforce B or reinforce L. We arbitrarily assume p B Ž t s 0. s 0.2 and p LŽ t s 0. s 0.8 with D b s 5 and D L s 0. Although arbitrary, this yields an expected value of D of 1 and reflects the fact that damage could be serious. The variance of D is 4. The expected variance declines with time, at faster rates for larger l Žsee w22x..6
IV. RESULTS The model described above has been implemented using time points at 10-year intervals beginning in 1965.7 The first three points Ž1965]1985. are used as calibration and control of emissions is fixed at zero. Optimal emission control levels are computed beginning in 1995. Learning occurs in 1995]2005 and 2005]2015. No learning occurs thereafter Žbut uncertainty persists.. The model described in the previous section is an infinite horizon model. Only finite horizon models can be solved as stochastic programs so we have chosen to approximate the infinite horizon model with a 20 periodr200 year finite horizon model. As has been shown elsewhere w22x, the control level in 1995 is essentially independent of horizon length, when the length is in excess of 20 periods. Model Ž1. ] Ž8. was solved as a function of d E and l. Focusing on the year 1995, with reversible emission control rates Ži.e., d E s 1., we find that optimal control levels for greenhouse gases are virtually unaffected by the rate of learning. If one over-controls today, then that error can be corrected in the future. Thus the fact that learning is taking place does not impact current decisions to control emission. Figure 2 shows the optimal 1995 control rate as a function of the learning rate Ž0 s no learning; 1 s complete resolution of uncertainty in one period.. The control rate is the fraction emissions have been reduced, relative to the uncontrolled level of emissions.8 Three curves are shown in the figure, corresponding to 5 Uncertainty in damage is commonly examined Žsee w16, 29x. although many other types of uncertainty are possible and are important. Clearly there is uncertainty in control costs. There is uncertainty in how the climate evolves. There is uncertainty in how the economy evolves. Peck and Teisberg w43x discuss some of the different types of uncertainty. 6 For instance, for l s 0.5, the expected variance declines from 4 to 2.3 in two periods. 7 The model is a stochastic programming model, solved using GAMSrMINOS. I am grateful to Lars Mathiesen for suggesting this approach. 8 The control rate is defined more precisely in the Appendix ŽEq. A-2..
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CHARLES D. KOLSTAD
FIG. 2. 1995 greenhouse gas control levels Žrate of time preference s 3%..
different values of d E , the abatement capital depreciation rate.9 The first thing to note is that the capital stock effect is real and has an effect on current control decisions. With no learning, there is no difference in the optimal decisions for the three d E values. In fact the control level is the same as it would be in the deterministic case. However, with more rapid learning, it makes sense to reduce control levels when abatement capital investment is irreversible. When abatement capital is perfectly reversible Ž d E s 1., then learning has virtually no effect on the optimal control level. In other words, the environmental stock effect, associated with greenhouse gas accumulation, does not appear. It could be that this failure to find an environmental stock effect is due to discounting the future at 3% per annum}a small rate but one which adds up over 100 years. When the rate of time preference is decreased to 1% per annum, the desirable level of control increases }from 7 to 15% for the case of no learning. However, the same pattern as found in Fig. 2 persists. Irreversible control capital has a significant influence on optimal levels of control in 1995 but the environmental stock effect just does not materialize. Why doesn’t the accumulation of greenhouse gas emissions lead to an environmental stock effect? To answer this question, we perturbed the optimal trajectory of control levels. In particular, we considered two cases with suboptimal control levels in 1995 and 2005 with a return to an optimal path after that. One is an approximate doubling of control to 15% for the 1995]2005 period and the other is no control for that period. Remarkably, this substantial deviation from optimality in 1995]2005 has little effect on subsequent emission control levels Žsee w20x.. Why is this? One reason is that costs and damages in the stochastic model are such that the optimal level of greenhouse gas abatement is very insensitive to the total stock of greenhouse gases. The simple reason is that while damage from greenhouse gases increases with the stock, and in fact is quadratic in temperature change, the present value of damage is quite linear over a fairly broad range in 9
To provide intuition on d E , abatement capital with a lifetime of 10, 20, 30, 40 or 50 years, depreciating at the same percentage per decade, leaving 10% residual value at the end of its life, would correspond to a d E of 0.90, 0.68, 0.54, 0.44 and 0.37 respectively.
THE OPTIMAL REGULATION OF GREENHOUSE GASES
13
current stocks of greenhouse gases. This is illustrated in Fig. 3 where the loss in the present value of utility Žin monetary units. is shown from an exogenous addition of 50 to 100 megatons of greenhouse gas in 1995. Keep in mind that greenhouse gas emissions are less than 10 megatons per year and that we allow future emissions subsequently to adjust optimally to a larger stock of gases. As is also illustrated in the figure, marginal damage is almost constant over a wide range of stocks of greenhouse gases. Figure 4 reproduces the marginal damage curve from Fig. 3 Žalthough writing it as a function of the emission control rate. and superimposes the marginal cost of control. Note from Figs. 3 and 4 that suboptimal emissions in one period will have virtually no effect on optimal emissions in the next period because of the insensitivity of marginal damage to the stock size and the fairly large slope of the marginal control cost function. This suggests that if damages were more nonlinear, the results might be different. To investigate this, we executed the model using damages proportional to the third, fourth, and fifth power of the temperature change, rather than just the square. We found negligible effect.10 We conclude that even if damages are nonlinear in the stock, and there are long time delays in stocks being translated into effects, then damages will still be roughly linear in current period emissions. This means that something like Fig. 4 applies. Getting back to the stock effect, recall that it has to do with the stock nature of greenhouse gases, and the fact that one cannot negatively emit; i.e., one cannot reverse emissions by negatively emitting in the future. Consequently, the effect can occur only if one might wish to negatively emit in the future. But because future emissions are so slightly influenced by today’s actions, there is no scenario under
FIG. 3. Marginal and total disutility as a function of stock of greenhouse gases.
