Production and Abatement Distortions under Noisy Green Taxes

Transcription

1 Prodution and Abatement Distortions under Noisy Green Taxes ongli Feng and David A. ennessy Working Paper 05-WP 409 Otober 005 Center for Agriultural and Rural Development Iowa State University Ames, Iowa ongli Feng and David ennessy are with the Department of Eonomis and the Center for Agriultural and Rural Development at Iowa State University. Senior authorship is not assigned. This paper is available online on the CARD Web site: Permission is granted to reprodue this information with appropriate attribution to the authors. Questions or omments about the ontents of this paper should be direted to ongli Feng, 560D eady all, Iowa State University, Ames, IA ; Ph: (515) ; Fax: (515) ; Iowa State University does not disriminate on the basis of rae, olor, age, religion, national origin, sexual orientation, gender identity, sex, marital status, disability, or status as a U.S. veteran. Inquiries an be direted to the Diretor of Equal Opportunity and Diversity, 3680 Beardshear all, (515)

2 Abstrat Pigouvian taxes are typially imposed in situations where there is imperfet knowledge on the extent of damage aused by a produing firm. A regulator imposing imperfetly informed Pigouvian taxes may ause the firms that should (should not) produe to shut down (produe). In this paper we use a Bayesian information framework to identify optimal signal-onditioned taxes in the presene of suh losses. The tax system involves reduing (inreasing) taxes on firms identified as ausing high (low) damage. Unfortunately, when an abatement deision has to be made, the tax system that minimizes prodution distortions also dampens the inentive to abate. In the absene of wrong-firm onerns, a regulator an solve the problem by not adjusting taxes for signal noise. When wrong-firm losses are a onern, the regulator has to trade off losses from distorted prodution inentives with losses from distorted abatement inentives. The most appropriate poliy may involve a ombination of instruments. Keywords: onditioning; heterogeneity; informativeness; Pigouvian tax; signaling JE lassifiation: D6, 0

3 Prodution and Abatement Distortions under Noisy Green Taxes 1. Introdution Nonpoint soure pollution of air and water is an important and partiularly vexing eonomi problem. When aurate monitoring is prohibitively expensive, the orretive regulation will inevitably involve errors in assessing ontributions to damage. A large literature, muh of whih is surveyed in Shortle and oran [13], has emerged with intent to identify optimal poliies. Some studies, as in Xepapadeas [14], and Millok, Sunding, and Zilberman [11], onsider the at of monitoring as a poliy hoie variable. Most of them, suh as the ambient taxes modeled in Segerson [1] and Cabe and erriges [3], use proxies for damage when assessing an imperfet Pigouvian tax. owever, as far as we an asertain, no studies have been done on how taxation in the presene of noisy signals onerning damage might affet inentives to abate. That is the matter of our artile. We pose a Bayesian model in whih external damage an take two levels and polluters an be of two types with regard to pollution ativities. The types differ in the probability of ausing low damage. Otherwise, types differ by prodution osts. A regulator an monitor a firm to better disern the firm s type, and then use this information to guide firm ativities (produe or not). The osts of monitoring are not relevant to the argument we make, and we ignore them. The signal reeived, though unbiased and not manipulable by the firm, is imperfet or noisy. In the absene of any opportunity to abate on the part of the firm, we show it is optimal for the regulator to use the signal to assess a tax equal to the firm s expeted damage onditional upon the signal. In this way, losses from what we all the wrong-firm problem (WFP) are minimized. The WFP is the sum of two onerns: that a noisy Pigouvian tax indues i) a firm that would on average generate positive soial surplus under prodution to not produe, and ii) a firm that would on average generate negative soial surplus under prodution to produe.

4 We then introdue an abatement opportunity, the ost of whih varies aross firms. In this setting, speifying the tax equal to expeted damage may not be optimal. This is beause the averaging involved in forming an expetation on damage redues the inentive to inur the abatement ost. A further issue is that in the Bayesian game there may be a unique equilibrium set of abaters, or multiple equilibrium sets of abaters. In all ases, however, too little abatement will our. The adverse onsequenes for abatement are shown to be most severe when the WFP is a signifiant problem, i.e., when one might otherwise have benefited most from setting taxes equal to expeted damage. This presents a poliy dilemma in that, without further information, prodution distortions may have to be traded off against abatement distortions. We disuss the judiious use of poliy instrument ombinations.. Model framework The model we develop onerns regulation over a set of firms suffiiently small that prodution onsequenes do not affet output market pries. A onsumption good has market value B and an be produed by a ontinuum of firms with eah supplying one unit of output. These firms differ by the ost 0, [, ], required to produe that single unit of output, and by the level of negative externality θ 0, θ [ θ, θ ], produed as a by-produt. 1 Entries and exits our at no ost. A signal x [ xx, ] an be obtained on θ at no ost to the viewer. The signal may be thought of as a tehnology indiator, field slope (nitrogen run-off), or proximity to a ity (hog odor). Or the signal may be the result of a siene-based test on air or water emissions. 1 Our model an also be interpreted as one of heterogeneity within a firm whereby different outputs are tagged with different and θ values. Then the firm s deision is not whether to produe a single output but rather how muh to produe. Various issues that lead to noise in a field level test for phosphorus runoff are provided in Mallarino et al. [9]. Klatt et al. [8] use five soil phosphorus tests on 33 field soil samples in an Iowa watershed. Result orrelations range from 0.97 down to Bearing in mind that none of the tests measures true soil phosphorus, this attests to the signifiane of noise in real-world signals that might be used.

