Price-Based vs. Quantity-Based Environmental Regulation under Knightian Uncertainty: An Info-Gap Robust Satisficing Perspective

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1 University of Massahusetts Amherst From the SeletedWorks of John K. Stranlund January, 2006 Prie-Based vs. Quantity-Based Environmental Regulation under Knightian Unertainty: An Info-Gap Robust Satisfiing Perspetive JOHN K STRANLUND, University of Massahusetts - Amherst Yakov Ben-Haim Available at:

2 University of Massahusetts Amherst Department of Resoure Eonomis Working Paper No Prie-Based vs. Quantity-Based Environmental Regulation under Knightian Unertainty: An Info-Gap Robust Satisfiing Perspetive Abstrat: John K. Stranlund 1 and Yakov Ben-Haim 2 Conventional wisdom among environmental eonomists is that the relative slopes of the marginal soial benefit and marginal soial ost funtions determine whether a prie-based or quantity-based environmental regulation leads to higher expeted soial welfare. We revisit the hoie between prie-based vs. quantity-based environmental regulation under Knightian unertainty; that is, when unertainty annot be modeled with known probability distributions. Under these irumstanes, the poliy objetive annot be to maximize the expeted net benefits of emissions ontrol. Instead, we evaluate an emissions tax and an aggregate abatement standard in terms of maximizing the range of unertainty under whih the welfare loss from error in the estimates of the marginal benefits and osts of emissions ontrol an be limited. The main result of our work is that the same riterion involving the relative slopes of the marginal benefit and ost funtions determines whether prie-based or quantity-based ontrol is more robust to unstrutured unertainty. Hene, not only does the relative slopes riterion lead to the poliy that maximizes the expeted net benefits of ontrol under strutured unertainty, it also leads to the poliy that maximizes robustness to unstrutured unertainty. Keywords: emissions ontrol, environmental regulation, info-gap, Knightian unertainty, robustness, satisfiing JEL Classifiation: D81, L51, Q58 1 John K. Stranlund, Department of Resoure Eonomis University of Massahusetts, 214 Stokbridge Hall, 80 Campus Center Way Amherst, MA E: stranlund@reseon.umass.edu P: F: Yakov Ben-Haim, Mehanial Engineering Tehnion Israel Institute of Tehnology, Haifa Israel E: yakov@tehunix.tehnion.a.il P: F:

3 July 2006 Prie-Based vs. Quantity-Based Environmental Regulation under Knightian Unertainty: An Info-Gap Robust Satisfiing Perspetive JOHN K. STRANLUND * Department of Resoure Eonomis University of Massahusetts-Amherst 214 Stokbridge Hall Amherst, MA 01003, USA (413) (phone) (413) (fax) stranlund@reseon.umass.edu YAKOV BEN-HAIM Yitzhak Moda'i Chair in Tehnology and Eonomis Faulty of Mehanial Engineering Tehnion - Israel Institute of Tehnology Haifa Israel yakov@tehunix.tehnion.a.il Abstrat: Conventional wisdom among environmental eonomists is that the relative slopes of the marginal soial benefit and marginal soial ost funtions determine whether a prie-based or quantity-based environmental regulation leads to higher expeted soial welfare. We revisit the hoie between prie-based vs. quantity-based environmental regulation under Knightian unertainty; that is, when unertainty annot be modeled with known probability distributions. Under these irumstanes, the poliy objetive annot be to maximize the expeted net benefits of emissions ontrol. Instead, we evaluate an emissions tax and an aggregate abatement standard in terms of maximizing the range of unertainty under whih the welfare loss from error in the estimates of the marginal benefits and osts of emissions ontrol an be limited. The main result of our work is that the same riterion involving the relative slopes of the marginal benefit and ost funtions determines whether prie-based or quantity-based ontrol is more robust to unstrutured unertainty. Hene, not only does the relative slopes riterion lead to the poliy that maximizes the expeted net benefits of ontrol under strutured unertainty, it also leads to the poliy that maximizes robustness to unstrutured unertainty. Keywords: emissions ontrol, environmental regulation, info-gap, Knightian unertainty, robustness, satisfiing. * Corresponding author. 1

