2 1. Introduction Under many of the maor U.S. environmental law e.g., the Comprehenive Environmental Repone, Compenation, and Liability Act (CERCLA, a

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1 EXTENDED LIABILITY FOR ENVIRONMENTAL ACCIDENTS: WHAT YOU SEE IS WHAT YOU GET Emma Hutchinon Klaa van 't Veld Department of Economic, Univerity of Michigan Abtract When a firm may be bankrupted by the liability from an environmental accident, current law often allow for the extenion of liability to third partie with whom the firm contract, with the aim of inducing full internalization of ocial cot. We find that, when the firm can take both obervable and unobervable care to reduce expected accident damage, extended liability indeed reult in full cot internalization, but not in firt-bet level of care. We alo find that, wherea without extended liability there i exce entry into environmentally hazardou indutrie, introducing extended liability lead to exit that, while exceive relative to the firt bet, i econd-bet optimal given firm' choice of care. Furthermore, we how that direct regulation of obervable care, when combined with extended liability, further ditort firm' incentive. However, when ued alone, uch regulation trictly dominate extended liability. Keyword: environmental rik, udgment proof problem, extended liability, lender liability, indutrial accident, bankruptcy, CERCLA

2 2 1. Introduction Under many of the maor U.S. environmental law e.g., the Comprehenive Environmental Repone, Compenation, and Liability Act (CERCLA, alo known a Superfund), the Oil Pollution Act, the Clean Water Act firm face trict liability for cleanup cot and other damage from environmental accident. From the law-and-economic literature (e.g., Shavell (1987)), it i well known that uch ex pot liability can provide firm with optimal incentive ex ante to take care i.e., to undertake afety meaure that reduce expected damage becaue it induce firm to internalize the full cot aociated with accident. Becaue environmental cleanup are often very cotly, however, it i not uncommon for environmental damage to exceed the market value of the firm reponible for an accident. In uch cae, the firm i bankrupted by it liability, and i aid to be udgment proof": becaue it cannot loe more than it market value in bankruptcy, it cannot be forced to pay the full accident damage, and therefore effectively externalize any reidual damage in exce of it value. One way in which environmental law attempt to remedy thi udgment proof problem" i by extending liability for reidual damage to partie with whom a firm interact. 1 Under CERCLA, for example, if a dipoal firm directly reponible for a hazardou-wate pill i udgment proof, reidual liability i frequently extended to firm that originally generated the wate or tranported it to the dipoal ite. The argument for extending liability to third partie (laid out for example by Segeron (1993)) i that, uing their contractual leverage, third partie will pa on their reidualliability cot to the udgment-proof firm, thereby retoring the firm' incentive to take optimal care. Formal analyi of thi argument ha hown that the degree to which it hold up in practice depend importantly on three feature of the intitutional etting, namely (i) whether afety meaure undertaken by the firm are obervable to third partie (Pitchford (1995), Boyd and Ingberman (1996, 1997), Heye (1996), Boyer and Laffont (1997), Fee and Hege (2000a), Lewi and Sappington (1999, 2001)), (ii) whether third partie have market power or not (Sunding and Zilberman (1998), Balkenborg (2001)), and (iii) whether the owner of the udgment-proof firm can increae 1 Other potential remedie include mandatory inurance and bonding requirement. While the model of extended liability developed below can be generalized to analyze thee alternative remedie, doing o i beyond the cope of our paper.

3 the amount of equity capital inveted in the firm (Pitchford (1995), Boyd and Ingberman (1996, 1997)). In thi paper, we abtract from the latter two iue by auming that third partie have no market power and the firm' owner have no equity. Thi allow u to focu on the firt iue, i.e., on how obervability of a firm' care deciion matter to the effectivene of extended liability. A tandard aumption in the exiting literature on extended liability i that care i unobervable 3 to third partie. 2 Equally tandard and cloely related i the aumption that care take the form of afety meaure that reduce the probability of an accident occurring. Becaue uch care tend to be of a procedural nature for example, making ure that worker follow tandard afety procedure it i reaonable to aume that it i difficult for third partie to oberve. In reality, of coure, firm can take meaure to reduce not ut the probability ofan accident, but alo the extent of damage if an accident occur. Moreover, uch damage-reducing meaure tend to take the form of tangible invetment, which are readily obervable to third partie. In the oil-hipping buine, for example, proper training and uperviion of crew reduce the probability of tanker grounding or colliion, but i difficult to oberve. On the other hand, fitting double hull to oil tanker reduce the extent of oil pill in the event of an accident, and i readily obervable. Similarly, in the hazardou-wate dipoal buine, day-to-day enforcement of procedure enuring that hazardou wate i tored correctly reduce the probability of leak, but i difficult to oberve. On the other hand, intalling monitoring well reduce the extent of leak (by enabling early intervention) and i readily obervable. In thi paper, we allow for both afety meaure that reduce the probability of an accident occurring and afety meaure that reduce the extent of accident damage. In addition, we aume that damage-reducing meaure are obervable to third partie, while probability-reducing meaure are not. We firt analyze the implication of thee aumption in the abence of extended liability. We find that, when accident damage are endogenou, the tate of being udgment proof become for ome firm namely relatively profitable one endogenou a well: uch firm can chooe whether or not to keep damage at a level exceeding their market value. We find alo that udgment-proof 2 An intereting exception i Boyd and Ingberman (2003), who abtract from the moral-hazard ditortion that can reult from care being unobervable to focu on a different iue, namely under what circumtance firm chooe to diolve rather than face their accident liability.