5 Cost is distributed independently of the signal and damage so that firms are distributed X aording to density f fθ, x( θ, x) on support 3 = [, ] [ θθ, ] [, ], a box in the spae S x x X of three-dimensional reals. The supersripted X on fθ, x ( θ, x) indiates the signal struture. If a different set of signals, labeled X, is available then we write the new joint density between X signals and damage as f, ( θ, x). For onveniene, we normalize suh that θ x S f f (, x) dd dx 1. In addition, and again for onveniene, we assume that X θ, x θ θ = f f (, x) 0 X θ, x θ > over the entire support. We do not onsider at this point one additional soure of heterogeneity, namely, the ost of an abatement ation. Write the mass density funtion for soial ost z = + θ as f ( z ), 3 the umulative mass z funtion as z F ( z) = f ( s) ds, the umulative marginal for as z 0 z F () = f () s ds, and the 0 X X X signal-onditioned mass density for θ as f x( θ x) = f, x( θ, x)/ f, x( θ, x) dθ. Consisteny X X requires f, x( θ, x) dx f, x( θ, x) dx f ( θ) X, X θ θ θ θ θ θ θ θ, so that we may omit the signal struture θ supersript in this ase. Other marginals, umulatives, and onditionals are represented in like manner. Desribe means by the average, or expetation, operator E[ ]. In partiular, the signal- X onditioned mean of θ under signal struture X is written as EX[ θ x] = θ f x( θ x) dθ. We first onsider a perfetly informed (i.e., knowing θ ) soial planner who wishes to implement a tax to maximize soial surplus. Another way of saying this is that the tax should optimally sort firms aording to prodution status. With soial ost per unit output written as z = + θ, the soial planner s problem is to hoose the value of z= zˆ so that when only firms with z values at or below ẑ produe then soial surplus is maximized, i.e., find the argument θ 3 Variable + θ has nothing to do with the signals, so we omit the supersript indiator of the signal struture when writing fz( z ). 3

6 that solves 4 max BF ( zˆ ) zf ( z) dz. (1) zˆ z ˆ z z z The optimality ondition is = 0, () * B z fb where * z fb is the marginal firm s value of + θ. A firm with + θ produes while a firm * z fb with + θ > does not. * z fb It is shown in the appendix that a tax at rate E[ θ] = θ f ( θ) dθ, i.e., average damage over the entire firm set, is optimal when the regulator has available no onditioning information. Under this tax, firms produe if B E[ θ ]. Different firms are ative under the first-best tax when ompared with under the uninformed tax. Partition parameter spae S into four sets. Set A ( A ) omprises firms that do (do not) produe under the average tax solution and should not sn d sn dn θ produe under no onditioning information. Set A ( A ) omprises firms that do (do not) s d s dn produe under the average tax solution and should produe under no onditioning information. The set measures are A = f f ( θ) ddθ; A = f f ( θ) ddθ; sn sn d B E[ θ], θ dn > B E[ θ], * + θ> zfb * + θ> zfb A = f f ( θ) ddθ; A = f f ( θ) ddθ. s s d B E[ θ], θ dn > B E[ θ], * * + θ zfb + θ zfb θ θ (3) Fig. 1 illustrates, where the shaded areas represent the two different types of errors. The problem with the uninformed tax solution is that, in general, the measure of firms in eah of sets sn A d and s A dn is likely to be nontrivial. Pigouvian taxation may improve performane, but the extent of suess will depend upon the statistial attributes of parameters and θ, and in partiular on how well a tax mathes the damage done. Fig. 1 provides a benhmark for our 4 We ould introdue a downward-sloping demand funtion and inlude onsumer surplus in the alulation. All the insights to follow still apply, but the demonstrations are longer. 4

7 analysis. It is, however, only a limiting ase for the problem we will address. In Fig. 1 it is assumed that all information on damage is available to the regulator. More generally, we will study the ase where a regulator reeives a signal and uses it to design signal-onditioned optimal taxes. 3. Pigouvian taxation The model in this setion does not address the abatement issue. It only asks how a regulator should tax in the presene of noisy signals on damage when the single deision a firm an make is whether to produe. Sine a firm s value of θ is not always available to the regulator, it is more realisti to ondition the tax on some other attribute. et the timeline be that 0) nature draws and type for a firm, 1) then the regulator reeives a signal about a firm s prodution environment and levies a tax on the unit in order to optimally deter damage, and ) then the produer makes the prodution deision. The signal reeived at time 1 is random; it is not manipulated in any way by the sender. Fig. provides a timeline. Our fous is on eonomi losses due to poliy issues. In order to be onrete about how poliy problems an arise and an be mitigated, we plae more struture on the firm set and on signal strutures. There are two damage levels, with value θ and with value θ < θ. There are two types of firms, type α and type α + δ, δ > 0. For type α (type α + δ ), the regulator observing the produer s operation will reeive an signal with probability α [0,1) (respetively, α + δ (0,1] ) and an signal with probability 1 α (respetively, 1 α δ ). The way to view α is as a lottery for the regulator whereby the true damage will turn out to be θ with probability α and θ with probability 1 α. The signal emitted by a firm is unbiased in that if the firm is an α then is emitted with probability α and damage is θ with probability α. But the signal realization is independent of the damage realization. Note that even if the regulator were to observe the firm type, she would still be exposed to this lottery risk. When we 5

