Regulation of Stock Externalities with Correlated Costs

Transcription

1 Regulation of Stock Externalities with Correlated Costs Larry Karp and Jiangfeng Zhang January 23, 2003 Astract We study a dynamic regulation model where firms actions contriute to a stock externality. The regulator and firms have asymmetric information aout serially correlated aatement costs. With price-ased policies such as taxes, or if firms trade quotas efficiently, the regulator learns aout the evolution of oth stock and costs. This aility to learn aout costs is important in determining the ranking of taxes and quotas, and in determining the value of a feedack rather than an open-loop policy. JEL Classification numers: C61, D8, H21, Q28 Key Words: Pollution control, asymmetric information, learning, correlated costs, choice of instruments A previous version of the paper was presented in the Fifth California Workshop on Environmental and Resource Economics, University of California, Santa Barara, CA, May 5-6, 2000, and at the annual meeting of the Canadian Agricultural Economics Society, University of British Columia, Vancouver, BC, Canada, June 1-3, Department of Agricultural and Resource Economics, University of California, Berkeley, CA karp@are.erkeley.edu Asian Development Bank, jzhang@ad.org The opinions expressed in this paper do not necessarily reflect the views of the Asian Development Bank.

2 1 Introduction The possiility of gloal warming has revived interest in comparing taxes and quotas when the regulator and firms have asymmetric information aout aatement costs. Weitzman (1974) showed that there is a simple criterion for ranking the policies when aatement costs and environmental damages are quadratic functions of the flow of pollution, uncertainty enters additively (i.e., it affects the level ut not the slope of the firm s marginal costs), and the optimal quantity restriction is inding with proaility one. 1 When the externality is caused y a stock rather than a flow the regulatory prolem is dynamic, and thecomparisonofpoliciesismorecomplicated. When stocks decay slowly, as do greenhouse gasses, current emissions may cause future environmental damages. Regulatory policies should alance current aatement costs and the stream of future environmental damages. In formulating these policies, the regulator should also consider the possiility of learning aout the firms aatement costs, therey reducing the future asymmetry of information. We show how the regulator s aility to update information aout serially correlated aatement costs affects oth the prolem of regulating a stock pollutant, and the comparison of taxes and quotas. In a dynamic model with serially correlated private information aout aatement costs, past oservations can provide information aout current costs. 2 In order to take advantage of this information, the regulator needs to use a feed-ack policy. Staring (1995) considers the simplest dynamic model where the regulator uses an open-loop policy, i.e. he announces the entire policy trajectory at the initial time. Weitzman s asic result, and the intuition for it, still holds: a steeper marginal damage function or a flatter marginal aatement cost function favor the use of quotas. A lower discount rate or a lower stock-decay rate oth features that increase the importance of future damages resulting from current pol- 1 There is a large literature that examines other aspects of the prolem of choosing policies under asymmetric information. Important contriutions to this literature include Dasgupta, Hammond, and Maskin (1980),Kwerel (1977),Malcomson (1978), Roerts and Spence (1976), Stavins (1996) and Watson and Ridker (1984). These papers are concerned with the prolem of flow rather than stock externalities. Non-linear policies can achieve higher payoffs than linear taxes or quotas. Roerts and Spence (1976) points out that a non-linear tax, with the marginal tax rate equals the marginal damage, achieves the first-est outcome in regulating a single firm and is superior to a quantity policy. 2 In addition to learning aout aatement costs, the regulator may learn aout the relation etween stocks and environmental damages. This second kind of learning leads to a different prolem, addressed in Karp and Zhang (2002) 1

3 lution favor the use of quotas. Hoel and Karp (2002) show that oth the aility to change the policy frequently and the use of a feedack rather than an open-loop policy favors the use of taxes. 3 They assume that cost are uncorrelated, so past oservations provide no information aout the current cost shock. The effect of all parameters on the policy ranking is qualitatively the same in the open-loop and feedack settings. Newell and Pizer (in press) extend the open-loop model y allowing costs to e serially correlated. They show that a more positive degree of autocorrelation favors the use of quotas. They study only the open-loop case, where the regulator learns nothing aout either the evolution of stocks or aatement costs. We consider the stock-regulation prolem with correlated costs when the regulator uses a feedack policy. Correlated costs increase the value of feedack rather than the open-loop decision rules, ecause the regulator has the opportunity to learn aout oth the evolution of stocks and costs. Not surprisingly, most of the intuition developed in the earlier papers survives in this more general setting. With a feedack policy, the distinction etween tradeale and nontradeale quotas is important, ecause of the two types of policies provide the regulator with different amounts of information. This paper thus extends our intuition for the stock regulation prolem and confirms that previous results hold in a more general setting. Although these theoretical insights are valuale, the implications for empirical work are proaly more important. For some pollution prolems (such as gloal warming) where we would like to compare taxes and quotas, we have only rough estimates (or guesses) of the slopes of marginal damages and aatement costs, and estimates of other parameters such as decay and discount rates and cost correlations. An empirical challenge is to use the existing data to rank policies. This challenge cannot e met merely y knowing the qualitative characteristics of a prolem. For example, Hoel and Karp (2002) show that despite the lack of a qualitative difference, there is a large quantitative difference in the criterion for ranking taxes and quotas, depending on whether the regulator uses open-loop or feedack policies; however, for all plausile parameter estimates, taxes dominate quotas for the control of greenhouse gasses under either open-loop or feedack policies. Newell and Pizer (in press) reach a similar conclusion with respect to changes in the cost correlation parameter, given that the regulator uses an openloop policy. The formulae that we derive enale us to rank taxes and quotas in a more realistic 3 The effect of the length of a period is the same in the more general setting discussed in this paper (where costs are serially correlated). Details are availale upon request. 2