4 4 firm take no damage-reducing care at all, but ome of thee firm overcompenate by taking too much probability-reducing care. 3 Finally, we find that in the abence of extended liability too many firm are in buine, in the ene that ome udgment-proof firm have poitive market value only becaue they externalize damage and take uboptimal care. When we introduce extended liability, we find that firm repond very differently depending on their profit before any accident-related cot (hereafter referred to a gro" profit). All firm are forced to fully internalize accident-related cot, but only firm with ufficiently high gro profit repond to thi by chooing to become olvent. Thee firm witch to taking ocially optimal level of both damage-reducing and probability-reducing care. Firm with intermediate gro profit, on the other hand, tay udgment proof, and witch from taking no damage-reducing care at all to taking too much of uch care, while at the ame time taking too little probability-reducing care. Latly, firm with ufficiently low gro profit are driven out of buine. We find moreover that too few firm remain in buine when extended liability iintroduced: ome firm become non-viable even though they would be viable were they forced to take ocially optimal level of care. 4 The welfare effect of extending liability depend on firm' gro profit a well. They are poitive for high-profit firm, ambiguou for intermediate-profit firm (the ign being dependent on the available care technologie), and poitive again for low-profit firm. An additional welfare reult i that direct regulation of obervable care, when combined with extended liability, perverely further ditort firm' incentive. However, when ued alone, uch regulation trictly dominate extended liability. What tie many of thee eemingly diparate reult together i that, under our aumption about obervability, firm can commit to damage-reducing afety meaure, but not to probabilityreducing one. Becaue extended liability force firm to fully internalize all accident-related cot, they would prefer, ex ante, to chooe the probability and damage level that minimize thee cot. Ex pot, however, once their contract with third partie have been fixed, udgment-proof firm chooe a higher accident probability, and they cannot commit to doing otherwie. A a reult, in order to minimize accident-related cot conditional on their too high choice of accident probability, 3 Beard (1990), Craig and Thiel (1990), and Poey (1993) alo find that udgment-proof firm may take toomuch probabilityreducing care. In their model, thi poibility arie for very different reaon, however, which moreover depend on treating time a dicrete. A note expanding on thi point i available from the author upon requet. 4 Thi reult lend ome upport to the often-made claim (e.g., Greenberg and Shaw (1992), Byrne and Greco (1994)) that extended liability will deter ocially productive invetment.

5 5 they chooe a damage level that i lower than would otherwie be optimal. In effect, the firm partially compenate for it underinvetment in unobervable care by overinveting in obervable care. The remainder of the paper i organized a follow. Section 2 and 3 introduce the model and determine the ocially optimal level of probability-reducing and damage-reducing care. Section 4 analyze firm' equilibrium choice of care in the abence of extended liability. Section 5 analyze the effect of introducing a pecific form of extended liability, namely lender liability. 5 Section 6 dicue direct regulation of damage-reducing care a a poible complement to, or ubtitute for, extended liability. Section 7 demontrate that, although we ue lender liability throughout the paper a a concrete example, our reult apply more generally to other form of extended liability. Section 8 conclude. 2. The Model The repreentative firm in our model i conidering engaging in an activity that carrie with it a rik of cotly environmental accident. The activity generate fixed profit Π, defined a revenue net of variable cot unrelated to accident. Accident occur at time generated by apoion proce with mean p (the continuou-time accident probability" of the firm), and each accident caue damage D. The firm face trict liability for all accident damage, but thi liability i limited by the firm' value V. If, therefore, the damage D from an accident exceed V, then the firm will be bankrupted, and there will be reidual damage D V. The main focu of our analyi i on how aigning liability for thee reidual damage to third partie will affect the firm' incentive to reduce the accident probability p and the magnitude of damage D. The firm can reduce p by incurring flow probability-reducing expenditure" c(p), and it can reduce D by undertaking damage-reducing invetment" that involve up-front capital cot of I(D). The function c(p) ha the following propertie: 5 Much of the literature on extended liability (e.g., Pitchford (1995), Heye Heye (1996), Boyer and Laffont (1997)) ha focued on the pecific example of lender liability. See Burke (1998) for an overview of the hitory of lender liability under CERCLA. See alo Fee (1999) for a dicuion of propoal to introduce lender liability in the European Union.

6 6 C1: c(p) =0atp = p. C2: c 0 (p) < 0;c 00 (p) > 0; 8p 2 (0; p). C3: c(p)!1a p! 0. If the firm pend nothing on probability reduction, then accident will occur at ome finite rate p (property C1). Reducing the probability of accident below p i cotly, and increaingly o a the accident probability decreae (property C2). Reducing the accident probability to zero i impoible (property C3). The damage-reduction cot function I(D) ha analogou propertie: I1: I(D) =0atD = D. I2: I 0 (D) < 0;I 00 (D) > 0; 8D 2 (0; D). I3: I(D)!1a D! 0. To tart up the activity, the firm mut incur up-front capital cot of K in addition to I(D). We aume that the firm ha no initial wealth and therefore ha to borrow tocover all up-front cot. 6 We aume that the lending ector i competitive and that L1: the lender' loan i ecured, with the firm' aet a collateral, L2: the firm' aet are not damaged or otherwie diminihed in value if an accident occur. Aumption L1 implie that if an accident occur that bankrupt the firm, the lender will have firt claim on the firm' aet, before any claim by accident victim. Aumption L2 implie that after repoeing the firm' aet, the lender can in principle re-lend them to a new firm under the ame term a before. 7 We further aume that the level of probability-reducing expenditure choen by the firm i not obervable to (or verifiable by) the lender or any other outide partie, but the level of damagereducing invetment it undertake i. Hence, we ometime ue unobervable care" and obervable care" a hort-hand term. 6 Our analyi goe through qualitatively unchanged if we allow for equity financing, a long a owner wealth i inufficient to make the firm olvent in the event of an accident. 7 Allowing for damage to capital ignificantly complicate the analyi, but doe not materially affect our reult.