4 Prie-Based vs. Quantity-Based Environmental Regulation under Knightian Unertainty: An Info-Gap Robust Satisfiing Perspetive 1. Introdution Environmental regulations are hosen in a world in whih authorities have only imperfet information about the underlying benefits and osts of environmental ontrol. In his seminal paper, Weitzman (1974) provides guidane for the optimal hoie between prie-based and quantity-based emissions ontrol (i.e., an emissions tax vs. a ompetitive market for transferable emissions quotas) that has beome onventional wisdom among environmental poliy sholars and pratitioners. Weitzman shows that when the marginal benefits and marginal osts of emissions ontrol are linear (at least to a loal approximation), when authorities are unertain about the interepts of these funtions (but not their slopes), and when the benefits and osts of emissions ontrol are unorrelated, then a prie-based regulation is more effiient than a quantity-based regulation if slope of the marginal ost funtion is greater than the absolute value of the slope of the marginal benefit funtion, and a quantity-based regulation dominates if the inequality is reversed. 1 This deision riterion is determined from a omparison of expeted soial welfare (expeted benefit minus ost) under the two types of regulations. 1 Weitzman (1974) demonstrates that this deision riterion must be modified when benefits and osts are orrelated. See Stavins (1996) for a further exposition of this ase. Weitzman, Laffont (1977), and Malomson (1978) examine modifiations to the relative-slopes riterion when these are also unertain. Linearity to a loal approximation of the marginal benefit and marginal ost funtions is also important for the determination of the pries vs. quantities riterion. Malomson (1978) and Weitzman (1978) reexamine Weitzman s original work when loal linear approximations to the benefit and ost funtion are not appropriate. In our paper, we stik to linear marginal benefit and ost funtions with unorrelated unertainty in their interepts, beause it is this problem that drives the onventional wisdom of environmental eonomists about the hoie between emissions taxes and aggregate abatement standards (Baumol and Oates, 1988). In fat, many reent ontributions to this literature (e.g., Hoel and Karp, 2002; Moledina et al., 2003; Montero, 2002; Quirion, 2004) assume linear marginal benefit and ost 2

5 Weitzman, as well as the small army of sholars who have ontributed to the pries vs. quantities debate sine 1974, assumed that unertainty about the marginal benefit and ost funtions ould be ompletely haraterized by known probability distribution funtions over the errors in estimates the marginal benefit and ost funtions. 2 Indeed, the riterion for hoosing between prie-based and quantity-based regulations is determined from a omparison of expeted soial welfare under the two regulatory instruments. While it has proven useful to assume that the error strutures of estimates of the marginal benefit and ost funtions are known, in many ases if not most, environmental authorities must design ontrol poliies even when they are not onfident of the error strutures. That is, the world of environmental deisionmaking may be usefully haraterized by Knightian unertainty in whih, not only are the benefit and ost funtions unknown, but the distributions of the estimates of these funtions are also unknown. 3 In this paper, we revisit the prie vs. quantity debate from the perspetive of robust satisfiing when the errors in the estimates of the benefits and osts of emissions ontrol are ompletely unstrutured. Info-gap deision theory (Ben-Haim, 2006) provides a useful way to think about the regulatory hoie of emissions ontrol instruments in the fae of true Knightian funtions with only interept unertainty. We note that the approah we take in this paper, that of robust satisfiing, an be applied to all of the ompliating fators mentioned in this footnote. 2 Reent ontributions to the pries vs. quantities debate in the area of environmental management inlude extensions to dynami environments (Hoel and Karp, 2002; Moledina et al., 2003), imperfetly enfored environmental regulations (Montero, 2002), and to eonomies with pre-existing distortions from labor or apital taxes (Quirion, 2004). 3 Knight was onerned "with situations whih are far too unique, generally speaking, for any sort of statistial tabulation to have any value for guidane. The oneption of an objetively measurable probability or hane is simply inappliable." This is what Knight alled "true unertainty" (Knight, 1921, pp ), and it dominates many eonomi poliy deisions beause loal or firm-speifi details and future ontingenies are poorly understood by poliymakers. 3