4 and general setting (e.g., with or without cost correlation, under open-loop or feedack policies, with or without trade in quotas). The literature to which this paper contriutes compares efficient quotas and efficient taxes. To e consistent with this literature we assume that when there is no trade in quotas, firms have the same marginal aatement costs. In this case, there would e no efficiency gain from trade. When firms have different cost shocks, we assume that there is trade in quotas, so the potential efficiency gain is realized. That is, in oth of these cases the quota is efficient. The difference etween the two cases is that with trade the equilirium quota price contains the same information aout the aggregate cost shock as does the equilirium response to a tax; without trade, the regulator learns nothing aout the cost shock when he uses a quota. Thus, we identify the role of trade in providing information (via the quota price). One way to interpret this model isthattradeinquotasisalwayspermitted. Inequilirium trade actually occurs if and only if firms are heterogenous. A third possiility, that we do not study (ut mention in footnote 9), is that firms are heterogenous and there is no trade under quotas. In this case the quota is inefficient. This model would illustrate the comined effects of the greater information and the greater allocative efficiency provided y taxes or y quotas with trade. Section 2 descries the model and Section 3 explains the intuition for our results. Section 4 generalizes Newell and Pizer s policy ranking under the open-loop assumption and shows that more positively correlated cost shocks always favors the use of quotas. Section 5 shows that under the feedack policy without quota trading,taxes tendtodominatequotaswhenthe cost shocks are highly positively or negatively correlated. Section 6 shows that efficient quota trading eliminates the informational advantage of taxes; as in the open-loop setting, higher autocorrelation of cost shocks then favors the use of quotas. Susequent sections assess the likelihood that the policy ranking is different in the three scenarios (open-loop, feedack with and without quota trading), and provide an empirical illustration. 2 The model We egin the analysis with the assumption that all firms are identical, so there is no incentive for firms to trade quotas. Firms ehave non-strategically towards the regulator. We fix unitsof time equal to years and assume that a period lasts for one year. All time-dependent variales 3

5 are constant within a period. At time t the stock of pollutant is S t and the flow of new pollution is x t. The fraction 1 of the stock decays within a period: In period t the flow of environmental damages is D(S t ): S t+1 = S t + x t. (1) D (S t )=cs t + g 2 S2 t, g > 0. The representative firm s usiness-as-usual (BAU) level of emissions in period t is x t = x + θ t where θ t is a random variale. With an actual emission level x t < x t,thefirm s aatement cost is a quadratic function of aatement A (x t )= 2 x 2 t x t with >0. The firm s enefit from higher emission equals the aatement costs that it avoids having to pay. Defining the cost shock θ t θ t,wewritetheenefit as a linear-quadratic function, concave in the emission with an additive cost shock. The enefit function for a representative firm is defined as the flow of cost saving due to more pollution (less aatement): 4 B (x t,θ t )=f +(a + θ t ) x t 2 x2 t, > 0. (2) At time t only the firm oserves θ t ; there is persistent asymmetric information. The regulator knows the parameters of the AR (1) process that determines the evolution of θ: for t 1. The regulator s sujective distriution for θ 0 is θ t = ρθ t 1 + µ t ; µ t iid 0,γ 2 (3) θ 0 θ0,σ 2 0. The random variale θ 0 has (sujective) mean θ 0 and variance σ 2 0. θ 0 and µ t are independent and the correlation coefficient ρ satisfies 1 <ρ<1. If the regulator uses quotas, we assume that these are always inding. 5 Depending on the choice of parameter values andtheinitialvalueofs, the proaility that the quota is inding 4 The parameters satisfy f = 2 x2 and a = x. We ignore the effect of θ on f since f has no effect on the regulator s control. 5 Costello and Karp (2002) study a dynamic model with flow pollution, in which the possiility that the quota is not inding enales the regulator to learn aout the firm s cost. Brozovic, Sunding, and Zilerman (2002) point out that even in the simplest static prolem, the regulator s payoff might not e gloally concave. In this case, a inding quota might e a local ut not a gloal maximum. 4

6 for an aritrarily large ut finite numer of periods can e made aritrarily close to one. Since the regulator discounts the future, this fact means that the loss from ignoring the possiility that the quota is slack is very small. Therefore, we view the assumption that the quota is always inding as an approximation. If the regulator uses a tax p t, firms in each period maximize the difference etween the savings in aatement cost and the tax payment: Max Π t = B (x t,θ t ) p t x t = f +(a + θ t ) x t x t 2 x2 t p t x t. The firms first order condition implies 6 x t = a p t + θ t. (4) Whentheregulatorusesatax,theflow of emissions and the evolution of the pollutant stock S t are stochastic. At time t the regulator knows the actual level of emissions and the tax at time t 1, and (using equation (4) ) he is ale to infer the value of θ t 1. The tax-setting regulator does etter using a feedack rather than an open-loop policy ecause he learns aout the cost variale and the pollution stock, and he conditions his policy on this information. Define zt i as the regulator s expected emission given the tax p t,fori = OL (open loop) or i = FB (feedack). Under the open-loop tax policy, zt OL = E 0 x t = a p t + 1 E 0θ t = a p t + 1 ρt θ0. Under the feedack tax policy, zt FB = E t x t = a p t + 1 E tθ t. The regulator s expectation of the cost variale (under the feedack policy) is E t θ t = θ 0 when t =0and E t θ t = ρθ t 1 when t 1. Choosing a tax p t is equivalent to choosing expected emissions zt. i Thefirm s actual level of emissions is ( z OL x t + 1 θt ρ t θ0 (open-loop) t (z t,θ t )= zt FB + 1 (θ (5) t E t θ t ) (feedack). Hereafter we model the tax-setting regulator as choosing zt; i we drop the superscript i (i = OL or i = FB) where the meaning is clear. 6 Non-strategic firmssolveasuccessionofstatic optimization prolems. If firms made investment decisions which affect their aatement costs, as in Karp and Zhang (2002a), firms would solve dynamic prolems. prolem requires the solution of a dynamic game. This 5

