0107_0800_1704 by jizhen1947

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									                            Dangerous Products and Spillover Liability

                                                     By:

                                             Kristin Roti Jones

                                         Department of Economics
                                            Hartwick College
                                           Oneonta, NY 13820


ABSTRACT:
       Eliciting socially optimal care choices from injurers facing limited liability is a difficult
challenge. Injurers who do not expect to face the full costs of their actions due to limited assets
do not have adequate incentives to take care. This issue has become of increasing relevance
recently as products such as breast implants, prescription drugs and asbestos have produced
thousands of lawsuits and cost defendants and insurance companies millions of dollars. In the
asbestos litigation alone, at least 78 firms have been pushed into bankruptcy due at least in part
to asbestos claims. 1 However, as more firms have entered bankruptcy, plaintiffs have shifted
their claims to the remaining non-bankrupt defendants, thus increasing the liability faced by
these more solvent producers. Increasing liability on potentially bankrupt injurers in hopes of
increasing care may not produce the desired response since it may simply push them farther into
bankruptcy. It is unclear whether allowing for such a shift could increase social welfare by
raising the expected liability that solvent firms face and thus inducing them to take more care or
if it will just push the formerly solvent firms into bankruptcy and distort their care incentives
further. In this paper, we develop a model of dangerous products to analyze this spillover
liability. We find that the case in which spillover would perhaps be easiest to use – when firms
are of very different sizes – is also that case in which it is most likely to be welfare-reducing.
1. Introduction

       The traditional result that limited liability induces injurers to invest in too little

precautionary care and to overengage in risky activities because they will not always pay the full

damages of their actions has been well-recognized in the literature.2 The impact of limited assets

on care incentives is a timely topic. In recent years, dangerous products such as asbestos,

prescription drugs and breast implants have spawned millions of lawsuits demanding billions of

dollars in damages. In the asbestos litigation alone, hundreds of thousands of claims have been

filed and the total costs to defendants and insurers have been estimated to be as high as $200

billion.3 The massive number of suits and the size of each claim have combined to send more

than 78 former producers of asbestos into bankruptcy. 4 With the care in a dangerous products

context being some sort of preproduction research – since many of these products cannot be

made any safer -- the potential for bankruptcy should indicate that firms will invest in too little

preproduction research and that there will be too much entry into such an industry.

       Overcoming the incentive problems of limited assets is not easy. If the firms did not

have limited assets, then increasing liability will induce more care. However, the effects of

increasing liability on potentially bankrupt firms are not clear. On the one hand, higher liability

will make bankruptcy more likely and may thus reduce incentives to take care. However,

increasing liability also may raise each firm’s expected payout when it is solvent and thus might

actually induce more care. It will be the net of these two counteracting effects that will

determine whether increasing liability on potentially bankrupt firms will be welfare-improving.

This net effect is directly applicable to the current asbestos litigation. As firms have become

bankrupt, plaintiffs have shifted their claims onto the solvent firms in the industry, sometimes

even forcing them into bankruptcy. USG CEO William Foote stated after USG’s June 2001




                                                                                                      1
Chapter 11 filing that “U. S. Gypsum can afford to pay for its own liability, but it cannot afford

to pay for the liability of other companies…yet that is exactly what is happening because of the

high volume of new cases and the other asbestos-related bankruptcies.”5 This spillover liability

is essentially an increase in liability for the more solvent firms in the industry. While the

spillover liability could push otherwise solvent firms into bankruptcy – as it did for USG -- it

might also induce more care from firms since they will face higher liability if they are solvent in

the face of spillover.

        The main focus of this paper is to determine when allowing for such spillover liability is

socially desirable. We develop a model similar to those found in the environmental liability

literature to deal with dangerous products such as asbestos, prescription drugs, cigarettes, and

breast implants with specific focus on the welfare effects of spillover liability. We find that

allowing for spillover liability has the expected effects of inducing a more solvent firm to invest

in more research and to produce less frequently but also induces a less solvent firm to engage in

less research and produce more frequently. While it might appear that higher research and less

frequent production would be welfare-increasing – since the traditional results of bankruptcy are

to induce less care and more risky behavior – we find that this is not necessarily the case. In fact,

the case in which spillover liability is perhaps most likely to be used – when firms are of very

different sizes – is exactly the case in which spillover is most likely to be welfare-reducing.