8 assert that a firm should produe we mean that the firm provides a lottery that makes the firm, in expetation, a (positive) surplus generator. Type-onditioned expeted damage levels are E [ θ] = αθ + (1 α) θ ; α E [ θ] = ( α+ δθ ) + (1 α δθ ) = E [ θ] δθ ( θ) < E [ θ]; α+ δ α α (4) on eah of the respetive types. These would be appealing tax assessments were firm type known. But firm type is not known, and the only information available to the regulator is signal ii, {, }. The share of firms that are α + δ is, for this setion, exogenous at φ (0,1). ater we will endogenize φ when we introdue an abatement hoie and identify inentives problems onerning abatement investments. In this setion the intent is to show that there are effiieny problems that have nothing to do with abatement ineffiienies. When the regulator reeives a signal ii, {, }, then she uses Bayes theorem to assess π π α+ δ α+ δ ( α + δ) φ ( α + δ) φ = = ; ( α + δ) φ + α(1 φ) α + δφ (1 α δ) φ (1 α δ) φ = = ; (1 α δ) φ + (1 α)(1 φ) 1 α δφ (5) where π α + δ i is the probability that the firm is a α + δ given signal i {, }. The expeted damage levels given signals are E [ θ] = π E [ θ] + (1 π ) E [ θ] α+ δ α+ δ α+ δ α α(1 φ) = ( α+ δθ ) + (1 α δθ ) + ( θ θ) δ ( α+ δθ ) + (1 α δθ ) ; α + δφ E [ θ] = π E [ θ] + (1 π ) E [ θ] E α+ δ α+ δ α+ δ α (1 α δ) φ = αθ + (1 α) θ + ( θ θ) δ αθ + (1 α) θ; 1 α δφ ( θ θ)(1 φ) φδ [ θ] E[ θ] = 0. (1 α δφ)( α + δφ) (6) The following is shown in the appendix. 6

9 Proposition 1. Optimal taxes are t = E [ θ ] under signal and t = E [ θ ] under signal, as provided in (6). eneforth we will label t = E [ θ ] and t = E [ θ ] as signal-onditioned expeted damage taxes, or SCED taxes. Note that, from (4) and (6), θ E [ θ] t t E [ θ] θ. The α+ δ α SCED taxes are squeezed between the taxes that would apply were firm type known to the regulator. This observation will be entral when we study abatement inentives. To this point we have assumed that the lotteries represented by firms of types α and α + δ are fixed. These lotteries represent the informativeness of the signals given. We will show next that the level of dispersion in signals determines the level of dispersion in optimal taxes. 5 This is relevant beause, as we will show later, dispersion in taxes enourages abatement. et a different onditioner be used so that the signal struture hanges, X X. The different onditioner may perhaps be a different sientifi test or attribute of an asset. The nature of the hange is that α α η where η 0 so that the fration of signals for a non-abating firm delines. Sine only the signal hanges, we must preserve unbiasedness in the unonditional probability of low damage. Thus, ( α + δ) φ + α(1 φ) = α + δφ, and we must have α + δ α + δ + η, η = (1 φ) η / φ, so that α(1 φ) + ( α + δ) φ ( α η)(1 φ) + ( α + δ + (1 φ) η / φ) φas required. Under SCED taxation and the new signal struture, t = E [ θ ] in (6) beomes where t E E ( η) = π α+ δ+ η α+ δ+ η[ θ] + (1 π α+ δ+ η ) α η[ θ], (7) 5 This is related to observations by Blakwell [1] and Epstein [5] that the value of information and utility of information in deision making are related to the dispersion of the signals the information onveys. 7

10 (1 φ) π α+ δ+ η = πα+ δ + η; α + δφ (1 φ)( θ θ) Eα η[ θ] = Eα[ θ] + ( θ θ) η; E α+ δ+ η[ θ] = Eα+ δ[ θ] η; φ (8) so that (1 φ) (1 φ)( θ θ) t( η) = πα+ δ + η Eα+ δ[ θ] η α + δφ φ (1 φ) + 1 πα+ δ η ( Eα[ θ] + ( θ θ) η) α + δφ δ(1 φ)( θ θ ) (1 φ)( θ θ ) = t = t = α + δφ ( α + δφ) φ ( η 0) η η ( η 0). (9) Similarly, under ex post taxation and using π = π (1 φ) η/(1 α δφ) beomes α+ δ+ η α+ δ, then t t ( η) = π E [ θ] + (1 π ) E [ θ] α+ δ+ η α+ δ+ η α+ δ+ η α η δ(1 φ)( θ θ ) (1 φ)( θ θ ) = t = + + t = 1 α δφ (1 α δφ) φ ( η 0) η η ( η 0), (10) so that δ (1 φ)( θ θ) t( η) t( η) = t( η = 0) t( η = 0) + η (1 α δφ )( α + δφ ) (1 φ)( θ θ) + η t( η = 0) t( η = 0). φ(1 α δφ)( α + δφ) (11) From equations (9)-(11), the following results an be easily obtained. Proposition. Under SCED taxation, all of t, t, and t t are inreasing and onvex in η. A more informative signaling system, as defined by an inrease in η, will render the SCED taxes more dispersed. We will defer omments on Proposition until we study abatement problems that SCED taxes may generate. Our next goal is to haraterize the losses that SCED taxes are optimal in guarding against. 8