7 3 The intuition for policy ranking The policy ranking otained in susequent sections depends on four considerations. The first two, which we refer to as the flexiility effect and the stochasticity effect, form the asis for policy ranking in Weitzman (1974) s static model, and in all of dynamic models previously cited. Under taxes, emissions and marginal aatement costs are positively correlated. This flexiility increases expected cost saving, favoring taxes. However, taxes result in a stochastic stock of pollution. Stochastic stocks increase expected damages ecause damages are convex in stocks. The stochasticity effect favors quotas. The third and fourth considerations are related to the autocorrelation parameter ρ. The change in stocks approximately equals the sum of flows. (A positive decay rate means that the two are not exactly the same.) Other things equal, the variation in the stock is smaller when the flows are negatively autocorrelated as occurs under taxes when costs are negatively correlated. Thus, negative autocorrelation of costs reduces the characteristic (stochasticity of stocks) that tends to make taxes unattractive. Similarly, positive autocorrelation of costs increases the characteristic that tends to make taxes unattractive. This relation the stock correlation effect explains why the preference for quotas is monotonically increasing in ρ under an open-loop policy. The stock correlation effect exists ut is less important under the feedack tax policy, ecause the regulator is ale to adjust the policy in every period to accommodate the previous shock. In choosing the current tax he need only consider next-period stock variaility. The fourth consideration is the learning effect. A higher asolute value of ρ means that knowledge of the previous cost provides more information aout the current cost. By oserving the level of lagged emissions and taxes, the tax-setting regulator learns the previous cost variale under a feedack policy. Under the feedack quota the regulator learns the previous cost variale only if quotas are traded. When quotas are traded, the learning effect is the same under taxes and quotas. This symmetry means that the only difference (related to ρ) etween taxes and quotas is the stock correlation effect. Consequently, a higher value of ρ favors (tradale) quotas under feedack policies. When quotas are not traded, the learning effect and the stock correlation reinforce each other for ρ<0, so a smaller value of ρ favors taxes; when ρ>0the two effects tend to counteract each other. The interplay of the two effects explains why the ranking of feedack taxes and non-tradale feedack quotas may e non-monotonic in ρ. 6

8 4 Open-loop policy With an open-loop policy, the regulator chooses an infinite sequence of policy levels, {x t } t=0 under quotas and {z t } t=0 under taxes, ased on the information he has at time 0. Thisdecision depends on the variance and covariance of the costs shocks, conditional on the information at time 0. Conditional on information at time 0, lim t var (θ t )= γ2. If the initial conditional 1 ρ 2 variance, σ 2 γ 0,alsoequals 2 (as in Newell and Pizer (in press)) the regulator s open-loop 1 ρ 2 prolem is stationary. For any other value of σ 2 0, the prolem is non-stationary. We consider the general case (an aritrary value of σ 2 0) in order to e ale to compare the policy ranking in the open-loop and feedack settings. In the feedack setting the regulator would have the prior σ 2 0 = γ2 after the firstperiodonlyifheeginswiththese eliefs and uses a non-tradale 1 ρ 2 quota. This prior is therefore not useful for comparing the open-loop and feedack settings. 7 The regulator wants to maximize the expectation of the present value of the difference etween aatement cost saving and pollution damages E 0 X t=0 β t {B (x t,θ t ) D (S t )}. β is a constant discount factor. The expectation is taken over the sequence of random variales {θ t } t=0 with respect to the information availale at t =0.Define T OL (S 0 ) as the maximized expected value of the regulator s open-loop (OL) program when he uses taxes and the initial stock of pollutant is S 0. Define Q(S 0 ) as the maximized expected value of the regulator s programwhenheusesaquota.(since asweexplaininsection5 thisvalueisthesameunder open-loop and feedack quotas without trading, we do not use a superscript on the function Q(S).) The expectation of the trajectories of the flow and the stock of pollution are the same for every scenario that we consider: open-loop and feedack, taxes and quotas, with and without quota trading. This fact is a consequence of the Principle of Certainty Equivalence of the linear-quadratic model with additive uncertainty. 8 Consequently, the policy ranking depends on the second moment of the random variale. 7 An earlier version of this paper explains a second reason for considering general priors. The variance of costs changes in a predictale manner, raising the possiility that the choice of policy instruments (as distinct from the choice of policy levels) might e time-inconsistent. Our earlier paper shows that this type of time-inconsistency does not occur. 8 This Principle states that in the linear-quadratic control prolem with additive random variales, the optimal 7

9 The payoff difference under taxes and quotas, given the open-loop policy, is 1 2(1 ρ 2 β) Ψ OL 0 T OL (S 0 ) Q (S 0 )= ³ n σ γ2 β 1 g 1 β β 1+ρβ 1 β 2 1 ρβ o. (6) (Details of this and other derivations, discussions of tangential issues, and some technical proofs, are contained in an appendix that is availale upon request.) Since the term outside the curly rackets is always positive, the policy ranking is independent of the regulator s priors ( θ 0, σ 2 0); that is, Newell and Pizer (in press) s criterion for ranking policies is correct for all priors. We restate this result as: Remark 1 The preference for quotas under the open-loop policy is monotonically increasing in g, β,, ρ, and monotonically decreasing in. Since the possiility of learning aout cost shocks depends on the value of ρ, we emphasize the role of this parameter. Proposition 1 Under an open-loop policy, the preference for quotas is monotonically increasing in ρ. Taxes dominates quotas iff ρ 1 β 2 β g β 1 β 2 + β g. 5 Feedack policy without quota trading The assumption that the optimal quota is always inding means that the quota-setting regulator learns nothing aout the previous cost shock and also means that the evolution of the stock of pollution is nonstochastic. Since no new information ecomes availale, the open-loop and feedack quota policies (and payoffs) are identical when quotas are not traded. When the regulator uses taxes, the evolution of S t is stochastic. By oserving the firms response to the tax, the regulator learns the value of the random cost. With taxes the regulator otains control rule does not depend on second moments of the random variale. Hoel and Karp (2001) examine the case where the slope rather than the intercept of firms marginal aatement cost is uncertain. In that case uncertainty is multiplicative, and the Principle of Certainty Equivalance does not hold. 8