6 unertainty. 4 The heart of info-gap analysis is the pursuit of deisions that maximize the range of unertainty about model parameters or funtions within whih the deisionmaker is ertain to ahieve a speified performane riterion. One deision is more robust than another if the range of unertainty under whih the performane riterion is met is larger. 5 In our ase, unertainty is about the true values of the interepts of the marginal benefits and osts of emissions ontrol and the performane riterion is to hold the loss assoiated with error in the estimates of these values to be no greater than some value. For a given loss assoiated with estimation error, one emissions ontrol poliy is more robust than another if the range of error in the estimates of the benefit and ost funtions under whih the performane riterion is met is larger. We apply info-gap analysis to the hoie between prie-based and quantity-based environmental regulation in the onventional model of environmental ontrol under unertainty. The main result of this paper is that the same riterion involving the relative slopes of the marginal benefit and ost funtions that Weitzman derived in the ase of strutured unertainty about the estimates of the benefit and ost funtions also reveals whether prie-based or quantitybased regulation is more robust to this unertainty when it is ompletely unstrutured. Although we ome to the same deision riterion, our motivation for deriving this riterion is very 4 Info-gap deision theory has been applied to a wide variety of problems, inluding finanial risk assessment (Ben-Haim, 2005), searh behavior in animal foraging models (Carmel and Ben- Haim, 2005), poliy deisions in marine reserve design (Halpern et al., 2006), natural resoure onservation deisions (Moilanen et al., 2006), inspetion deisions by port authorities to detet terrorist weapons (Moffitt et al., 2005a) and invasive speies (Moffitt et al., 2005b), tehnologial fault diagnosis (Piere et al., 2006) and engineering model-testing (Vinot et al., 2005). 5 Info-gap robust-satisfiing is a quantitative ombination of Knightian unertainty with Simon's onept of bounded rationality. The robust-satisfiing poliy-maker seeks a deision whih satisfies the performane and is robust to unertainty. Simon (1983) stresses the importane of satisfiing doing good enough; meeting ritial requirements when deisions must be made with defiient or erroneous information (i.e. Knightian unertainty) and limited omputational resoures. 4

7 different. While the deision riterion for hoosing between prie-based and quantity-based emissions ontrol maximizes the expeted net soial benefits of emissions ontrol, we show that this deision riterion also leads to the hoie of poliy that is most robust to estimation error when authorities do not have reliable information about the probability struture of this error. Hene, we provide a further justifiation for the use of the relative-slopes riterion that is onventional wisdom among environmental eonomists not only does this riterion lead to the poliy that maximizes the expeted net benefits of ontrol under strutured unertainty, it also leads to the poliy that maximizes robustness to unstrutured unertainty. The rest of the paper proeeds as follows: In the next setion we lay out the struture of the info-gap approah to hoosing between a prie-based and a quantity-based environmental regulation under true unertainty. In the third setion we alulate the robustness to unertainty of eah poliy type. In the fourth setion we derive the ondition under whih the prie approah is more robust to unertainty than the quantity approah, and vie versa, and disuss the poliy signifiane of our findings. We onlude in the fifth setion. 2. Info-Gap Robust-Satisfiing 2.1 Info-gap unertainty about the benefits and osts of pollution ontrol Let q denote aggregate redution (abatement) in the emissions of some pollutant. The aggregate benefits and osts of abatement take on the familiar quadrati forms. The benefit and osts funtions are, respetively: 2 Bq ( ) = bq Bq /2; [1] 2 Cq ( ) = q+ Cq /2, [2] 5