2. Model

        Suppose that two firms are considering producing a product that may or may not be

dangerous. The probability that it is dangerous is given by q. If it is dangerous, it will generate

damages for which the firms will be held strictly liable. Neither firm knows whether or not the

product is dangerous but can invest in some level of research before producing (given by r1




                                                                                                      2
and r2 ) that may reveal the product’s true nature. A firm might run a series of animal tests, for

example. However, this research is not perfect. There is some probability (given by p(r1,r2 )) that

a dangerous product will not be uncovered by research levels r1 and r2 . p(r1,r2 ) is the probability

of a false negative. Even though asbestos is dangerous, not every animal exposed to it will

develop a disease. Each firm recognizes the fallibility of research and so must update its beliefs

about the product based on the research results and decide whether or not to produce based on

these updated beliefs.

       We model firm behavior in a three-stage game. In the first stage (stage 0), nature decides

whether or not the product is dangerous. In the second stage, the firms each choose a research

level. With probability q[ p(r1, r2 )], the research will reveal that the product is dangerous.
                         1−

With probability (1− q) + qp(r1,r2 ), the research will yield no evidence of danger, in which case

the firms realize that the product may or may not be dangerous. We assume that after the

research levels have been chosen and the research results found each firm is able to observe both

its own research results and the research level and results of the other firm. This public good

characteristic of research is acknowledged frequently in the literature.6 One reason that research

might be publicly available would be if government institutions required that it be, as the FDA

currently requires that clinical trials for certain drugs be submitted to a public data bank.7 This

does of course create an opportunity for free-riding. Because investing in research is costly, each

firm has an incentive to reduce its research expenditures and instead simply wait to observe the

outcome of the other firm’s research at the end of the stage.

       In the third stage, the firms decide whether or not to produce, understanding that they will

be held strictly liable for any damages. Production yields a benefit to each firm of V. If the

product is dangerous, production also generates damages of ND where N is the number of people



                                                                                                      3
who develop an illness or disease and D is the damages per injured person. While we assume

that the firms do know the damage level D, we assume that they do not know the number of

victims. In the case of asbestos for example, the damages to a person who develops lung cancer

may be estimable by considering the likelihood of death and the medical care needed. However,

the number of exposed people who will actually develop lung cancer will depend on where and

how the product is installed as well as personal factors such as whether the exposed person

smokes. This figure is likely very difficult for the firm to estimate. We assume that N is

                            [ ]
uniformly distributed on N,N . We also assume that the firms have different asset levels. Firm

1 is the larger firm, whose assets are given by A1 . Firm 2 is the smaller firm and its assets are

given by A2 , where A1 > A2 . If the actual damages are at the maximum level, both firms will be

bankrupt ( ND > A1 ). However, if the actual damages are at the minimum level, both will be

solvent ( ND < A2 ).

         Our model is a sequential game of incomplete information and so our equilibrium notion

is that of a weak perfect Bayesian equilibrium (hereafter referred to as a WPBE). A WPBE has

three requirements. First, when each player makes a decision, it must have a belief about which

decision node it is on. In this game, after observing the research results of the first stage, the

firms form beliefs about whether or not the product is dangerous. Second, these beliefs are

formed using Bayes’ Rule. If no evidence of danger is found by the first stage research, the

probability that the product is dangerous, given that the research did not discover that it was

is (1− q )+(qp(2r1),r2 ) . Third, a WPBE requires that the firm’s strategies be sequentially rational, meaning
    qp r ,r 1




that at each decision node, the firm’s action is optimal given the information available at that

point. This requirement applies to both the production decision and the research decision. When

the firm makes the research decision, it has no information. However, when it makes its


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production decision, it must do so accounting for whatever it learned from the first stage

research.

       A WPBE will consist of a set of two strategies and beliefs. Each strategy will take the

form (r; X, Y) where r is the first stage research choice, X is the firm’s second stage action if

danger is revealed and Y is the firm’s second stage action if danger is not revealed. There are

only two possible actions in the second stage, produce (P) or do not produce (DNP). While

technically there are an infinite number of different possible r choices, we will show that for

each X, Y combination, there is only one possible r value that would make the strategy

sequentially rational. There are therefore only four possible WPBEs: [([(r1; DNP, P), (r2; DNP,

P)], [(r1; DNP, P), (r2 DNP, DNP)], [(r1; DNP, DNP), (r2 DNP, P)], and [(r1; DNP, DNP), (r2

DNP, DNP)] where the research levels will be specific to each WPBE.