10 informationovertime,soafeedacktaxpolicyresultsinahigherpayoffthandoesanopenloop policy. We compare the payoffs under the two policies y comparing their respective value functions. With correlated costs, there are two state variales, the stock of pollution S t and the current expected value of the cost variale y t E t θ t. When necessary to avoid confusion, we use superscripts to distinguish state variales under tax or quota policies, e.g. yt tax and y quota t. The value functions under oth taxes and quotas are quadratic in the state, i.e. they oth have the form Ã! Ã! Ã! V0,t i St +(v 1 v 2 ) + 1 {z } y t 2 (S V11 V 12 St t y t ). (7) V 12 V 22 y t V 1 {z } V 2 The first term of the value function V0,t, i i = tax, quota, depends on t ecause Var t (θ t ) changes exogenously in the first period. The appendix contains explicit expressions for the parameters of the value functions. The matrices V 1 and V 2 are the same under taxes and quotas. For a given state, the optimal control, zt under taxes and x t under quotas are equal and are given y a (1 β ) βc ( βv 11 ) β + 1 ρβ (1 ρβ ) βv 11 y t + β V 11 βv 11 S t, (8) with q V 11 = (β 2 + βg ) (β 2 + βg ) 2 +4βg < 0. (9) 2β The function V 11 is independent of ρ; the correlation parameter affects only the slope of the control rule with respect to the state y t. Equation (8) implies that an increase in current expected costs, y t, increases the current (expected) flow of pollution, as in the static model. Thevaluesofzt and x t differ over time ecause the actual trajectories of the state vector differ under taxes and quotas. However, as we remarked aove, the first moments of pollution flows and stocks are the same in all of our scenarios, ecause of the Principle of Certainty Equivalence. Taking expectations at time 0, wehavee 0 yt tax This relation and equation (8) imply = E 0 (ρθ t 1 )=ρ t θ0 = y quota t. E 0 z t = x t. (10) The expected values of the programs are different ecause of the second moments of the cost shocks. It makes sense to compare the values of these programs only under the same 9

11 information set, e.g. at time t =0. The same comparison holds at an aritrary time, provided that the regulator evaluates the two policy instruments using the same information set. In the initial period, y0 tax = y quota 0 = θ 0. The payoff difference under taxes and quotas, given the feedack policy, is due to the difference in the term V0,t i (evaluated at t =0)inequation(7) : Ψ FB 0 T FB S 0, θ 0 Q S0, θ 0 = ½ ¾ ³ 1 σ 2 (1 β 2(1 ρ 2 β) 0 + γ2 β V 11) (1 ρβ ) 2 β2 2 V 11(1 ρ 2 2 β) (11) 1 β (1 ρβ β V 11) 2. The ranking of feedack tax and quota policies depends only on the sign of the term in curly rackets. Hoel and Karp (2002) show that for ρ =0and σ 2 0 = γ 2, the parameters, g, β and have qualitatively the same effect on the policy ranking under oth open-loop and feedack policies. The comparative statics of the ranking with respect to, g, and are unchanged when ρ 6= 0: Remark 2 The preference for quotas under the feedack policy is monotonically increasing in g,, and monotonically decreasing in. However, the effect of ρ and β is different in the open-loop and feedack setting. Equation (11) implies Proposition 2 Feedack taxes dominate non-tradale feedack quotas iff µ f(ρ) ρ 2 β β V ρβ +1 β2 V (12) The function f (ρ) is convex in ρ and for some parameter values is nonmonotonic in ρ over ρ ( 1, 1). For such parameter values, the preference for quotas is non-monotonic in ρ. The function f (ρ) reaches a minimum at ρ 0; the minimum occurs at ρ =0if and only if =0,i.e. foraflow externality. We define ρ 1 and ρ 2 as the smaller and the larger roots of f (ρ) =0, provided that these roots are real. If oth of these roots are in the interval ( 1, 1), then for low values of ρ an increase in ρ makes quotas more attractive, and the opposite holds for high values of ρ. The following are sufficient conditions for either taxes or quotas to dominate: Corollary 1 A sufficient condition for taxes to dominate quotas is 1 β 2 β > g. 10

12 A sufficient condition for quotas to dominate taxes is ρ 1 < 1 and ρ 2 > 1. Under an open-loop policy, a larger value of β favors the use of quotas, ecause a larger β increases the importance of the future stock variaility arising from the current flow variaility. Under feedack policies, the comparative statics of β is amiguous. We have: Proposition 3 Under a feedack policy, a higher discount factor favors quotas (i.e., it decreases Ψ FB 0 )iff 2 + g β V 11 + g q β 2 + β g β g ρ (ρ2 β 2 + ρ). (13) 2(1 ρ 2 β) 3 2 A sufficient condition for inequality (13) is β 2 ρ ρ 2 satisfied if 0 ρ. and ρ 0. These two inequalities are A higher discount factor favors the use of quotas if the gain from the informational advantage under taxes is not great enough to offset the higher expected damage from future stock variaility. The sufficient condition 0 ρ meansthatequation(13)isverylikelytohold for stock pollutants that decay slowly. For example, when a period is one year a half-life of 15 years corresponds to =0.9548; greenhousegasseshaveahalf-lifeofover80years,for =0.99 (see Section 8). In the limiting case where =0the externality is a flow pollutant. Define g = βg, the present value of the slope of marginal damages and set σ 2 0 = γ, the variance of the innovation to costs. The difference etween payoffs under taxes and quotas for a flow pollutant is " µ Ψ FB σ 2 0 βρ 2 0 = 2 (1 β) (1 βρ 2 ) + 1 g #. 1+ g This expression shows how the learning that is made possile y a nonzero value of ρ favors the use of taxes for the case of a flow pollutant. When ρ =0we otain Weitzman (1974) s criterion. 11