7 7 3. The Social Planner' Problem In thi ection, we conider the ocial planner' problem, which ito chooe the accident damage level D and the accident probability p that maximize welfare. To keep the welfare analyi imple, we aume that the firm' activity generate no conumer urplu. 8 In flow term, the welfare generated by the firm acro an infiniteimal time interval dt i then wdt ßdt [c(p)+ri(d)+pd] dt; (1) where ß Π rk repreent the firm' flow profit Π le the flow opportunity cot rk to ociety from having productive capital K tied up in the firm. The firt term on the right-hand ide of (1) therefore capture what would be the net benefit of the activity in which the firm i engaged, were thi activity rikle. The activity i not rikle, however, and the econd term capture all cot related to the poibility ofenvironmental accident: the flow cot of unobervable care c(p), the flow opportunity cot ri(d) of the firm' invetment in damage-reducing capital, and expected accident damage pd. 9 For an infinitely-lived firm, the preent value of thi welfare flow i 10 W ß c(p) ri(d) pd : (2) r A firm that i udgment-proof, however, i by definition not infinitely-lived: it goe bankrupt a oon a an accident occur. To facilitate welfare and other comparion between olvent and udgment-proof firm, we therefore aume that a bankrupt udgment-proof firm i alway immediately replaced with a new, identical ucceor firm. Expreion (2) then repreent the welfare generated by the reulting infinite equence of identical firm. 8 Thi could be motivated by the exitence of a cotly, but non-hazardou, outide option available to the firm' cutomer that i in infinitely elatic upply. A an example, the productive activity may be that of providing wate-dipoal ervice for generator of hazardou wate. The non-hazardou outide option available to the generator could then be to ue a clean" technology that doe not generate any wate. 9 By the Poion aumption, during any infiniteimal time interval of length dt at mot one accident will occur with probability pdt. Expected damage acro the interval are therefore p dtd. 10 Uing Campbell' Theorem for Poion procee, it can be hown that if accident occur at random time t 1 ;t 2 ; ::: generated by apoion proce with mean p, and if each accident caue damage equal to D, then the expected preent value of thoe damage for all future accident combined equal pd=r.

8 8 We aume that the ocially optimal damage level D S and accident probability p S are uniquely defined by the following firt-order condition: ri 0 (D) p =0; (3) c 0 (p) D =0: (4) The firt condition equate the marginal flow cot ri 0 (D) of damage-reducing care to the marginal ocial benefit p. The econd condition equate the marginal cot c 0 (p) of probability-reducing care to the marginal ocial benefit D. For reaon of expoition, it will be ueful to define a function ( ) that i equal to the invere of the function c 0 ( ) wherever thi invere exit (i.e., for p<p), and equal to p otherwie. Condition (4) can then be rewritten a p = (D): (5) A hown in figure 1, defining the ( ) function allow u to repreent the ocial optimum (D S ;p S ) graphically in (D; p) pace a the point where the ri 0 (D) and (D) curve cro. Alo hown in the figure are two additional critical value of D and p, namely D, which i the damage level that would be ocially optimal if probability-reducing care were zero, and p, which i the probability that would be ocially optimal if damage-reducing care were zero. It hould be noted that, while figure 1 identifie the D and p value that minimize accidentrelated cot, it tell u nothing about the level of welfare generated by any given firm. For thi, we need to conider in addition the net benefit ß aociated with the firm' activity. Let ß S denote the level of net benefit that ut equal minimized accident-related cot, i.e., ß S c(p S )+ri(d S )+p S D S : Clearly, firm with ß le than ß S will not be in buine at the ocial optimum, a they would generate negative ocial welfare. For firm with ß greater than ß S, ocial welfare i poitive and

9 9 p p ri 0 (D) (D) p S p N p C A 0 0 D D S V N (ß 3 ) D B D Figure 1. Marginal cot ri 0 (D) of reducing accident damage and (invere) marginal cot c 0 1 (D) (D) of reducing the accident probability. Without extended liability, firm with value V N (ß 3 ) are indifferent between chooing damage D and being udgment proof and chooing damage D S and being olvent. given by equation (2) evaluated at (D S ;p S ). For later reference, let W S (ß) denote thi level of welfare expreed a a function of ß.

10 10 4. The Firm' Problem Without Extended Liability The firm' problem differ from the ocial planner' in two repect. Firt, wherea the ocial planner take into account all damage aociated with environmental accident, the firm, acting in the bet interet of it hareholder, only take into account thoe damage for which itcanbe held liable. Second, the firm mut borrow capital, in return for a loan fee that allow the lender to at leat break even in expectation. By aumption, the lender can oberve the firm' invetment in damage reduction, o that the term of the loan contract can be conditioned on D. In contrat, the lender cannot oberve the firm' expenditure on probability reduction, o that the loan contract cannot be conditioned on p. Thi implie that, although the firm might want to promie the lender a high level of unobervable care if doing o would reduce it borrowing cot ex ante (before the loan contract i igned), it cannot credibly make uch a promie if that level of care i not privately optimal to the firm ex pot (after the loan fee ha been fixed). We capture thee obervability aumption by modeling the firm' and the lender' deciion a if they take place in three conecutive tage. In tage 1, the firm chooe the level of damage D. In tage 2, lender oberve D and the loan fee ` i determined competitively. In tage 3, the firm chooe the accident probability p conditional on it earlier choice of D and the now contractually fixed loan fee `. In both tage 1 and 3, the firm' obective i to maximize it hareholder value, V, which i implicitly defined a follow: rv dt =[Π c(p) ` p minfv;dg] dt: (6) Equation (6) i a no-arbitrage condition that mut be atified in order for rik-neutral hareholder to be willing to hold equity in the firm acro any infiniteimal time interval dt. On the left-hand ide, rv repreent the opportunity cot to the hareholder of keeping their equity in the firm, which i equal to return that they could enoy if they intead old their hare and inveted the proceed V at a rikle rate r. On the right-hand ide are the expected return to hareholder if they do keep their equity in the firm. Thee return conit of a dividend flow Π c(p) ` and and