8 where b,, B, and C are all positive onstants, restrited to guarantee that optimal abatement levels are stritly positive throughout. 6 The soial welfare funtion is simply the differene between the benefits and osts of abatement: 2 W( q) = B( q) C( q) = ( b ) q ( B + C ) q /2. [3] We assume that the quadrati forms for the benefit and ost funtions and the numerial values of B and C have been verified, but that the values of b and are highly unertain. Note that b and are the interepts of the marginal benefit and marginal ost funtions, respetively. Thus, this set-up is onsistent with most of the literature on the pries versus quantities debate, in whih it is typial to assume that the marginal benefit and ost funtions are known exept for unertainty about their interepts. Assume that the environmental authority possesses estimates of the interepts of the marginal benefit and ost funtions, denoted b and, but the fators that modify these parameters are omplex and poorly understood, so we fae true Knightian unertainty: no reliable probabilisti model is available for the size of the estimation errors. That is, we must deal with the unertainty in the estimates of these parameters without knowledge of the probability distribution funtions that underlie them. Instead, we desribe the unertainty in the estimates of b and with a frational-error info-gap model for unertainty in the following unbounded family of nested sets of the marginal funtions: b and, whih is b b U( α, b, ) = B ( q) = b B q, C ( q) = + C q: α, α, α 0. [4] b 6 Stritly speaking the abatement ost funtion is the minimum abatement ost funtion. Impliitly, we assume that abatement responsibilities are alloated among the pollution soures to minimize the aggregate osts of ahieving eah level of aggregate abatement. This an be ahieved with an emissions tax or with a ompetitive tradable emissions permit poliy. 6

9 It is lear that α represents the unknown frational error in the estimates of the interepts of the marginal benefit and ost funtions. When the horizon of unertainty, α, is zero, then the true interepts, b and, preisely equal the estimated values, b and. The range of possible interept values inreases as the horizon of unertainty inreases. The horizon of unertainty is unbounded, so there is no known worst ase. These properties are harateristi of all info-gap models (Ben-Haim, 2006). 2.2 Robustness Generally, info-gap robustness is the greatest level of unertainty with whih a deision maker is ertain to meet a pre-determined performane riterion. In our ase, we look for the robustnesses of an emissions tax and an aggregate abatement standard, and our performane riterion is to limit the welfare loss from error in estimating the marginal benefit and ost funtions. The emissions tax and the aggregate abatement standard are determined by maximizing the welfare funtion [3], given the best available estimates of b and. Let the tax be t and the abatement quota be q, whih are hosen to maximize [3] with b= b and =. We will alulate t and q in the next setion. 7 The welfare loss inurred with poliy p (, t q ), when b and differ from b and is denoted L( p, b, ). The poliy maker would like low welfare loss, and therefore speifies L to be the largest aeptable loss. An important result of our analysis will be to evaluate the reliability of attaining any speified level of loss. 7 It is worth noting that t and q are the same prie and quantity poliies that are derived in onventional analyses of pries vs. quantities. Under the onventional approah to this problem, the expeted values of the errors in the interepts of the marginal benefit and ost funtions are assumed to be zero. Maximizing expeted welfare then leads to t and q. 7