13 6 Ranking with quota trading Previous sections assume that all firms are identical, so firms have no incentive to trade emissions permits. When firms are heterogeneous, emissions trade increases efficiency and also reveals industry-wide costs with a one-period lag. The informational advantage of taxes disappears in this case. Suppose there are n firms, where n is large. Let x i,t e firm i s emissions at time t. The enefitforfirm i of emitting x it is B i (x i,t,θ t, i,t )= f n +(a + θ t + i,t ) x i,t n 2 x2 i,t, (14) where i,t is the firm-specific deviation from the industry-wide cost θ t. These firm-specific deviations are i.i.d. over time with mean 0 and constant variance σ 2. They are uncorrelated with each other and are independent of the industry-wide average θ t,whichfollowsthear(1) process definedinequation(3). We use p t (p tax t or p quota t ) to denote either the tax level or the quota price from trading. The first order condition to firm i s prolem gives its emission response as x i,t = a p t n + θ t + i,t n. (15) Summing over x i,t gives the aggregate industry level emission x t = nx x i,t = 1 i=1 µ (a p t )+θ t + P it n. (16) ³ The last term in equation (16) is iid 0, σ2 n ;sincenis large, we replace σ2 n with 0. Thus, once the regulator knows p t and x t,heknowsθ t. Under quotas the regulator chooses x t,andp t is endogenous; under taxes the regulator chooses p t, and x t is endogenous. Quotas. Under quotas, in each period the aggregate emissions, and consequently the pollutant stock, are deterministic. Equation (16) implies the equilirium quota price p quota t = a + θ t x t. By oserving p quota t, the regulator learns θ t. Sustituting this price into equation (15) gives firm i s emission level as x t + it. Sustituting this expression into equation n n (14), summing over i, and taking expectations gives the aggregate expected cost saving under 12

14 quotas 9 : E t nx i=1 B i x quota i,t,θ t, i,t = f +(a + Et θ t ) x t 2 x2 t σ2. (17) Taxes. A tax policy results in a stochastic level of aggregate emissions. Choosing a tax is equivalent to choosing the expected aggregate emissions z t E t x t = 1 (a ptax t )+ 1E tθ t. The actual level of aggregate emissions is x t = z t + 1 (θ t E t θ t ). By oserving x t the regulator learns θ t. Firmi s emission level is 1 z n t + 1 (θ n t E t θ t )+ 1 n i,t. Sustituting this expression into equation (14), summing over i, and taking expectations gives the aggregate expected cost saving under taxes: nx E t B i x tax i,t,θ t, i,t = f +(a + Et θ t ) z t 2 2 z σ Var t (θ t ). (18) i=1 Taxes vs. Quotas. The firms individual deviations have the same effect on the aggregate expected cost saving under taxes and quotas. These firm-specific deviations do not affect the policy ranking. As efore, the regulator maximizes the expectation of the difference etween firms aggregate cost saving from polluting and environmental damage: X ( nx ) β t B i (x i,t,θ t, i,t ) D (S t ). i=1 E 0 t=0 Under quotas, the control variale is x t and the expected aggregate enefit is (17). Under taxes, the control variale is z t and the expected aggregate enefit is (18). The expectation is taken over sequences of random variales {θ t } t=0 and { i,t} t=0 with respect to the information availale at the current time, t =0. With the open-loop policy, the regulator decides his future policy trajectory at the initial period t =0and commits to it. There is no learning. All of the conclusions in Section 4 also hold when quotas are traded. The payoff difference under taxes and quotas, Ψ OL+Trade 0,isthe same as in equation (6). With the feedack policy, the regulator adjusts the instrument level as he learns aout cost shocks. Under oth taxes and quotas, the regulator s posterior is θ t ρθ t 1,γ 2, t 1. 9 If firms are heterogeneous and each firm receives the same allocation of quotas, ut there is no trade in quotas, the expected single period enefit of emissions is given y f +(a + E 0 θ t ) x t 2 x2 t. The difference etween this expression with the right side of equation (17) identifies the informational advantage of trade (the fact that with tradewehavee t θ t rather than E 0 θ t ) and the allocative efficiency (the presence of the term 1 2 σ2 ). 13 t

15 The corresponding law of motion for y t E t θ t under oth quotas and taxes is y t+1 = ρy t + ρµ t. (Without quota trading, y t+1 = ρy t ; see the appendix.) This difference identifies the informational advantage of tradeale quotas (relative to non-traded quotas). The optimal control, z t under taxes and x t under quotas, oeys the linear control rule (8). The regulator has the same amount of information aout costs under taxes and quotas, although in general the realization of S is different under the two policies. Consequently, the optimal controls are identical, for a given value of S: zt = x t. (19) Note the qualitative difference etween equations (10) (E 0 zt = x t )and(19)(zt = x t ). By oserving the price of emissions permits, the regulator has the same information aout costs under taxes and quotas in every period, not just at time t =0. Although as previously noted the first moment of stocks and flowsarethesameunder taxes and quotas, their second moments differ, leading to different expected payoffs: Ψ FB+Trade 0 = J0 T ³ 1 σ γ2 β 1 β S0, θ 0 J Q S0, θ 0 = 0 (1 β V 11)(1 ρβ + β V 11) 1 ρβ β V 11 V 11 < 0 isgiveninequation(9). The payoff under taxes is higher than under quotas if and only if. (20) 1 ρβ + β V This inequality implies Remark 3 With efficient quota trading, changes in parameters, g, β,, and ρ have qualitatively the same effect on the policy ranking under open-loop and feedack policies. The qualitative differences in policy ranking etween open-loop and feedack policies depend only on the informational advantage of taxes under feedack policies. Emissions trading eliminates this informational advantage. Again, we emphasize the effect of cost autocorrelation: Proposition 4 With quota trading, more positively autocorrelated cost shocks (higher ρ) favors the use of quotas under oth open-loop and feedack strategies. Feedack taxes dominate tradeale feedack quotas iff ρ 1 µ 1+ β β V