11 4. osses from the WFP Under the SCED taxes given in (6), we will develop a loss funtion apturing statistial type I and II errors, namely, the losses when a tax indues firms that I) should produe (under the lottery in damages the firm provides) to shut down, and II) should shut down to produe. Consider I) first. This an our when the tax is too high, i.e., when the firm is type α + δ and either the signal is i) or ii). 6 For i), the expeted benefit from prodution would be B E α δ [ θ ] 0 sine the firm should produe. But B t < 0, so that the firm does not + produe. Therefore, the error ours on firm set ( B t, B E α δ [ θ ]]. The expeted loss from non-prodution is B E α δ [ θ ] 0 over this set of firms. Finally, the probability of the error is + 1 α δ on fration φ of firms. For ii), the expeted benefit from prodution is as before. But B t < 0 so that the error ours on firm set ( B t, B E α δ [ θ ]]. The probability of the error is α + δ on fration φ of firms. Summing the two ases and integrating over, aggregate loss from this sort of error is + + I ( B t, B Eα+ δ [ θ ]] ( B t, B Eα+ δ[ θ ]] = (1 α δ) φ B E [ θ] df α+ δ + ( α + δ) φ B E [ θ] df. α+ δ (1) Now onsider prodution by firms that should shut down but the tax is too low, i.e., when the firm is an α and the signal is either i) or ii). For ase i), the expeted benefit from prodution would be B E α [ θ ] < 0 sine the firm should not produe. But B t 0, so that the firm produes. Therefore, ( B E [ ], B t ]. The expeted loss from prodution is α θ ( B E α [ θ ]) 0 over this firm set. The probability of misdiagnosis is α on fration 1 φ of firms. For ase ii), the expeted benefit from prodution is as before. Now, though, B t 6 When the signal is and the type is α then satisfation of the inequalities B E α [ θ ] 0 and B t < 0 would lead to type I error. But (6) relates that t > B E α [ θ ] annot our. 9

12 0 so that the firm set in question is ( B E [ ], B t ] with probability of misdiagnosis 1 α on fration 1 φ of firms. Summing the two ases and integrating over, the aggregate loss from this sort of error is α θ II ( B Eα [ θ ], B t ] = α(1 φ) + E [ θ] B df ( B Eα [ θ ], B t ] + (1 α)(1 φ) + E [ θ] B df. α α (13) Summing, total expeted loss from misdireted firms is 7 = +. (14) I II The loss funtion is desribed in Fig. 3, with B on the horizontal axis and onditional loss on the vertial axis. Intervals Ia and Ib are where type I errors our; see (1). In the loss funtion, two weightings are applied. One is a ost weighting as represented through the integration in (1) and the downward-sloping line segment in Fig. 3. The other is a weighting by the overall probability that wrong signals of this sort our, and this seond weighting differs aross the intervals Ia and Ib. Interval IV represents firms that do not produe and should not produe under both signals, so there are no losses for suh firms. Intervals IIa and IIb are where type II errors our, and the struture in (13) is similar to that for type I error. On the right is interval III, representing firms suh that prodution should and does our (given limited information as represented by the lotteries) under either signal so there ould be no loss. 8 The possibility of type I and II errors does not disappear when the signal is ompletely informative beause it informs only on firm type and the types ontinue to emit heterogeneous signals. 5. Abatement, absent the WFP From this point on we extend the - damage model by assuming a firm an alter the extent 7 Optimizing over hoies of t and t provides an alternative proof of Proposition 1; see the appendix. 8 Obviously a firm annot be subjet to both error types. While the intervals overlap, they are for different firm types and that additional dimension is omitted in the figure. 10