10 The robustness-against-unertainty of poliy p is the greatest horizon of unertainty in the interepts of the marginal benefit and ost funtions (i.e., α ) up to whih all realizations of these funtions result in loss no greater than L. Formally: ˆ( α p, L ) = max α : max L( p, b, ) b, U( α, b, ) L. [5] For poliy p, we will demonstrate that ˆ( α pl, ) = 0 for L = 0. This implies zero robustness to zero loss; that is, only zero error in the estimates of the b and an produe zero welfare loss. This is true of any regulation that is based on best estimates of model parameters. Furthermore, we will show that ˆ( α p, L ) is stritly inreasing in L, whih implies that greater robustness to estimation error is only ahievable by tolerating a greater loss; that is, robustness to error trades off against loss, and ˆ( α p, L ) quantifies this tradeoff. The main result of this paper is a determination of the relative robustness of the emissions tax, t, and the abatement standard, q. Speifially, t is more robust to estimation error than q if ˆ α(, t L ) ˆ > α(, q L ); if the inequality is reversed, the abatement standard is more robust. In the first ase, ˆ α(, t L ) ˆ > α(, q L ) reveals that the range of error in the estimates of the interept of the marginal benefit and ost funtion for whih an environmental authority is ertain that the loss from error will not exeed a ritial value, L, is greater under the emissions tax than under the abatement standard. Thus, the authority is more onfident about limiting the loss from error under the emissions tax than under the abatement standard. Of ourse, if ˆ α(, t L ) < ˆ α(, q L ), then the authority is more onfident about limiting the welfare loss from estimation error under the abatement standard. 8

11 3. Info-Gap Analysis of Prie-Based vs. Quantity-Based Environmental Regulation In this setion we derive the robustness funtions for an emissions tax and an aggregate abatement standard, both of whih are determined with the best-available estimates of the interepts of the marginal benefit and ost funtions and the info-gap models for unertainty in those estimates. 3.1 Robustness of an aggregate abatement standard If the interepts of the marginal benefit and ost funtions were known, the environmental authority would hoose an aggregate abatement standard to maximize the welfare funtion 2 W( qb ;, ) = ( b q ) ( B + C ) q /2, whih leads to q = ( b ) ( B + C ). However, with only its best estimates of b and, the authority sets the abatement standard to maximize W( q; b, ), and therefore, sets the abatement standard at q = ( b ) ( B + C ). The welfare loss when b and differ from b and is * then Lqb ( ;, ) = W( q; b, ) W( qb ;, ), whih is alulated to be 2 Lqb ( ;, ) = [( b b ) ( )] 2( B + C ). The unertainty model [4] speifies the ranges of error in the estimates of the interepts of the marginal benefit and ost funtions at any horizon of unertainty, α ; that is, b b [ αb, αb] and [ α, α ]. (Reall that b and are positive onstants). The inner maximum of the definition of robustness [5] requires maximization of Lqb ( ;, ) with respet to b and. In turn, this requires either b b =αb and = α, or b b = αb and =α. That is, for some frational error in the estimates of b and, α, Lqb ( ;, ) is maximized with 9

12 respet to b and if b is over-estimated and is under-estimated, or if b is under-estimated while is over-estimated. In either ase, 2 2 L( α, q ) = α ( b + ) 2( B + C ), [6] is the maximum welfare loss of q when the frational error in the estimates of the interepts of the marginal benefit and marginal ost funtions is α. Now, from [5], the robustness of abatement standard q is the highest α for whih L( α, q ) L. Sine L( α, q ) is monotonially inreasing in α, we hoose α to satisfy L( α, q ) = L. Solving this equation for α provides the robustness of the abatement standard q : ˆ( α ql, ) = 2 L( B + C ) ( b + ). [7] Sine b,, B, and C are all positive onstants and L is non-negative, ˆ( α ql, ) is nonnegative. The robustness funtion, ˆ( α ql, ), quantifies the greatest frational error in the estimates of the interepts of the marginal benefit and marginal ost funtions for whih the welfare loss assoiated with the abatement standard q is not greater than some ritial welfare loss L. Note that, with respet to L, ˆ( α ql, ) has a zero interept, is montonially inreasing, and stritly onave. The zero interept implies that there is zero robustness to zero loss; that is, the environmental authority an have no onfidene in limiting the loss from estimation error to zero. That robustness is inreasing in L implies that greater robustness only omes with tolerating a greater loss: the authority gains onfidene in meeting the performane riterion as this riterion is relaxed. Finally, the onavity of ˆ( α ql, ) suggests that the marginal ost (i.e., the marginal inrease in ritial loss) of attaining greater robustness is inreasing. Put another way, onfidene in limiting the welfare loss inreases with higher aeptable loss, but at a 10