16 7 Sensitivity of the ranking This section identifies the region of parameter space where the open-loop and feedack assumptions lead to different policy rankings We define the critical ratio of g as the value of the ratio that makes the regulator indifferent etween taxes and quotas. The critical ratio is otained y setting the differences in payoffs equal to 0 and solving for g. The preference for quotas is monotonically increasing in g under oth open-loop and feedack policies, so the quota is the right instrument if and only if the actual value of the ratio of slopes exceeds the critical value. The critical values in the open-loop and feedack cases are respectively ³ g OL ³ µ µ g OL+Trade 1 ρβ 1 = = 1+ρβ β 2, n ³ g FB (1 ρβ ) (1 ρβ )+(1 β 2 ) p o 1 ρ 2 β = n β (1 ρ 2 β)+(1 ρβ ) p o, 1 ρ 2 β ³ µ g FB+Trade 1 =(1 ρβ ) β 2. 2 ρβ In the static model, the critical ratio of g is 1. When oth ρ =0and =0, the critical ratio under oth open-loop and feedack policies is β 1 rather than 1, since (y assumption) the current flow of pollution causes damages in the next period. The following Propositions descrie the relation etween the critical ratio of g and ρ. Proposition 5 The critical ratio of g is monotonically decreasing in ρ under oth an open-loop policy and a feedack policy with tradale quotas. Under a feedack policy without tradale quotas, the critical ratio of g is nonmonotonic in ρ: g FB < 0, if ρ < = 0, if ρ = ρ > 0, if ρ >. Proposition 6 Without efficient quota trading, the critical ratio of g under feedack policies is always greater than or equal to the open-loop level: ³ g FB ³ g OL, ρ. (21) 15

17 Ratio of slopes g/ Ratio of slopes g/ 9 Without Quota Trading 3 With Quota Trading open-loop D 4 C 3 open-loop A B C feedack Autocorrelation of cost variales ( ρ ) 0.5 A B feedack Autocorrelation of cost variales ( ρ ) Figure 1: With efficient quota trading, the critical ratio of g under feedack policies is greater than the open-loop level only for a suset of parameters: ³ g FB+Trade ³ g OL+Trade, iff ρ ρ; (22) where ρ is a function (of only β and ) that satisfies 1 ρ <0. The equality in (21) holds only if ρ = = 0, i.e. withaflow externality and uncorrelated cost shocks. The function ρ (β, ) used in equation (22) equals 1 only if β = =1, i.e. where oth discount rate and stock decay rate are zero. The proposition implies Corollary 2 Taxes will necessarily e the right instrument choice in the feedack setting if taxes dominate quotas in the open-loop setting, provided that there is no quota trading; or there is quota trading, and cost shocks are non-negatively autocorrelated. With negatively correlated shocks and quota trading, there exist parameter values such that it is optimal to choose taxes under the open-loop policy ut quotas under the feedack policy or vice versa. Figure 1 plots the critical ratios of g against ρ, holding β = and = Since a period is one year, these values imply a continuous discount rate of 0.05 and a half-life of 15 16

18 years. The left panel shows the case without quota trading, and the right panel shows the case with quota trading. In oth panels, the solid curve graphsthecriticalratio under the feedack policy and the dotted curve graphs the ratio under the open-loop policy. The tax is etter than the quota if and only if the actual ratio of g lies elow the critical ratio. The left panel of Figure optimal instrument choice without quota trading with quota trading region open-loop feedack region open-loop feedack A tax tax A tax tax B quota tax B quota tax C quota quota C quota quota D tax quota Tale 1: Feedack vs. open-loop optimal instrument choice. 1 illustrates equation (21) and the right panel illustrates equation (22). Tale 1 summarizes the different ranking possiilities. 8 An Application to Gloal Warming There is little agreement aout the likely magnitude of aatement costs and environmental damages related to greenhouse gasses. Using a linear-quadratic formulation, we can construct a simple model that incorporates in a transparent manner a particular elief aout these magnitudes. If we had a good knowledge of the physics and economics of gloal warming, it would e worth constructing complex models. Atthisstageweknowlittlemorethanthat there is a proale connection etween greenhouse gasses and gloal warming, and that the economic consequences of this relation may e important. The extent of the uncertainty and disagreement aout these magnitudes makes the simplicity of the model very important. It is easy to see how policy conclusions depend on eliefs aout the unknown parameters. For example, we can determine whether a particular conclusion would change if we increase our estimate of environmental damages y a factor of 10 or 100. Of course, since these experiments maintain the assumption of the linear-quadratic structure, they tell us nothing aout whether the policy conclusions are sensitive to functional form. 17