13 of damage done through an abatement ativity. That is, the firm beomes type α or type α + δ by making an abatement hoie. The timeline is now that 0) nature draws the prodution ost and the abatement ost, all it e, for a firm, 1) then the firm makes a non-observable investment deision on abatement that determines firm type, ) then the regulator reeives a signal ii, {, }, about a firm s prodution environment and levies a SCED tax on the unit, and 3) then the produer makes the prodution deision. Fig. 4 provides the timeline. In this setion, however, we assume that firms always operate (i.e., B is suffiiently high) sine our intent is to understand how SCED taxation in the presene of noisy signals on damage an affet inentives even absent the WFP. We will later show that the problem we identify here and the WFP have similar effets on inentives. Signals emitted by firms are affeted by the abatement ation. Under no ation, the regulator observing (ex post) the produer s operation will reeive an signal with probability α (0,1) and an signal otherwise. Under the ation, denoted as letter a, the regulator observing the produer s operation will reeive an signal with probability α + δ, and an signal otherwise. The ost of the ation, e, is heterogeneous aross firms, having stritly inreasing distribution Je :[ ee, ] [0,1]. This distribution is known to the regulator, but no more is known to the regulator apart from the observed signal. So a firm is now haraterized by the ouple ( e, ), and the firm s lottery haraterization of α or α + δ is an endogenous hoie. For onveniene we will assume that e and are independent. The regulator s assessed SCED tax upon reeiving a signal i {, } is t i. The sum of expeted tax and ation ost to the firm upon not taking the ation is set at αt + (1 α) t sine t ours with probability α and t ours with probability 1 α. The expeted tax upon taking the ation is ( α + δ) t + (1 α δ) t + e, so that the ation will be taken if ( t t ) δ e. The indifferent produer is that with e value eˆ = ( t t ) δ, (15) 11

14 while the set that hooses to inur the abatement sunk ost has measure Je. ( ˆ) For later omparison, the effiient solution is when the threshold abatement ost satisfies ( α + δ θ + (1 α δ) θ + e = αθ + (1 α) θ, with solution e = ( θ θ ) δ. (16) To identify the problem with SCED taxation, assume that taxes are as in (6), exept that endogenous Je ( ˆ) substitutes for exogenous φ. Thus, taxes are ) t t t α[1 J( eˆ )] = ( α + δ) θ + (1 α δ) θ + ( θ θ ) δ θ ; α + δj( eˆ ) (1 α δ) J( eˆ ) = αθ + (1 α) θ + ( θ θ ) δ θ ; 1 α δj( eˆ ) ( θ θ)[1 J( eˆ)] J( eˆ) δ t = 0; [1 α δj( eˆ)][ α + δj( eˆ)] (17) so that eˆ = ( t t ) δ ( θ θ ) δ = e. Notie that t is dereasing in e beause the probability the ation is taken given is inreasing in the fration taking the ation. Similarly, t is dereasing in e. This means that the taxes are (weakly at any rate) too high. Noting that ê [ ] e, means damage is given as JeE ( ˆ) α+ δ[ θ] + 1 Je ( ˆ) Eα[ θ] = αθ + (1 α) θ δje ( ˆ)( θ θ) αθ + (1 α ) θ δ J( e )( θ θ ), (18) and is soially exessive. In summary, Proposition 3. et the tax imposed on a firm be the SCED. Assume that e and are independently distributed and the support of is suh that no firm will shut down upon emitting signal. Then, relative to perfet signals on firm type, noisy signals lead to underinvestment in abatement and an inrease in mean damage. The entral point of this paper is the ontrast between Proposition 1 (where abatement opportunities were not available) and Proposition 3. When in the presene of abatement 1

15 opportunities, setting taxes equal to SCED will distort inentives. From (15) and (17) we have 3 J( eˆ)[1 J( eˆ)]( θ θ) δ eˆ =. [1 α δj( eˆ)][ α + δj( eˆ)] (19) Sine t t is not monotone in e we annot be sure from (19) whether there is a unique fixedpoint solution. Before going further, let us onsider what the ondition relates. Nature moves first in desribing the distribution of firm harateristis. The regulator s rule of setting taxes equal to SCED is known to firms. Firms make prodution and abatement deisions that are onsistent with that rule and in light of the noise externalities that firms impose on eah other. The game is Bayesian where the regulator interprets signals to arrive at fixed-point problem (19), solves it for the set abating, and then sets tax values using (17). Thus, (19) and (17) represent a perfet Bayesian equilibrium in a game of imperfet information as explained in Fudenberg and Tirole [6]. Three ases illustrate muh of the ontent in (17) and (19). Example 1. At one extreme suppose that α = 0 so that those not taking the ation annot attain the signal. Then π a = 1, i.e., those attaining the signal do damage θ + δ( θ θ) and are assessed a tax at that amount. On the other hand, t = θ + ( θ θ )(1 δ) J( e) δ /[1 δj( e)] < θ beause some ation takers reeive an signal. In equilibrium, ê = ˆ ˆ where the right-hand side is dereasing in ê. Thus, in this ase [1 J( e)]( θ θ) δ /[1 δj( e)] there is a unique solution to the equilibrium fration of ation takers. The value of inreases with an inrease in either δ or θ and dereases with an inrease in θ. Je ( ˆ) Example. At the other extreme, suppose that α + δ = 1 so that those taking the ation always attain the signal. Then t = αθ + (1 α) θ beause those emitting the signal ertainly do not take the ation. For signal emitters, t = θ + ( θ θ ) δ(1 δ)[1 J( e)]/[1 δ + δj( e)] > 13