13 dereasing rate. Robustness funtions are always monotoni, expressing a fundamental trade-off between robustness and performane. They are not always onave. 3.2 Robustness of an emissions tax We now turn to alulating the robustness of an emissions tax. We demonstrate that it has the same basi harateristis of robustness to an abatement standard. Firms in an industry will reat to an emissions tax, t, by hoosing their levels of abatement so that their individual marginal abatement ost funtions are equal to the tax. In aggregate, then, the intersetion of the tax and the aggregate marginal abatement ost funtion determines aggregate abatement. From t = + C q we have qt ( ) = ( t ) / C. Soial welfare [3], in terms of an emissions tax, is then Wtb b qt B C qt 2 (;, ) = ( ) () ( + )( ()) /2. The tax that maximizes this soial welfare funtion is t = ( bc + B ) ( B + C ). With only its best estimates of b and, however, the regulatory authority hooses t = ( bc + B ) ( B + C ). (It is easy to demonstrate that t produes q abatement, and t produes q abatement). The welfare loss when b and differ from b * and is Lt (; b, ) = Wt ( ; b, ) Wt (; b, ), whih is alulated to be Lt (; b, ) = [( b bc ) + ( B ) ] 2( B + C )( C ) 2 2. As with the abatement standard, the unertainty model [4] speifies the ranges of error in the estimates of b and, b b [ αb, αb ] and α [, α ], at any horizon of unertainty, α. In ontrast to the abatement standard, however, the inner maximum of the definition of robustness [5] requires either b b =αb and = α, or b b = αb and = α. That is, 11

14 Lt (; b, ) is maximized if both b and are over-estimated, or if both are under-estimated. In either ase, 2 2 L( α, t ) = α [ bc + B ] 2( B + C )( C ) 2. Robustness of t is then the solution to arg max α L( α, q ) L ; that is, ˆ( α t, L ) = C 2 L ( B + C ) ( bc + B ). [8] Again, ˆ( α t, L ) is the greatest frational error in the estimates of the interepts of the marginal benefit and marginal ost funtions for whih the welfare loss assoiated with the emissions tax t is not greater than some ritial welfare loss L. With respet to L, ˆ( α t, L ) has a zero interept, is montonially inreasing, and stritly onave. Therefore, ˆ( α t, L ) has the same basi struture as ˆ( α ql, ). However, for some L, ˆ( α t, L ) and ˆ( α ql, ) will normally take on different values. 4. The Relative Robustness of Prie-Based and Quantity-Based Emissions Control We are now able to ompare the robustness of the emissions tax, t, to the robustness of the aggregate abatement standard, q. Using the robustness funtions [7] and [8], alulate ˆ α(, t L ) ˆ α(, q L ) = ( C B ) 2 L ( B + C ) ( b + )( bc + B ). [9] Clearly, the relative robustness of t and q to any ritial loss, L, depends solely on the relative slopes of the marginal benefit and ost funtions, C and B. Speifially, t is more robust to estimation error than q if and only if the marginal ost funtion is more steeply sloped than the marginal benefit funtion; that is, C > B. On the other hand, if C < B, the abatement standard is more robust than the emissions tax. Note further that the differene in the 12