19 A numer of papers use the linear-quadratic structure, together with existing estimates of costs and enefits of greenhouse gas aatement, to examine particular policy issues. We riefly review these papers. Under the assumption of zero autocorrelation of cost shocks, Hoel and Karp (2002) show that taxes dominate quotas even if environmental damages associated with greenhouse gasses are much more severe than is commonly elieved. This conclusion holds under open-loop and feedack policies, with a period of commitment of anywhere from one to ten years. Newell and Pizer (in press) find that taxes dominate quotas under open loop policies with positive autocorrelation. Hoel and Karp (2001) compare feedack taxes and quotas with multiplicative (rather than additive) and serially uncorrelated cost shocks. The multiplicative structure means that the Principle of Certainty Equivalence does not hold, and thus raises issues that are not present in the previous papers. Susequent papers examine more complicated (feedack) models, where the regulator learns aout environmental damages (Karp and Zhang 2002) or where there is endogenous investment that reduces aatement costs (Karp and Zhang 2002a). Those papers review previous integrated assessment models that have een used to study greenhouse gas aatement. We use a time period of one year. Many economic studies (Kolstad 1996), (Nordhaus 1994) use an annual decay rate of ( a half-life of 83years)foratmospheric concentrations of the primary greenhouse gas, CO 2, implying = We set β = (a continuous discount rate of 0.03). Using these values, Tale 2 shows the critical values of the ratio g for five values of ρ, under open-loop and feedack policies with either homogenous or heterogenous firms. 10 This tale shows that for ρ 1 the ranking depends primarily on whether taxes have an informational advantage over quotas as they do only if firms are homogenous so that there is no trade in quotas. The critical ratios under open-loop and under feedack with heterogenous firms and tradale quotas are similar; these ratios are quite different than the critical ratio under feedack without tradale quotas. That is, when costs are highly (positively or negatively) autocorrelated, the ranking of policies is not particularly sensitive to the open-loop versus feedack distinction provided that quotas are traded; if quotas are not traded, this distinction is important for ranking policies. For moderate values of ρ (i.e. ρ.5), the ranking depends 10 Recall that in the asence of trade there is no allocative inefficiency ecause (y assumption) firms are homogenous; the quota is always efficient. The reader interested in comparing taxes and inefficient quotas can carry out the calculations using the single period enefit function descried in footnote 9. This calculation requires an additional parameter that measures the extent of firm heterogeneity, given y σ 2. 18

20 open-loop feedack & homogenous firms feedack & heterogenous firms (no trade) (with trade) ρ = ρ = ρ = ρ = ρ = Tale 2: The critical ratio g/ primarily on whether the regulator uses a feedack or an open-loop policy; allowing trade in quotas causes a relatively small change in the critical ratio. That is, when costs are not highly autocorrelated, the open-loop versus feedack distinction is important in ranking the policies; current information on cost shocks is not particularly important, so the possile informational advantage of taxes has little effect on the ranking. The survey in Hoel and Karp (2002) suggests a point estimate of g = (ased ontheestimatethatdoulingcaronstockscausesa5%reductioningrossworldproduct (GWP) and that a 50% reduction in emissions causes a 1% reduction in GWP). 11 The estimated value of g is linearly related to the estimate of environmental damages. (A ten-fold increase in the estimated reduction in GWP associated with a douling stocks leads to a ten-fold increase in the estimate of g.) This evidence suggests that in the case of greenhouse gasses, the serial correlation of cost shocks does not overturn the preference for taxes. This conclusion does not depend on whether the regulator uses feedack policies and learns aout the cost shock. 9 Summary This paper provides a criteria for ranking taxes and quotas for the control of a stock pollutant in a linear-quadratic model. We extended previous results y including serially correlated aate- 11 We cannot assess the plausiility of these estimates. A reader who thinks that the damage estimate understates actual damages y, for example, a factor of 10, should magnify the point estimate of g y a factor of 10. The lack of reliale estimates of costs and damages is, as we have emphasized, one of the main attractions of using a simple linear-quadratic model. 19

21 ment costs under either open-loop or feedack policies, with or without trading in emissions quotas. Under a tax policy, or under a quota policy with efficient quota trading among polluting firms, the regulator learns aout the industry s aatement cost schedule y oserving the aggregate emission response. The feedack strategy, unlike the open-loop strategy, enales the regulator to use new information to adjust his policy level, leading to higher welfare. With feedack policies, the ranking of taxes and non-traded quotas may e nonmonotonic with respect to oth the autocorrelation parameter and the discount factor. In contrast, the effects of the other parameters (the relative slopes of marginal costs and damages, and the decay rate) on the ranking are monotonic and are qualitatively the same under the open-loop and feedack assumptions. A large asolute value of autocorrelation increases the potential for learning, thus increasing the advantage of feedack taxes relative to oth the non-traded quota and to open-loop taxes. When quota trading occurs, the feedack tax loses its informational advantage over feedack quotas, and more autocorrelated cost shocks favor the use of quotas. Without quota trading, taxes will certainly e the right instrument choice in the feedack setting if taxes dominate quotas in the open-loop setting. However, this conclusion does not hold if firms can trade quotas. With tradeale permits, endogenous learning occurs under quotas. In this case, a regulator who is required to use an open-loop policy might want to use one instrument, where a regulator who is ale to use a feedack policy would use the other instrument. Using estimates of greenhouse gas-related damages and aatement costs, we provide evidence that taxes dominate quotas for the control of greenhouse gasses regardless of the opportunities for learning. 20

22 References BROZOVIC, N., D. SUNDING, AND D. ZILBERMAN (2002): Prices vs. Quantities Reconsidered, Giannini Foundation Working paper. COSTELLO, C., AND L. KARP (2002): Dynamic taxes and quotas with learning, Working Paper, Giannini Foundation of Agricultural Economics, University of California, Berkeley. DASGUPTA, P.,P.HAMMOND, AND E. MASKIN (1980): On Imperfect Information and Optimal Pollution Control, Review of Economic Studies, 47(5), HOEL, M.,AND L. KARP (2001): Taxes and Quotas for a Stock Pollutant with Multiplicative Uncertainty, Journal of Pulic Economics, 82, HOEL, M., AND L. KARP (2002): Taxes Versus Quotas for a Stock Pollutant, Resource and Energy Economics, 24, KARP, L., AND J. ZHANG (2002a): Controlling a Stock Pollutant with Endogenous Aatement Capital and Asymmetric Information, Giannini Foundation of Agricultural Economics, Working Paper No. 928, University of California, Berkeley; karp/. (2002): Regulation with Anticipated Learning aout Environmental Damage, Giannini Foundation of Agricultural Economics, Working Paper No. 926, University of California, Berkeley; karp/. KOLSTAD, C. D. (1996): Learning and Stock Effects in Environmental Regulation: The Case of Greenhouse Gas Emissions, Journal of Environmental Economics and Management, 31(1), KWEREL, E. (1977): To tell the truth: imperfect information and optimal pollution control, Review of economic studies, 44(3), MALCOMSON, J. (1978): Prices vs. Quantities: A Critical Note on the Use of Approximations, Review of Economic Studies, 45(1), NEWELL, R., AND W. PIZER (in press): Stock Externality Regulation Under Uncertainty, Journal of Environmental Economics and Management. 21