16 ( α + δ) θ + (1 α δ) θ. This is beause some not taking the ation do emit a low signal. If α + δ = 1 then (19) redues to eˆ J eˆ θ ˆ θ δ δ δj e = ( ) /[1 + ]. The right-hand side is inreasing in ê and, furthermore, the derivative may have value exeeding 1. Indeed, with Je= ( 0) = 0 then the value e ˆ = 0 is a solution. In that ase any stritly positive solution will be unique whenever Je /[1 δ + δje ] is onave, i.e., whenever J e θ θ δ δ e( = 0)( )(1 ) > [1 ( 0)] δ + δj e= and + <. Notie that J ( e ) < 0 is (1 δ) Jee( e) δ{ J( e) Jee( e) [ Je( e)] } 0 ee suffiient to ensure a unique positive solution when α + δ = 1, but this is a strong requirement. In the terminology of Caplin and Nalebuff [4], the distribution funtion Je is 1 -onvex if [ J( e)] 1 is onvex. For suffiiently smooth distributions, [ J( e)] 1 is onvex whenever JJ ee <. Thus, a δ -weighted average between onavity on Je and 1 -onvexity ( J e ) 0 on Je ensures a unique positive solution when α + δ = 1. Example 3. Finally, suppose for simpliity that Je is the uniform distribution on [0,1] while 3 3 α = 0.5. Then (19) resolves to δ eˆ ( θ θ ) δ eˆ+ ( θ θ ) δ 0.5= 0 with solutions ( θ θ) δ 0.5 δ ( θ θ) δ 4( θ θ) δ 1 ± +. If and only if > θ θ δ, then there is a unique (stritly) positive root. 9 Suppose that ( θ θ ) δ = 1 so that, by (16), the effiient solution is for all to at. Then δ eˆ δ eˆ+ δ 0.5 = 0 with solutions 1 0.5(1 δ 1 3 δ ) ±. Sine δ (0,0.5] when α = 0.5, there is no solution on e ˆ (0,1]. 10 Thus, for δ (0,0.5) the only solution on e [0,1] is e ˆ = 0. That is, the Nash equilibrium is for no one to abate when the optimal solution is for all to abate. Is it a stable solution? From (19) and 9 e ˆ = 0 is one of three roots, but we have already anelled through by it Sine δ 1 3δ < 1 if and only if 0.5 < δ, whih is not possible, we need not 1 onsider the positive root. On the other hand, δ 1 3δ > 0 if and only if δ >

17 ( θ θ ) δ = 1, private benefit less private ost has value (1 )(0.5 δ ) δ. A 1 e e e e deviation to Δ e > 0 at the limit, im Δ e, gives a hange in private benefit of Δ e e+ δ e δ δ (0.5 δ e ) e= 0 1= (0) If δ (0,0.5) then a small Δ e > 0 inrease in e away from 0 dereases private gains so that the solution is stable. If instead α = 0, then (19) resolves to δeˆ [1 + ( θ θ ) δ ] eˆ+ ( θ θ ) δ = 0. If, as before, ( θ θ ) δ = 1 so that the effiient solution is for all to at then (19) beomes ˆ δe δ e δ (1 + ) ˆ+ = 0 with solutions [(1 + δ) /( δ)] ( 1 ± (1 δ)(1 + 3 δ)/(1 + δ) ). The positive root is outside e ˆ [0,1]. 11 The negative root is inside the unit interval. In that ase, private 1 benefits are δ(1 e)(1 δe) e and the derivate at that root is δ δ 1 (1 δ e) e= [(1 + δ) /( δ)] 1 (1 δ)(1+ 3 δ)/(1 + δ ) 1< 0, (1) so that the equilibrium is stable under a small deviation. Problems related to that in (15)-(16) have arisen elsewhere. Studying a ompetitive labor market, MGuire and Ruhm [10] showed that the inability to observe drug abuse diretly, rather than only through aidents, eliited an insuffiient wage inentive to ure drug dependeny. ennessy [7] and Bogetoft and Olesen [] showed that noisy grading of produe in the presene of ompetitive trading after prodution eliited insuffiient inentive to invest in quality enhanement. In our ase, the problem arises with SCED taxation and not with ompetitive The roots are on either side of the unit interval. 11 The larger root is at most one if and only if (1 δ)(1 + 3 δ) δ 1, whih is not possible. The smaller root is non-negative if and only if δ 0. Finally, the smaller root is at most one if and only if 0 δ so the smaller root is on the unit interval. 15

18 trading. As suh, the problem is one the regulator has muh ontrol over. Indeed, the problem originates from a regulatory attempt to inrease soial welfare. 6. Abatement, with the WFP In this ase B might possibly be so small as to deter prodution. It is under this possibility, but without abatement opportunities, that SCEDs give optimal taxes. Firm profit under signal i {, } and ost is max[0, B t ] so that the expeted revenue enhanement effet of investing in abatement is ( α + δ) max[0, B t ] + (1 α δ) max[0, B t ] αmax[0, B t ] (1 α) max[0, B t ] = δ(max[0, B t ] max[0, B t ]). When this exeeds the ost of investing in abatement, then investment ours, i.e., (15) beomes eˆ = max[0, B t ] max[0, B t ] δ, () where independene between and e ensures that the t i are as in Proposition 1. Fig. 5 graphs the right-hand side of () as the value of B i hanges. It is the ontinuous, piee-wise linear funtion labeled EPBA, for expeted private benefit from abatement. It is flat to t, with slope δ on ( t, t ], and flat on ( t, ). Graphed also is the expeted soial benefit from abatement, ( max[0, B ] max[0, B ] ) θ θ δ and labeled ESBA. It is flat to θ, with slope δ on ( θ, θ ], and flat on ( θ, ). Observe that θ θ t t max[0, B t ] max[0, B t ]. (3) This inequality, together with the logi underpinning Proposition 3, leads immediately to Proposition 4. Assume that e and are independently distributed. Assume too that SCED taxes are imposed. Relative to the ase without the WFP under noisy signals, under the WFP and noisy signals there is even more underinvestment in abatement and even larger mean damage. 16