11 an expected capital lo p minfv;dg. The capital lo arie if an accident occur, in which cae the hareholder will either loe V, through bankruptcy, orpay the full damage D. An important point to note about equation (6) i that the firm' value V directly depend on it choice of D and p, a well a on the loan fee `. A a reult, we cannot peak of a firm that i" either olvent or udgment proof. Rather, a given firm, characterized by parameter value Π and K, may be olvent at ome value of D and p, but udgment proof at other. Whenever we peak below of a olvent or udgment-proof firm, thi hould therefore be undertood a horthand for a firm that i olvent or udgment proof at the value of D and p under conideration." From equation (6), the value of a firm that i olvent in the above ene, o that minfv;dg = D, i 11 V = Π c(p) ` pd : (7) r Similarly, the value of a udgment-proof firm, for which minfv;dg = V,i 11 V = Π c(p) ` : (8) r + p Given our aumption of a competitive lending ector, the equilibrium loan fee i given by Ldt= `dt r[k + I(D)] dt =0; (9) where L repreent the lender' flow return. That i, the loan fee ` mut cover the opportunity cot r[k + I(D)] of the loan that it make to the firm. Note that aumption L1 and L2 imply that the lender doe not incur any lo in the event of an accident. Uing the above equation, we find that the olution to the firm' optimization problem can be characterized a follow: Lemma 1. Without extended liability, the only locally optimal D and p value are (a) D S and p S for firm that are olvent at that combination, (b) D and p = (V ) for firm that are udgment proof at that combination. 11 The right-hand ide of thi equation can be hown to equal the expected preent value of the udgment-proof firm' dividend tream Π c(p) ` up to the time when it i bankrupted by an accident.

12 12 Proof: All proof are given in appendix A. The eaiet way to undertand thi reult i to ubtitute equation (9) for the loan fee into the right-hand ide of equation (6), yielding the following reduced-form expreion for hareholder return [Π c(p) rk ri(d) p minfv;dg] dt; or, rearranging and ubtituting ß Π rk: ß dt [c(p) +ri(d) +p minfv;dg] dt: Comparing the latter expreion with equation (1) for flow welfare how that at (D; p) combination where a firm i olvent, o that minfv;dg = D, the firm fully internalize all accident-related cot. A a reult, any (D; p) combination other than (D S ;p S ) that leave a firm olvent will fail to minimize it accident-related cot, and therefore cannot be optimal, even locally. In contrat, at (D; p) combination where a firm i udgment proof, o that minfv;dg = V, it only face an expected capital lo of pv from an accident. Since it doe not pay the full damage D, any mall reduction in D that till leave the firm udgment proof will alway decreae it hareholder return, by increaing the damage-reduction-cot portion ri(d) of it loan fee without any offetting (firt-order) decreae in it liability. The only damage level conitent with equilibrium for a udgment-proof firm i therefore D, where it incur no damage-reduction cot at all. Conditional on thi damage level, the ocial planner would chooe p = (D), where the marginal cot c 0 (p) of further reduction in p ut balance the ocial marginal benefit D. For the udgment-proof firm, however, the private marginal benefit of reducing p i only V < D, equal to it lo of hareholder value in cae of an accident. A a reult, it will chooe a higher privately optimal accident probability p = (V ). The reult of lemma 1 i only partial, in that it doe not etablih whether any given firm i in fact olvent at (D S ;p S ) or udgment proof at (D; (V )), or poibly both. Nor doe the lemma etablih how (V ) compare to the ocially optimal accident probability p S.

13 The anwer to thee quetion turn out to depend on the value of ß, which we identified in ection 3 a repreenting the ocial benefit of the activity in which the firm i engaged, gro of accident-related ocial cot c(p)+ri(d)+pd. From the firm' point of view, ß repreent it private benefit (i.e., profit), gro of accident-related private cot c(p) + ri(d) + p minfv;dg (though net of all other cot, including the capital-cot component rk of the loan fee). We hereafter refer to ß a the firm' gro" profit. 13 Propoition 1. Let V N (ß) denote the value of a firm with gro profit ß without extended liability, if it chooe D S and p S and i olvent at that combination. Similarly, let V N (ß) denote the firm' value if it chooe D and p = (V ) and i udgment proof at that combination. Alo, define critical gro profit level ß N and ß 1 through ß 4 implicitly a follow: ß N : ß 1 : ß 2 : ß 3 : ß 4 : V N (ß) =0 V N (ß) =D S V N (ß) =D S V N (ß) =V N (ß) V N (ß) =D Then (a) 0=ß N <ß 1 <ß 2 <ß 3 <ß 4. (b) Firm with ß 2 [0;ß 3 ) are udgment proof in equilibrium, while firm with ß 2 [ß 3 ; 1) are olvent. (c) Firm with ß 2 [ß 2 ;ß 4 ) can chooe" to be either olvent or udgment proof, in the ene that for thee firm both (D S ;p S ) and (D; (V )) are locally optimal. (d) Of the firm that are udgment proof in equilibrium, thoe with ß 2 [0;ß 1 ) take too little unobervable care ( (V ) >p S ), while thoe with ß 2 (ß 1 ;ß 3 ) take too much ( (V ) <p S ). The propoition i eaiet to undertand with the help of figure 2. The proof of the propoition in appendix A etablihe that the lope and relative poition of the V N (ß) and V N (ß) curve in the figure do not depend on any particular parameterization of the model, and therefore neither

14 14 $ V N = W S D D S V N W S 0 0 ß S ß 1 ß 2 ß 3 ß 4 ß Figure 2. Firm value a a function of gro profit without extended liability. doe the ordering of the critical gro-profit level in part (a). Once thi ha been etablihed, part (b) through (d) of the propoition can eentially be read off from the figure. To undertand part (b) and (c) of the propoition, note from the figure that for all firm with ß in the range [0;ß 2 ), V N (ß) < D S and V N (ß) < D, implying that all uch firm are udgment proof at both (D S ;p S ) and (D; (V )). A a reult, by lemma 1, (D S ;p S ) i for thee firm not even locally optimal only (D; (V )) i, and thi combination i therefore alo the global optimum for thee firm. At the oppoite extreme, only (D S ;p S ) i both locally and globally optimal for firm with ß 2 [ß 4 ; 1), becaue thee firm are olvent atboth(d S ;p S ) and (D; (V )). In between ß 2 and ß 4, firm are olvent if they chooe (D S ;p S ), but udgment proof if they chooe (D; (V )). Both (D; p) combination are therefore locally optimal, and which of the two i globally optimal depend on which ofthe two yield higher firm value: it i (D; (V )) for firm with ß in the range [ß 2 ;ß 3 ) and (D S ;p S ) for firm with ß in the range (ß 3 ;ß 4 ). Firm with ß = ß 3, finally, are indifferent between the two local optima.