23 NORDHAUS, W. D. (1994): Expert Opinion on Climate Change, American Scientist, pp ROBERTS, M., AND A. SPENCE (1976): Effluent charges and licenses under uncertainty, Journal of Pulic Economics, 5(3-4), STARING, K. (1995): Regulering av forurensing under usikkerhet - en dynamisk modell - Regulation of pollution under uncertainty - a dynamic model, Master degree thesis, Department of Economics, University of Oslo. STAVINS, R. (1996): Correlated uncertainty and policy instrument choice, Journal of Environmental Economics and Management, 30(2), WATSON, W., AND R. RIDKER (1984): Losses from effluent taxes and quotas under uncertainty, Journal of Environmental Economics and Management, 11(4), WEITZMAN, M. (1974): Prices versus quantities, Review of Economic Studies, 41(4),

24 A Appendix A.1 General Solution for a Linear Quadratic Dynamic Programming Prolem We set up the dynamic programming equations in general matrix notation as: ½ J t (X t )=Max U Y t 2 X0 tqx t + Yt 0 WX t + M 0 X t + U1Y 0 t 1 2 Y t 0 U 2 Y t +U 3 Var t (θ t )+βe t J t+1 (X t+1 )}, (23) s.t. X t+1 = AX t + BY t + Cµ t + D. The suscript t in J t denotes the change in Var t (θ t ): Var t (θ t ) = σ 2 0 when t = 0;and Var t (θ t )=σ 2 µ when t>0. X t is a n 1 vector of state variales; Y t is a m 1 vector of control variales; µ t isawhite noise. Dimensions for those coefficient matrices are: Q is n n symmetric; W is m n; M is n 1; U 0 is 1 1; U 1 is m 1; U 2 is m m; U 3 is 1 1; A is n n; B is n m; C is n 1; D is n 1. Given the quadratic value function J t (X t )=V 0,t +V1 0X t+ 1 2 X0 tv 2 X t,thefirst order condition with respect to Y t is WX t + U 1 U 2 Y t + βe t B 0 V 1 + B 0 V 2 (AX t + BY t + Cµ t + D) =0. The optimal feedack control rule is Y t = U 2 βb 0 V 2 B 1 U1 + βb 0 (V 1 + V 2 D)+ W + βb 0 V 2 A X t, (24) a linear function of state variales. Sustituting Yt ack into the dynamic programming equation and equating coefficients gives the algeraic Riccati matrix equation for V 2 : V 2 = Q + βa 0 V 2 A + W 0 + βa 0 V 2 B U 2 βb 0 V 2 B 1 W + βb 0 V 2 A. (25) V 2 is a n n symmetric negative-semidefinite matrix. After otaining V 2,wecansolveforthe n 1 coefficient matrix V 1 : V 1 = hi βa 0 β (W 0 + βa 0 V 2 B)(U 2 βb 0 V 2 B) 1 B 0i 1 h i M + βa 0 V 2 D +(W 0 + βa 0 V 2 B)(U 2 βb 0 V 2 B) 1 (U 1 + βb 0 V 2 D), (26) and the constant term V 0,t V 0,t = η t µu 3 + β 2 C0 V 2 C + 1 ½ U 0 + βv 0 1 β 1D + β 2 D0 V 2 D + 1 ¾ 2 [U 1 + B 0 (V 1 + V 2 D)] 0 (U 2 βb 0 V 2 B) 1 [U 1 + B 0 (V 1 + V 2 D)] (27) 1

25 where η t depends on the second moment of cost shocks: η t = σ β 1 β σ2 µ when t =0,and η t = 1 1 β σ2 µ when t>0. Thefirst moment of the cost shock affects oth the value function and the optimal control, ut the second moment affects only the constant term of the value function. A.2 Feedack Policy First we write the law of motion for y t E t (θ t ). For the feedack quota policy, no new information ecomes availale over time, and y t+1 = E t+1 θ t+1 = E 0 (E t+1 θ t+1 )=E 0 θ t+1 = ρ t+1 θ0 = ρy t. Under the feedack tax policy, the regulator infers θ t y oserving the firm s emissions, and ( ρθ0 = ρy 0 + ρµ y t+1 = E t+1 θ t+1 = E t+1 ρθt + µ t+1 = ρθt = 0, t =0 ρ (ρθ t 1 + µ t )=ρy t + ρµ t, t 1 (28) with µ 0 θ 0 θ 0. The distriution of cost shocks in the initial and susequent periods have different variances: ( σ 2 Var t (θ t )= 0, t =0 (29) Var t (µ t )=γ 2, t 1. Under oth feedack taxes and quotas, the equation of motion for the state variale y t E t θ t is independent of the regulator s actions. However, there is a qualitative difference in y t under taxes and quotas ecause of endogenous learning. Under feedack tax policies, using equation (5), the expected enefitinperiodt,is E t {B [x t (z t,θ t ),θ t ]} = f +(a + E t θ t ) z t 2 z2 t Var t (θ t ). The regulator s value function under taxes, T FB (S t,y t ), solves the dynamic programming equation T FB (S t,y t )=Max Et B [x t (z t,θ t ),θ t ] D (S t )+βe t T FB (S t+1,y t+1 ) ª z t ½ =Max f +(a + y t ) z t z t 2 z2 t Var t (θ t ) ³ cs t + g o 2 S2 t + βe t T FB (S t+1,y t+1 ) s.t. S t+1 = S t + z t + 1 µ t y t+1 = ρy t + ρµ t. 2