       Different assumptions about firm assets will make certain WPBEs possible and eliminate

others. When the firms are assumed identical (or when the firms are always solvent), the only

possible WPBEs are [(r1; DNP, P), (r2; DNP, P)] and [(r1; DNP, DNP), (r2 DNP, DNP)]. With

symmetric firms, only the symmetric WPBEs exist. When we allow for potential bankruptcy, we

add the possibility of [(r1; DNP, DNP), (r2 DNP, P)]. However, in our model, it is never

possible for [(r1; DNP, P), (r2 DNP, DNP)] to be a WPBE because we have assumed that firm 1

is the more solvent firm and thus always faces higher expected liability. Throughout our

analysis, we speak of the firms’ research decisions and production decisions. The research

decision is the research choice under the [(r1; DNP, P), (r2; DNP, P)] WPBE (since the research

choice under [(r1; DNP, DNP), (r2 DNP, DNP)] is always 0 for both firms). The production

decision is the existence criteria for the [(r1; DNP, P), (r2; DNP, P)] WPBE. This criteria tells




                                                                                                    5
under what parameter values it is possible to see both firms producing when the product is not

revealed to be dangerous.

        We reserve the details of our analysis for a mathematical appendix and here present only

the basic methodology and results. We first derive the socially optimal research and production

decisions. Then, we address the benchmark case in which firms are always solvent, comparing

the research and production decisions to the socially optimal choices. Next, we allow for limited

liability (or potential bankruptcy) without spillover liability. In this case, if a firm is bankrupted

by the liability, it loses only its assets. Finally, we turn to the case in which spillover liability is

allowed. In this case, if one firm is bankrupted and the other solvent, the unpaid liability of the

bankrupt firm is transferred to the solvent firm. The solvent firm then faces more than its initial

assigned liability.



Social Optimality

        In defining the socially optimal decisions, it is important to recognize that these decisions

are twofold. First, we consider whether it is socially optimal for the firms to produce, given the

research results. Second, we define that socially optimal research levels, knowing the socially

optimal second stage research decisions.

        If the research shows evidence of a dangerous product, it is never socially optimal for the

firms to produce because we have assumed that V < N D . However, it may be socially optimal

for both firms to produce if danger is not revealed. The social payoff from both firms

researching and producing if no evidence of danger is found is given

by A1 + A2 − r1 − r2 + qp(r1 , r2 )[2V − 2 ND ] + (1 − q )2V while the social payoff if neither firm




                                                                                                           6
researches or produces is given by A1 + A2 . It is only socially optimal for the firms to produce

when no danger is discovered if A1 + A2 − r1 − r2 + qp (r1 , r2 )[2V − 2 ND ] + (1 − q)2V > A1 + A2 .

       However, this production decision is truly socially optimal if the research levels on which

the production decision is based are those that maximize the social payoff. These levels –

                                  -- are given by − 1 + qp1 (r1 , r2 )[2V − 2 ED] = 0 and
             SO              SO                              SO     SO
denoted r1          and r2

−1 + qp2 (r1SO ,r2 SO )[2V − 2ED] = 0 . Note that because the firms are identical except for their asset

levels, these research choices are equal. These socially optimal research levels are given in

Proposition 1.



Proposition 1: If both firms research and produce when no evidence of danger is found, the

                                                            SO            SO
socially optimal research levels are given by r1                 and r2        .



          The socially optimal production decision is found by comparing the maximum social

benefit when both firms research at the socially optimal levels and produce and produce when no

evidence of danger is found to the maximum social benefit when neither researches or produces.

                                                              SO               SO
If both firms research and produce and choose r1                   and r2           , the social payoff is given by

A1 + A2 − r1         − r2        + qp(r1 , r2 )[2V − 2 ED ] + (1 − q)2V . Therefore, it is socially optimal for
               SO           SO          SO   SO




both firms to research and produce when no danger is revealed if

A1 + A2 − r1        − r2 SO + qp(r1SO ,r2 SO )[2V − 2ED] + (1 − q)2V > A1 + A2 . We summarize this socially
               SO




optimal condition in Proposition 2.




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Proposition 2: It is socially optimal for both firms to research and produce when no evidence of

danger is found if A1 + A2 − r1
                                  SO
                                       − r2 SO + qp(r1SO ,r2 SO )[2V − 2ED] + (1 − q)2V > A1 + A2 .8



        Notice from Proposition 1 that in order for our firms to choose the socially optimal

research levels, each must consider the full social costs of failing to uncover a dangerous

product, or 2V − 2 ED . There is a discrepancy between the private costs of not discovering

( V − ED ) and the social cost because each firm’s failure to discover can induce both firms to

produce the product. It is also clear from Proposition 2 that it may be socially to have both firms

produce or to have neither firm produce when no danger is discovered but never to have one

produce and one not produce. Therefore, the asymmetric WPBE [(r1; DNP, DNP), (r2 DNP, P)]

is never socially optimal.