15 Figure 1 illutrate in (D; p) pace the ituation of a firm with ß = ß 3. Suppoe firt that thi firm chooe to be udgment proof, by chooing damage level D. It conditionally optimal accident probability i then p N = (V ), and it unobervable care cot c(p N ) correpond to the area between p N and p to the left of the (D) curve. 12 In addition to thee cot, the firm will face expected liability cot given by the rectangle p N V. Suppoe intead that the firm chooe damage level D S. In thi cae, it conditionally optimal accident probability ip S = (D S ), o it unobervable care 15 cot correpond to the area between p S and p to the left of the (D) curve, while it expected liability cot correpond to the rectangle p S D S. In addition, the firm will incur obervable care cot ri(d S ) paed on through it loan fee. Thee cot correpond to the area under the ri 0 (D) curve between D S and D. Aggregating thee area for the two cae how that if the firm chooe D S rather than D, it obervable-care cot are greater by area A + B, but it unobervable-care and liability cot are maller by area A + C. For a firm with gro profit ß 3, thee area are identical in ize, o that the firm i indifferent between the two damage level. Firm with gro profit omewhat above ß 3, however, will have omewhat higher value V N (ß), making area B maller than area C. Such firm will therefore trictly prefer damage level D S, although D will remain an alternative local optimum a long a V N (ß) i maller than D. Similarly, firm with gro profit omewhat below ß 3 will trictly prefer damage level D, although D S will remain an alternative local optimum. To undertand part (d) of the propoition, note from figure 2 that firm with ß 2 (ß 1 ;ß 3 ) have value V N between D S and D. Becaue thee relatively profitable udgment-proof firm do not invet in damage reduction and are able to externalize reidual damage, they end up with a market value V N that exceed D S. It i thi market value that the firm have at take in the event of an accident, wherea the ocial planner ha only D S at take. Thee firm therefore have a greater incentive to avoid accident than doe the ocial planner, and o chooe an accident probability lower than p S. 13 Firm with gro profit below ß 1 are alo udgment proof, but have value V N le than D S, and o chooe an accident probability greater than p S. In the extreme cae, firm that have zero 12 The horizontal ditance between the vertical axi and the (D) curve atany p i equal to c 0 (p), the marginal cot of care. Thu, the total care cot c(p) ofachieving any p are repreented by the area to the left of the (D) curve, between p and p. 13 Note however, that the ocially optimal accident probability p S i defined conditional on damage having been reduced to the ocially optimal level D S. A dicued after lemma 1, the ocially optimal accident probability conditional on udgment-proof firm' privately optimal damage level D i p = (D), which i in fact lower than the accident probability p = (V N )choen by uch firm.

16 16 value chooe p, i.e., take no unobervable care at all. Becaue they take no obervable care either, equation (8) reduce to V = ß=(r + p) for thee firm. Thi implie that the lowet gro-profit level at which firm are viable (in the ene that they have non-negative value) in the abence of extended liability iß N =0. Welfare Without Extended Liability To evaluate the welfare lo reulting from the udgment proof problem without extended liability, we compare the welfare generated by firm (a given by equation (2)) when they chooe their privately optimal value of D and (V ), to the welfare generated at the ocial optimum. Recall from ection 3 that firm with ß 2 [0;ß S )would not be in buine at the ocial optimum, a the welfare they would generate at (D S ;p S ) i negative. It follow that the welfare lo reulting from thee firm' choice in the abence of extended liability correpond to the negative welfare they actually generate at (D; (V )). Firm with ß 2 [ß S ;ß 3 ), in contrat, would be in buine at the ocial optimum. For thee firm, the welfare lo from the udgment proof problem i therefore ut the deadweight lo reulting from their ditorted choice of D and (V ). Figure 3 illutrate thi deadweight lo for an arbitrary udgment-proof firm. At the ocial optimum, the firm' full accident-related cot are equal to c(p S )+p S D S + ri(d S ). At it privately optimal choice of D and p N = (V ), the firm' full accident-related cot (including the expected damage p N (D V ) that it externalize) are c(p N )+p N D. The difference between thee two cot correpond to the haded area in the figure labeled DWL N. 5. The Firm' Problem With Extended Liability In thi ection, we conider how extending liability for reidual damage to a third party change the incentive of the firm. Becaue the literature on extended liability ha largely focued on the pecific example of extending liability to lender, we ue thi example in much of the remainder of thi paper. However, a we demontrate in ection 7, our reult generalize to other form of extended liability.

17 17 p p ri 0 (D) (D) p L DWL L p S p N DWL N 0 0 V L D L D S V N D Figure 3. Deadweight lo of a udgment-proof firm without extended liability. D If lender are liable for environmental damage in exce of what the firm i able to pay, then, all ele equal, they will require a higher loan fee in order to break even. However, all ele will not be equal: in anticipation of thi repone by the lender, the firm will in general chooe a different (D; p) combination.