10
We considered the extreme cases of complete learning in one period and no learning. More nonlinear damages resulted in increased current period emissions. For the 3% rate of time preference case, current period emission control increased from 7 to 18%. We also found that learning had an effect in the high curvature case when there was no stock effect, consistent with the results of Ulph and Ulph w50x. To eliminate this effect, we modified the no-learning case to be exactly the same as the learning case except that control levels at any point in time were constrained to be the same over all learning histories.
14
CHARLES D. KOLSTAD
FIG. 4. Marginal cost, marginal damage vs 1995 control rate.
which it would be optimal to negatively emit in the future to correct over-emissions today. This is the crux of why there is no such effect with regard to emissions.11 This point is also made in w50x. Finally, we can fall back on Malinvaud’s w25x result that when uncertainty is small, it makes no difference in optimal decisions. This seems to apply here since learning and uncertainty have very modest effects on outcomes. It is interesting to compare this result to that of Manne and Richels w29x. While their model is substantially different from ours, they show that immediate resolution of uncertainty Žvery rapid learning. results in lower emission control rates Žhigher emissions. than when uncertainty is not resolved. This is qualitatively the same as our result. V. POLICY IMPLICATIONS AND CONCLUSIONS In this paper we have addressed the question of whether the stock nature of greenhouse gas emissions or the sunk nature of control costs lead to a bias in today’s decisions regarding the control of greenhouse gases. Certainly the political debate has emphasized irreversibilities, either in control capital or climate change, and has suggested these irreversibilities have profound effect on control decisions made today. To answer the question, we have developed a stochastic model of greenhouse gas control and parameterized the rate of learning. We find only qualified support for such positions. In particular, we find no evidence of a stock effect from greenhouse gases affecting today’s control decisions. Of course the stock effect is only one type of environmental irreversibility. We have not examined irreversible changes in the climate or irreversibilities in damage. Such irreversibilities are of real concern to many concerned with climate policy. Only when emission control investments are very long-lived and irreversible is there a stock effect associated with control capital}and that effect calls for a 11
The fact that marginal damage is increasing reflects the fact that damage is not completely linear.
THE OPTIMAL REGULATION OF GREENHOUSE GASES
15
downward bias in control levels when learning is occurring rapidly, relative to the case of no learning or no uncertainty. One interpretation of these results is that rapid rates of learning should cause today’s societies to ‘‘go slow’’ on controlling greenhouse gas emissions. This is certainly one interpretation although go slow means undertaking some control of greenhouse gases, just not as much as in the case with no learning. Another interpretation is that we should vigorously pursue emission control policies that are reversible, that do not involve large, long-lived sunk costs since irreversible emission control decisions unnecessarily tie our hands. To be specific, it might be appropriate to adopt temporary taxes on carbon in lieu of permanent taxes. With learning, our model suggests that a temporary Že.g., 5]10 years. tax on carbon might dominate a similar permanent tax on carbon. Of course we have not analyzed this fully; all we can say is that more flexibility in control may be desirable and a temporary tax is more likely to induce flexibility. In any event, the results here should be qualified as arising from the specific model used and the specific nature of uncertainty Žin damages. adopted in the analysis. Further research should definitely investigate other types of uncertainty. APPENDIX: DETAILS ON STOCHASTIC GROWTH MODEL The stochastic growth model is fairly completely described in the text. Parameter values are taken from and described more fully in w36x, although there may be modest differences in the values of specific parameters due to our use of a different version of the DICE model. The variables L and s are parameters that vary with time. The populationrlabor supply Ž L. is 3.324 billion in 1965 and grows initially at a rate of 23.5% per decade, with the growth rate declining by 19.5% per decade Ži.e., 19.5% of 23.5% is the decline.. This leads to an asymptotic population of 11.09 billion. The uncontrolled greenhouse gas emissions]output ratio Ž s . starts at 0.5368 in 1965, declining by 12.5% per decade thereafter. The production function ŽEq. Ž2c.. is Y Ž t . s 1 y b1 m Ž t .
b2
g
AŽ t . K Ž t . L Ž t .
1y g
,
Ž A-1.
where
mŽ t . s 1 y
EŽ t .
s Ž t.YŽ t.
.
Ž A-2.
Thus m Ž t . is the emission control rate, defined by Eq. ŽA-2.. The parameters b1 , b 2 , and g are 0.0686, 2.887, and 0.25, respectively, and AŽ t . grows with time reflecting increases in total factor productivity. AŽ1965. s 0.00852 and grows initially at 20% per decade, with this growth rate declining by 11% per decade. The term in brackets in Eq. ŽA-1. reduces gross output by the cost of emission control. The per capita damage function ŽEq. Ž2d.. is given by u
dŽ t . s
u 1T Ž t . 2 Y Ž t . L Ž t . 1 q u 1T Ž t .
u2
,
Ž A-3.
16
CHARLES D. KOLSTAD
where u 1 and u 2 take the values 0.001478 and 2, respectively.12 This is equivalent to damage being quadratic in temperature with a 38C temperature rise yielding a 1.3% decline in output. The climate model ŽEqs. Ž3b. and Ž3c.. is T Ž t q 1 . s T Ž t . q r 1 ln M Ž t . r590 q r 2 T Ž t . q r 3 O Ž t .
Ž A-4.
O Ž t q 1 . s O Ž t . q r4 T Ž t . y O Ž t . ,
Ž A-5.
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