19 The ontrast between Proposition 1 and Proposition 3 identified a problem with SCED taxes when abatement deisions are to be made. But it assumed away the problem motivating the SCED taxes, leaving the potential loophole that the abatement problem beomes less relevant in the presene of the WFP. Proposition 4 shows that the abatement problem beomes worse in this ase. Optimal use of signals to mitigate WFP losses involves guarding against a tax that is too severe when the signal is and too permissive when the signal is. But then the inentive to invest in ahieving a lower level of expeted damage is dulled, and so fewer firms make the investment. 7. Naïve taxation and other poliy issues We have seen that levying Pigouvian taxes at the SCED levels redues the inentive to abate and, furthermore, the extent of redution is stronger when the WFP is relevant. If the WFP never ours, then optimal taxes are easy to identify. Just ompare (15) with (16) to see that one should set t = θ and t θ =. 1 We all this naïve taxation in that the regulator may be viewed as naïvely assuming that an signal means that the monitored firm always auses damage to extent θ. In this ase, a little knowledge is a dangerous thing in that a omparatively sophistiated regulator who uses Bayes rule reates a problem that a regulator who taxes θ i upon seeing i {, } would avoid. owever, in strething out the penalties like that then WFP losses likely inrease. From (14), loss due to the wrong-firms problem beomes ( B θ, B Eα+ δ[ θ]] ( B θ, B Eα+ δ[ θ]] ( B Eα [ θ], B θ ] ( B Eα [ θ], B θ ] = (1 α δ) φ B E [ θ] df α+ δ + ( α + δ) φ B E [ θ] df α+ δ + α(1 φ) + E [ θ] B df + (1 α)(1 φ) + E [ θ] B df. α α (4) 1 Bogetoft and Olesen [] make a related point when motivating the use of ompetition through ontrats rather than ompetition through ompetitive trading after prodution in markets for 17

20 It may, however, be that strething the taxes delivers no additional abatement. This would our were Je ( ) Je ( ˆ) = 0, i.e., were no firms deterred from abating in any ase. Suppose that equilibrium e is ompletely inelasti to the shift from SCED taxation to naïve taxation so that we may write Je as φ, exogenous. Then the hange in losses due to the WFP is ( B θ, B t ] ( B t, B θ ] ( B t, B θ ] Δ = (1 α δ) φ B E [ θ] df + ( α + δ) φ B E [ θ] df + α(1 φ) + E [ θ] B df ( B θ, B t ] + (1 α)(1 φ) + E [ θ] B df. α α+ δ α α+ δ (5) Sine B E α δ [ θ ] on ( B θ, B t ] and B E α [ θ ] on ( B t, B θ ], the first + and third right-hand terms are non-negative. The other two right-hand terms are of ambiguous sign and would be non-negative if df = df = 0. Therefore, the ( θ, Eα+ δ [ θ]] ( Eα[ θ], θ ] unsuessful attempt to eliit more abatement an be at the real ost of further distortions in the firm sets produing and not produing. Cirumstanes are readily identified where SCED taxation and a subsidy on abatement would be an improvement over either naïve taxation or SCED taxation by themselves. Viewing Fig. 5, we an break the analysis into five ases. Case i), B θ : This is not an interesting ase beause prodution should never our and would never our under naïve taxation or SCED taxation. Case ii), [ B t, B θ ): In this ase, the gap between soial and private benefit from abatement is ( max[0, B ] max[0, B ] max[0, B t ] + max[0, B t ] ) = ( B θ ) δ. θ θ δ (6) graded produe. 18

21 Were it possible, and in addition to SCED taxation, providing a small diret subsidy on abatement to this amount would be optimal. Sine abatement is not observable, the subsidy might be in the form of publi involvement in tehnology awareness and/or skill development. Note that the subsidy in (6) depends on so that, even within the Case ii) interval, the value of would need to be known in order to implement the subsidy. Case iii), [ B t, B t ): Equation (6) assumes the value ( E [ θ] θ ) δ. Were it feasible, and in addition to SCED taxation, this ost-independent amount would be the optimal subsidy on abatement. Case iv), [ B θ, B t ): The subsidy would be ( B θ + E [ θ] E [ θ]) δ in this ase, and quite diffiult to implement. Case v), < B θ : This ase has already been onsidered in detail in Setion 5. The abatement subsidy would be ( θ θ + E [ θ] E [ θ]) δ, but naïve Pigouvian taxation without an abatement subsidy would also work. When osts span two or more of these ases, no single subsidy level will support first best. Finally, a further approah would be to takle the problem diretly by subsidizing researh and development of a more informative test. Proposition, when viewed together with (15)-(16), shows that the level of abatement that ours under imposition of SCED taxes is inreasing and onvex in η, implying inreasing marginal benefits from an innovation that inreases the value of test informativeness. 8. Conlusion The intent of this paper has been to tease out onsequenes of Pigouvian taxation in the presene of noisy signals. For poliy guidane, we are of the view that a bundle of poliies may prove most pratial in remedying losses from distorted prodution and abatement levels. This 19