18 18 A in ection 4, we begin our analyi by conidering the locally optimal choice of D and p for firm that are either olvent or udgment proof at thoe choice, without initially characterizing uch firm in term of exogenou parameter value. For firm that are olvent, the analyi of ection 4 goe through unchanged. Becaue uch firm fully internalize all accident damage, there i no reidual liability to pa on to lender. The lender' break-even contraint i therefore unchanged, and the firm will till optimally chooe D S and p S. For firm that are udgment proof, the tage-3 problem of optimally chooing p conditional on any given damage level D and fixed loan fee ` i unchanged a well. The firm will therefore till optimally chooe p = (V ). The tage-1 problem of optimally chooing D doe, however, change under extended liability, becaue the lender' tage-2 break-even contraint become Ldt= `dt [rk + ri(d)+p maxfd V;0g] dt: (10) With extended liability, the loan fee ` ha to cover more than ut the opportunity cot of the loan. Thi i becaue, if the firm experience an accident, the lender will be held liable for any reidual damage maxfd V;0g. For the lender to break even in expectation, it loan fee will therefore have to alo cover the expected value of thi reidual liability. 14 Given the change in the lender' break-even contraint, the olution to the firm' optimization problem change a follow: Lemma 2. With extended liability, the only locally optimal D and p value are (a) D S and p S for firm that are olvent at that combination, 14 Note that in equation (10) both p and V are themelve function of the loan fee `. Thi implie that, for udgment-proof firm, increaing the loan fee now ha two oppoing effect on lender return. On the one hand, raiing ` clearly increae the lender' flow receipt; on the other hand, it alo increae the lender' reidual liability p(d V ), both by reducing the firm' value V and by increaing the accident probability p = (V ) that the firm will chooe after the loan fee ha been fixed. Nothing in the aumption of our model guarantee that the firt effect outweigh the econd. Thi ha two conequence. Firt, for any given ß and choice of D, more than one level of the loan fee may atify the break-even contraint. In uch cae, market competition between lender will enure that the firm i charged the lowet break-even fee, o that the equilibrium loan fee i till unique. Second, the minimum loan fee may be dicontinuou in ß and D, in which cae the equilibrium value V may be dicontinuou in ß. So a not to unnecearily complicate the expoition of our model, we rule out uch dicontinuitie by aumption. An appendix howing exactly what aumption i needed to enure continuity ofv in ß, and the (very minor) implication for our reult if thi aumption fail, i available from the author upon requet.

19 (b) D L <D S and p L >p S uch that ri 0 (D L )=p L = (V ) for firm that are udgment proof at that combination. 19 The intuition for thee reult i again eaiet to grap if we ubtitute the new break-even contraint into the right-hand ide of equation (6) to obtain the following expreion for hareholder return: [Π c(p) rk ri(d) pd] dt: Equivalently, we can write thee return a ßdt [c(p) +ri(d) +pd] dt: Note that thee expreion apply regardle of whether the firm i olvent or udgment proof: if it i olvent, then it loan fee will be r[k + I(D)] and it expected accident liability will be pd; if it i udgment proof, it loan fee will be r[k + I(D)] + p(d V ) and it expected accident liability will be pv. Either way, the firm end up fully internalizing all accident related cot c(p)+ ri(d)+ pd. An immediate implication i that, ex ante, both olvent and udgment-proof firm would prefer to chooe the D and p combination that minimize thee accident cot, namely D S and p S. What differentiate a olvent firm from a udgment-proof one, however, i that only a olvent firm can credibly commit to the lender that it will chooe p S in tage 3 after having choen D S in tage 1. Thi i becaue, if a olvent firm chooe D S in tage 1, and the loan fee i et at ` = rk + ri(d S ) in tage 2, the firm i left with hareholder return [Π c(p) ` pd S ] dt: in tage 3. Chooing p S in tage 3 i therefore in fact optimal for the olvent firm, a it maximize thee return, and the loan fee ` = rk + ri(d S ) in fact allow the lender to break even. In contrat, a firm that i udgment proof at D S and p S cannot commit to chooing p S in tage 3 even if it were to chooe D S in tage 1. Any loan fee that leave the lender breaking even will leave thi firm with a value V <D S at p S, and, once the loan fee ha been fixed in tage 2, with

20 20 anticipated hareholder return [Π c(p) ` pv ] dt in tage 3. It will therefore in tage 3 chooe an accident probability ~p L = (V ) >p S that maximize thee return. Anticipating thi, the lender will et the reidual-liability component of it loan fee at a level conitent with ~p L, and a a reult the firm will end up fully internalizing accident-related cot ~p L D S rather than p S D S. Thi in turn implie, however, that D S i no longer optimal for the firm in tage 1: at D S, the marginal cot ri 0 (D S ) of further reduction in damage will be trictly lower than the marginal benefit ~p L. The firm will therefore chooe a damage level D L trictly below D S, and a different accident probability p L above p S, uch that ri 0 (D L )=p L = (V ). In effect, in order to keep it overall expected damage down, the udgment-proof firm partially compenate for pending too little" on unobervable care ex pot, by pending what would otherwie be too much" on obervable care ex ante. Figure 3 illutrate thi equilibrium in (D; p) pace. The haded area labeled DWL L in the figure repreent the amount by which the firm' accident-related cot exceed thoe at the ocial optimum. The next propoition how which firm will in fact be olvent or udgment proof in the equilibrium with extended liability. Thi again turn out to depend on their gro profit ß. Propoition 2. Let V L (ß) denote the value of a firm with gro profit ß with extended liability, if it chooe D S and p S and i olvent at that combination. Similarly, let V L (ß) denote the firm' value if it chooe D L and p L uch that ri 0 (D L ) = p L = (V ) and i udgment proof at that combination. Alo, define critical gro profit level ß L implicitly a follow: ß L : V L (ß) =0: Then (a) ß S <ß L <ß 2. (b) Firm with ß 2 [ß L ;ß 2 ) are udgment proof in equilibrium, while firm with ß 2 [ß 2 ; 1) are olvent.