15 robustness of the two poliies inreases as L inreases, Thus, the value of one poliy over the other (in terms of immunity to loss) inreases as the potential loss is allowed to inrease. Figures 1 and 2 illustrate the robustness funtions for t and q when C > B and B > C, respetively. [Plae Figures 1 and 2 about here]. The main result of this work is that the same relative-slopes riterion that Weitzman (1974) derived to hoose between prie-based and quantity-based regulation applies in our ase as well. However, we have demonstrated a omplementary purpose for this riterion. In the onventional ase, the relative-slopes riterion provides guidane about the hoie between an emissions tax and an aggregate abatement standard when the poliy objetive is to maximize expeted welfare from emissions ontrol when the errors in the estimates of the interepts of the marginal benefit and ost funtions are haraterized by known probability distribution funtions. We have shown that the relative-slopes riterion also determines whih poliy is more robust to unstrutured Knightian estimation error. Therefore, our work omplements the onventional approah to the pries versus quantities debate, beause it provides a further justifiation for the relative slopes riterion. Not only will this riterion lead to the poliy that maximizes the expeted net benefits of ontrol under strutured unertainty, it also leads to the poliy that maximizes robustness to unstrutured unertainty. To illustrate the results of our work, suppose that the marginal benefit and ost parameters take on the following values: b = 100, = 40, B = 1, and C = 2. In this ase q = 20 and t = 80. Beause C > B, t is always more robust that q (Figure 1). To verify this for allowable loss L = 100, use [7] to alulate ˆ( α ql, = 100) = and use [8] to alulate ˆ( α t, L = 100) = That is, welfare loss from estimation error no greater than L = 100 is guaranteed with the tax poliy if the estimated interepts err by no more than ± 20.4%. In L 13

16 ontrast, welfare loss no greater than 100 is guaranteed with the quantity poliy if the estimated interepts err by no more than ± 17.5%. That is, the robustness premium of the tax poliy is ± 2.9%. For a higher allowable welfare loss, say L = 300, ˆ( α ql, = 300) = and ˆ( α t, L = 300) = At this higher level of allowed welfare loss, the robustness premium of the tax poliy is ± 5.1%. In both of these ases, as well as any L > 0, the range of error in the estimates of the interepts of the marginal benefit and marginal ost funtions within whih the welfare loss is limited to be no more than standard. L is greater under the tax than under the abatement We an use our hypothetial numerial example to illustrate the trade-off between robustness-to-unertainty and welfare loss, expressed by the positive slopes of the urves in Figures 1 and 2. Whihever poliy is hosen, greater robustness an only be obtained in exhange for greater welfare loss. Furthermore, this trade-off is evaluated quantitatively. For instane, a robustness of ˆ α = 0.2 orresponds to immunity to 20% error in the estimated interepts, b and. For the tax poliy, this orresponds to a ritial welfare loss of L = 96. Pursuing greater robustness, say 40%, requires a greater ritial welfare loss of L = Conlusion We laim two ontributions of this work. First, we have illustrated a poliy evaluation tool, infogap robust-satisfiing, whih is an alternative to expeted welfare maximization. Info-gap robust-satisfiing is useful when errors in model parameters are unstrutured in the sense that they annot be modeled with known or onfidently estimated probability distribution funtions. In these ases, one annot evaluate poliies in terms of their affet on expeted benefits and 14

17 osts. Info-gap robust satisfiing requires no knowledge of underlying probability models, and evaluates poliies on the basis of robustness to loss when unertainty is unstrutured. Our seond ontribution is that we have reinterpreted the onventional wisdom of environmental eonomists that the relative slopes of the marginal benefits and marginal osts of pollution ontrol determine whether prie-based or quantity-based poliies should be pursued. It is onventional wisdom that the relative-slopes riterion leads to the poliy type that maximizes the expeted net soial benefits of environmental ontrol under strutured unertainty about soial benefits and osts. We have shown that this riterion also leads to the poliy hoie that is more robust to unstrutured unertainty about soial benefits and osts. We applied the info-gap robust satisfiing approah to the anonial model of environmental ontrol under unertainty that drives the onventional wisdom about pries versus quantities. It is well-known, however, that the relative-slopes riterion must be modified when there is also unertainty in the slopes of the marginal benefit and ost funtions, when the unertainty in benefits and osts are orrelated, when the marginal benefit and ost funtions are nonlinear, as well as when many of the other simplifying assumptions that underlie the onventional model are relaxed. (See footnotes 1 and 2). The approah we take in this paper an easily be applied to all of these more ompliated situations. We believe that doing so would be a fruitful area for future researh, and will lead to a deeper understanding of the design of environmental poliies under unertainty, and possibly to different poliy reommendations. 15