22 may involve signal-onditioned expeted damage taxes, subsidies where possible on abatement ativities, and subsidies to develop more effetive testing tehnologies. 0

23 Figure 1. Sets of open and shut firms under average damage tax. Figure. Timeline under Pigouvian taxes. 1

24 Figure 3. osses from WFP aross firm osts. Figure 4. Timeline with opportunity to abate.

25 Figure 5. Set of abaters under noise and WFP. 3

26 Appendix Demonstration that tax t = E[ θ ] is optimal under no onditioning information: If tax t is imposed suh that B < t, then the firm shuts down and no eonomi surplus is generated. If tax t is imposed suh that B t, then prodution ours. But soial surplus ould be positive or negative. The regulator s task is to hoose t to maximize soial surplus, or B t (A1) max t ( B θ) df df θ ( θ). Rearranging provides θ B t B t B t t θ θ t (A) max ( B θ) df df ( θ) = max ( B ) df E[ θ] df. The differential is E[ θ ] t f ( = B t) = 0, (A3) with solution t = E[ θ ] sine f ( = B t) 0 on [, ]. To show that this is indeed the maximum, note that the seond-order ondition is f ( = B t) + E[ θ ] t df ( = B t)/ dt = f ( = B t) < 0 when evaluated at t = E[ θ ]. Proof of Proposition 1. The proof is as in the demonstration above, exept that the measure F θ ( θ ) is disrete and beomes a signal-onditioned measure. (A3), then, beomes E [ θ ] t = 0 under signal and E [ θ ] t = 0 under signal. 4

27 Alternative Proof of Proposition 1. From (1), (13), and (14), ( B t, B Eα+ δ[ θ ]] ( B t, B Eα+ δ [ θ ]] ( B Eα [ θ ], B t ] ( B Eα [ θ ], B t ] = (1 α δ) φ B E [ θ] df α+ δ + ( α + δ) φ B E [ θ] df α+ δ + α(1 φ) + E [ θ] B df + (1 α)(1 φ) + E [ θ] B df. α α (A4) Differentiate and set equal to 0: d = [( α + δ) φ( t Eα+ δ[ θ]) α(1 φ)( Eα[ θ] t) ] f( = B t) = 0; dt d dt [ α δ φ θ α φ θ ] = (1 ) ( t E [ ]) (1 )(1 )( E [ ] t ) f ( = B t ) = 0. α+ δ α (A5) Given positive support for on [, ], (A5) resolves to the tax levels as given in (6). Convexity at the solution to (A5) is immediate upon applying (A5) to seond derivatives of (A5). 5

28 Referenes [1] D. Blakwell, Equivalent omparisons of experiments, Annals Math. Stat. 4 (, June) (1953) [] P. Bogetoft,.B. Olesen, Inentives, information systems, and ompetition, Amer. J. Agr. Eon. 85 (1, February) (003) [3] R. Cabe, J.A. erriges, The regulation of non-point-soure pollution under imperfet and asymmetri information, J. Environ. Eon. Manage. (, Marh) (199) [4] A. Caplin, B. Nalebuff, Aggregation and imperfet ompetition: On the existene of equilibrium, Eonometria 59 (1, January) (1991) [5].G. Epstein, Deision making and the temporal resolution of unertainty, Int. Eon. Rev. 1 (, June) (1980) [6] D. Fudenberg, J. Tirole, Game Theory, MIT Press, Cambridge MA, [7] D.A. ennessy, Information asymmetry as a reason for food industry vertial integration, Amer. J. Agr. Eon. 78 (4, November) (1996) [8] J.G. Klatt, A.P. Mallarino, J.A. Downing, J.A. Kopaska, D.J. Wittry, Soil phosphorus, management praties, and their relationship to phosphorus delivery in the Iowa Clear ake agriultural watershed, J. Environ. Qual. 3 (6, November/Deember) (003) [9] A.P. Mallarino, B.M. Stewart, J.. Baker, J.D. Downing, J.E. Sawyer, Phosphorus indexing for ropland: Overview and basi onepts of the Iowa phosphorus index, J. Soil Water Conserv. 57 (6, November/Deember) (00) [10] T.G. MGuire, C.G. Ruhm, Workplae drug abuse poliy, J. ealth Eon. 1 (1, April) (1993) [11] K. Millok, D. Sunding, D. Zilberman, Regulating pollution with endogenous monitoring, J. Environ. Eon. Manage. 44 (, September) (00) [1] K. Segerson, Unertainty and inentives for nonpoint pollution ontrol, J. Environ. Eon. Manage. 15 (1, Marh) (1988) [13] J.S. Shortle, R.D. oran, The eonomis of nonpoint pollution ontrol, J. Eon. Surveys 15 (3, July) (001) [14] A.P. Xepapadeas, Observability and hoie of instrument mix in the ontrol of externalities, J. Pub. Eon. 56 (3, Marh) (1995)