21 21 $ V L = W S D D S V N W S V L 0 0 ß S ß L ß 1 ß 2 ß 3 ß 4 ß Figure 4. Firm value a a function of gro profit with extended liability. In Figure 4, the V L (ß) function correpond to the the ection of the bold curve to the left of ß 2 (below D S ), while the V L (ß) function correpond to the ection to the right ofß 2 (above D S ). Alo hown, a dahed curve, are the V N (ß) and W S (ß) curve from figure 2. Comparing figure 2 and 4 (or propoition 1 and 2) how that extending liability to the lender affect firm with different gro profit ß very differently. Working from right to left in the figure, and thereby from high to low gro-profit level, we firt have firm with ß ß 3. Thee firm are not affected at all, ince they chooe to be olvent at (D S ;p S ) even in the abence of extended liability. In the figure, the V L V N curve for thi range of ß value. The V L and V N curve coincide with the curve coincide a well for firm with ß 2 [ß 2 ;ß 3 ). Thee firm chooe to be udgment proof in the abence of extended liability, but witch to being olvent once extended liability iintroduced. Recall from ection 4 that all firm with ß ß 2 are olvent if they chooe (D S ;p S ). However, in the abence of extended liability, (D S ;p S ) i only a local optimum for firm with ß 2 [ß 2 ;ß 3 ), and the alternative local optimum (D; (V N )) i for thee firm more attractive,

22 22 given that they can externalize damage p(d V N ). Introducing extended liability, by forcing firm to internalize all damage, effectively remove the econd local optimum, thereby inducing thee firm to witch to the firt. A a reult, they witch from taking too little obervable care (D >D S ) and too much unobervable care (p <p S ) to taking the ocially optimal amount of both. Next down are firm with ß 2 [maxfß 1 ;ß L g;ß 2 ), which are udgment proof both before and after introducing extended liability. 15 In the abence of extended liability, the ability to externalize damage left thee firm with a value V N greater than D S, and thereby with a greater incentive to avoid accident than the ocial planner. Once forced to internalize all damage through extended liability, however, the value of thee firm drop below D S for any (D; p) choice. In particular, thi i true at (D S ;p S ), o that the commitment problem dicued above, following lemma 2, kick in. The firm' optimal choice therefore become (D L ;p L ), a defined in part (b) of the lemma. A a reult, they witch from taking too little obervable care to taking too much, and converely from taking too much unobervable care to taking too little. If model parameter are uch that ß L < ß 1, there will alo be firm with ß 2 [ß L ;ß 1 ). Thee firm, too, are udgment proof both before and after extended liability i introduced. However, ince V N < D S for thee firm, they take too little unobervable care (p = (V N) > ps ) in the abence of extended liability. The drop in value of thee firm when extended liability i introduced induce them to take even le unobervable care. For all firm with ß 2 [ß L ;ß 2 ), accident-related cot exceed thoe at the ocial optimum by an area uch adwl L in figure 3. Thi area increae (and the vertical ditance between the V L W S curve therefore increae a well) the further ß fall below ß 2, ince udgment-proof firm with lower ß have a lower private value and therefore chooe a higher accident probability. At ß L, the value fall to zero, and firm with thi level of gro profit therefore chooe p = (0) and D L = D. From equation (8), thi implie that ß L mut ut equal the accident-related cot c(p)+ri(d)+pd incurred by thee firm. Since thee cot exceed minimized accident-related cot, which in turn are by definition equal to ß S, it follow that ß L mut exceed ß S. Latly, firm with ß 2 [0;ß L ) are driven out of buine when extended liability iintroduced. Any loan fee that leave the lender breaking even will leave thee firm with negative value. and 15 Our model aumption are conitent with ß L being either greater than or maller than ß 1.

23 For all firm that are udgment proof without extended liability, i.e., all firm with ß < ß 3, thee eemingly diparate reult have three feature in common. Firt, all uch firm face a higher loan fee ` when extended liability iintroduced. 16 Second, the private value of all uch firm fall: V L < V N. Third, provided they remain in buine, the amount of unobervable care taken by uch firm fall a well: (V L ) > (V N ). On the face of it, thee three feature eem to mirror the reult of Pitchford (1995). Pitchford how that introducing extended liability increae the loan fee of a udgment-proof firm, thereby reducing it value (given by ß ` in hi ingle-period model), and thereby perverely reducing the firm' incentive totake unobervable care. Upon cloer inpection, however, our reult turn out to be quite different from Pitchford'. Firt, for firm with ß 2 [ß 2 ;ß 3 ), introducing extended liability doe indeed reduce firm value. However, becaue it alo induce thee firm to increae obervable care, damage which are fixed in Pitchford' model are reduced by a greater amount than firm value. A a reult, the firm witch from being udgment proof to being olvent. Second, for firm with ß 2 [ß 1 ;ß 3 ), introducing extended liability doe indeed reduce the incentive to take unobervable care. However, all thee firm took too much unobervable care without extended liability. A a reult, for the ubet of thee firm with ß 2 [ß 2 ;ß 3 ), the level of unobervable care after extended liability iintroduced end up being ocially optimal. Latly, amentioned in footnote 15, model parameter may be uch that ß L ß 1. In thi cae, no firm in our model will exhibit the pervere behavior that Pitchford identifie, namely a witch from taking too little unobervable care to taking even le. 23 Welfare Effect of Introducing Extended Liability Not urpriingly, in light of propoition 1 and 2, the welfare effect of introducing extended liability depend again on firm' gro profit: Propoition Recall that all uch firm tart borrowing to finance invetment in damage reduction. Moreover, for the ubet of firm with ß<ß 2, which are till udgment proof after extended liabilityiintroduced, the new loan fee alo include a reidual-liability component.