18 Aknowledgements: Professor Stranlund gratefully aknowledges funding for this researh by the U. S. Department of Agriulture under USDA/ERS/PREISM Cooperative Agreement No. 43-3AEM , and the Cooperative State Researh Extension, Eduation Servie, U. S. Department of Agriulture, Massahusetts Agriultural Experiment Station under Projet No. MAS

19 Referenes Baumol, William J. and Wallae E. Oates The Theory of Environmental Poliy, Cambridge University Press. Ben-Haim, Yakov Info-Gap Deision Theory: Deisions Under Severe Unertainty. Seond Edition, Aademi Press. Ben-Haim, Yakov Value at risk with info-gap unertainty. Journal of Risk Finane 6(5), Carmel, Yohay and Yakov Ben-Haim Info-gap robust-satisfiing model of foraging behavior: Do foragers optimize or satisfie? Amerian Naturalist 166(5), Halpern, Benjamin S.; Helen M. Regan, Hugh P. Possingham and Mihael A. MCarthy Aounting for unertainty in marine reserve design. Eology Letters 9, Hoel, Mihael and Larry Karp Taxes versus quotas for a stok pollutant. Resoure and Energy Eonomis 24(4) Knight, Frank H Risk, Unertainty and Profit. Houghton Mifflin Co. Re-issued by University of Chiago Press, Laffont, Jean Jaques More on pries vs. quantities. Review of Eonomi Studies 44(1), Malomson, James Pries vs. quantities: A ritial note on the use of approximations. Review of Eonomi Studies 45(1), Moffitt, L. Joe; John K. Stranlund and Barry C. Field. 2005a. Inspetions to avert terrorism: Robustness under severe unertainty. Journal of Homeland Seurity and Emergeny Management 2(3), Artile 3. Moffitt, L. Joe; John K. Stranlund, Barry C. Field, and Craig D. Osteen. 2005b. "Robust inspetion for invasive speies with a limited budget." Forthoming in The Eonomis of Plant Health. Alfons Oude Lansink (ed.). Springer. Moilanen, Atte and Brendan A. Wintle Unertainty analysis favours seletion of spatially aggregated reserve strutures. Biologial Conservation 129(3), Moledina, Amyaz A.; Jay S. Coggins, Stephen Polasky and Christopher Costello Dynami environmental poliy with strategi firms: Pries versus quantities. Journal of Environmental Eonomis and Management 45(2S), Montero, Juan-Pablo Pries versus quantities with inomplete enforement. Journal of Publi Eonomis 85: Piere, S.G.; K. Worden and G. Manson A novel information-gap tehnique to assess reliability of neural network-based damage detetion. Journal of Sound and Vibration 293(1-2), Quirion, Phillippe Pries versus quantities in a seond-best setting. Environmental & Resoure Eonomis 29(3): Simon, Herbert A Reason in Human Affairs. Stanford University Press, Stanford. 17

20 Stavins, Robert Correlated unertainty and poliy instrument hoie. Journal of Environmental Eonomis and Management 30(2), Vinot, P.; S. Cogan and V. Cipolla A robust model-based test planning proedure. Journal of Sound and Vibration 288(3), Weitzman, Martin Pries vs. quantities. Review of Eonomi Studies 41(4), Weitzman, Martin Reply to Pries vs. quantities: A ritial note on the use of approximations by James M. Malomson. Review of Eonomi Studies 45(1),

21 ˆ( α p, L ) ˆ( α pl, ) ˆ( α t, L ) ˆ( α ql, ) ˆ( α ql, ) ˆ( α t, L ) Figure 1: Relative robustness of t and q when C > B. L Figure 2: Relative robustness of t and q when B > C. L 19