24 24 (a) For firm with either low gro profit ß 2 [0;ß S ) or high gro profit ß 2 [ß 2 ;ß 3 ) the welfare effect of introducing extended liability i unambiguouly poitive. (b) For firm with intermediate gro profit ß 2 (ß S ;ß 2 ) the welfare effect i ambiguou. (c) Without extended liability, there iexce entry of firm, but extending liability may reult in exce exit. For firm with ß 2 [0;ß S ), welfare unambiguouly improve; thee firm generate negative welfare in the abence of extended liability, and are driven out of buine when extended liability i introduced. For firm with ß 2 [ß 2 ;ß 3 ), welfare unambiguouly improve a well; thee firm witch from being udgment proof to being olvent, and thereby witch to making ocially optimal deciion when extended liability iintroduced. For firm with ß 2 (ß S ;ß 2 ), however, introducing extended liability replace one ource of welfare lo with a different one. The net welfare effect turn out to be ambiguou, and depend on the hape of the cot curve. To undertand thi ambiguity, divide the interval (ß S ;ß 2 )into the two ub-interval (ß S ;ß L ) and [ß L ;ß 2 ), and conider firt firm with ß 2 [ß L ;ß 2 ), i.e., firm that tay in buine after extended liability i introduced. For thee firm, introducing extended liability implie replacing deadweight lo DWL N, caued by internalizing too few accident-related cot, with deadweight lo DWL L, caued by internalizing too many. To how that introducing extended liability may either decreae or increae welfare for thee firm, it i therefore ufficient to how that for ome configuration of the cot curve DWL N will be zero and DWL L poitive, while for other configuration the oppoite will be true. Thi i eaiet to etablih for firm with ß 2 (ß 1 ;ß 2 ), which have value V N of extended liability. To ee that the deadweight lo DWL N >D S in the abence can be zero for uch firm, note in figure 3 that the area repreenting thi deadweight lo will be maller, the cloer the ri 0 (D) curve i to the (D) curve for all D D S (o that ociety gain little from reducing damage from D to D S ) and the cloer to horizontal at p S the (D) curve i for all D V N (o that ociety gain little from changing the accident probability from (V N )top S ). In the limit where both the (D) and ri 0 (D) curve become arbitrarily cloe to horizontal at level p S everywhere to the right of D S, the deadweight lo become vanihingly mall. It i alo clear from the figure, however, that DWL L will be poitive a long a the (D) and ri 0 (D) curve diverge everywhere to the left of

25 D S. Converely, if the curve coincide to the left of D S, but not to the right, DWL L will be zero, but DWL N poitive. 17 Similar argument can be ued to how that the welfare effect of introducing extended liability i ambiguou alo for firm with ß 2 (ß S ;ß L ), i.e., firm that are driven out of buine when extended liability iintroduced. More pecifically, we prove in the appendix that the welfare generated by thee firm in the abence of extended liability can be either poitive ornegative, o that driving thee firm out of buine can either decreae or increae welfare. Thi again turn out to depend on the magnitude of DWL N and thereby on the relationhip between the (D) and ri 0 (D) curve. A for part (c) of the propoition, recall that, in the abence of extended liability, firm with ß 2 [0;ß S ) generate negative welfare, but have poitive market value becaue they externalize damage and do not take firt-bet level of care. In thi ene, the abence of extended liability reult in exce entry. When extended liability iintroduced, firm with ß 2 [0;ß S ) are driven out of buine, but o are firm with ß 2 (ß S ;ß L ). Thee latter firm would have poitive market value and generate poitive welfare if the commitment problem dicued above did not prevent them from taking firt-bet level of care. In thi ene, extended liability reult in exce exit. Of coure, from a policy tandpoint, the firt-bet level of care are irrelevant aabenchmark if no policy exit that actually implement the firt bet. The more relevant quetion in that cae i whether ome other, feaible policy dominate extended liability, in that it allow a greater range of firm to both tay in buine and generate poitive (though not necearily maximal) welfare. The next ection dicue one uch policy, namely direct regulation of damage-reducing invetment Direct Regulation of Damage-Reducing Invetment Given our aumption that lender can oberve a firm' effort to reduce accident damage, but not it effort to reduce the probability of accident, it i reaonable to aume that the ame would be 17 For firm with ß 2 [ß L ;ß 1 ) (hould they exit), the condition are only lightly different. Specifically, for DWL N to be zero but DWL L poitive, the (D) and ri 0 (D) curve mut converge at p S to the right ofv N (rather than D S ), and diverge to the left.

26 26 true of regulator. If o, a natural quetion to ak i whether direct regulation of damage-reducing invetment might either (1) complement extended liability, or (2) be a uperior alternative to it. Taking thee two quetion in turn, we find the following: Propoition 4. (a) Direct D-regulation doe not complement extended liability: any binding retriction on firm' choice of D can only reduce welfare if extended liability i in place. (b) Direct D-regulation i a uperior alternative to extended liability if it can be tailored to firm' gro profit level; if it cannot, then either policy may dominate the other. To undertand part (a) of the propoition, recall the dicuion following lemma 2. There, we howed that the firm' hareholder return under extended liability coincide with the ocial return, becaue the firm i forced to internalize all cot. That the udgment-proof firm' choice of D neverthele differ from the ocial planner' i becaue the firm i contrained by it own ubequent choice of p conditional on D. Subect to thi contraint, it choice of D maximize it hareholder return. But ince we are auming that direct regulation on probability-reducing expenditure are infeaible, the regulator, too, i contrained by the firm' conditional p choice, in exactly the ame manner. The firm' choice of D i, in other word, econd-bet optimal under extended liability, and any binding D-regulation can therefore only reduce welfare. To undertand part (b) of the propoition, conider the following thought experiment. Suppoe that, intead of introducing extended liability, the regulator directly mandate damage level D L, i.e., the damage level that a udgment-proof firm would have choen under extended liability. The firm will then face a lower loan fee than it would with extended liability, namely `0 = rk + ri(d L ) intead of ` = rk + ri(d L )+p L (D L V L ). A a reult, the firm' private value will be higher, and the accident probability it chooe will therefore be lower. Overall, then, damage are the ame a with extended liability, but with the accident probability now lower, ocial cot will be lower. The above reaoning hold for any firm that i both viable and udgment proof under extended liability, i.e., any firm with ß 2 [ß L ;ß 2 ). For all uch firm, direct D-regulation trictly dominate extended liability from a welfare tandpoint. To how that direct D-regulation i a uperior alternative to extended liability overall, it i therefore ufficient to how that it can replicate the outcome of extended liability when applied to firm with gro profit outide of the range [ß L ;ß 2 ). Clearly,