Full Solvency

        We begin by addressing the benchmark case in which both firms are fully solvent. In this

case, each firm always faces the full damages if the product is dangerous. The payoff from

producing is given by Ai − ri + qp(r1 , r2 )[V − ND ] + (1 − q )V for i = 1, 2. If firm i chooses to

produce, it chooses the research level that maximizes this payoff. This is defined by

− 1 + qp i (r1 , r2 )[V − ND ] = 0 for i = 1, 2. We show in the appendix that this research level is

actually less than the socially optimal level. Even though the firms face the full damages

associated with their production, they actually underinvest in research. This is stated in

Proposition 3.



Proposition 3: Fully solvent firms invest in less than the socially optimal research levels.



                                                                                                       8
The results of Proposition 3 derive from the public good characteristics of research. Because

each firm can observe the other’s research before choosing whether or not to produce, each has

an incentive to reduce its own investment in research and instead wait and observe the other’s

research for free. This is the classic free-rider problem.

       The more interesting result from this model is that free-riding on research actually

induces the firms to produce less frequently than is socially optimal. 9 This result is summarized

in Proposition 4.



Proposition 4: Fully solvent firms produce less frequently than is socially optimal.



This production effect is due to the firms’ underinvestment in research. Because the firms decide

whether or not to produce after observing the research levels and results of both firms, they

realize that they have invested in too little research and so the danger of inadvertently producing

a dangerous product is higher than it would have been if they had chosen the socially optimal

research levels. The firms compensate for this increased risk by producing under a more limited

parameter range.

       The result of Proposition 4 is somewhat counter to traditional free-riding results. When

firms do not face the full costs of their actions – which they do not in this case since both firms

suffer when one fails to uncover a dangerous product – we expect that they will invest in too

little care and overengage in risky activities. While our model has yielded the former result, we

actually find that the firms underengage in risky activity, in this case production.




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        This analysis has shown that even the benchmark case we are using for comparison

throughout our paper implies suboptimal research and production decisions. Therefore, any

distortions related to limited liability are not necessarily welfare-reducing as they would be if the

fully solvent firms made socially optimal choices. When we add spillover liability to the model

and ask whether it is welfare-improving, we ask whether the net effects of potential bankruptcy

and spillover are to induce higher research levels and more production and thus offset these

effects of free-riding.

Limited Liability

        When firm i’s liability is limited by its asset level, its expected liability is given by

ELi = p(ND < Ai ) DE [N | ND < Ai ] + p(ND > Ai ) Ai . Only when the firm’s assets exceed the

actual damages does the firm face the full damages of its production. Otherwise, it only pays out

its assets (which are less than the damages). It is easy to show that this function is increasing in

the asset level, which implies both that EL1 > EL2 and that limited liability reduces a firm’s

expected liability. We assume that V < ELi , or that the damages are so high that even the

expected liability exceeds the benefits of producing.10 The payoffs from producing are now

Ai − ri + qp(r1 , r2 )[V − ELi ] + (1 − q )V for i = 1, 2. The research levels that jointly maximize

these expressions are given by − 1 + qp1 (r1 , r2 )[V − EL1 ] = 0 and − 1 + qp 2 (r1 , r2 )[V − EL2 ] = 0 .

        Limited liability has two different effects on each firm’s research decision. In the

appendix, we derive the total effect on firm 1’s research and firm 2’s research to be

                                        64 + 4 } 64 + 4 }
                                                  7 8 −                        7 8 −
                                        q 2 [ p1 p22 ][V − D2 ]       q 2 [ p2 p12 ][V − D1 ]
                                  dr1 =            +            dD1 −            +            dD2     (1)

                                       64 + 4 } 64 + 4 }
                                                7 8 −                          7 8 −
                                       q 2 [ p2 p11 ][V − D1 ]       q 2 [ p1 p21 ][V − D2 ]
                                 dr2 =            +            dD2 −            +            dD1




                                                                                                              10
dD1 refers to the reduction in firm 1’s expected liability due to limited liability while dD2 refers

to the reduction in firm 2’s expected liability due to limited liability. The first term in each

expression shows the effect of the own firm’s lower expected liability on its research decision.

The lower expected liability gives each firm less incentive to engage in research since they suffer

less if they fail to uncover a dangerous product. The second term, however, shows the effect of a

change in firm 2’s expected liability on the research choice of firm 1. Since firm 2 engages in

less research due to a reduction in its expected liability, firm 1 will be induced to engage in more

research because there is less opportunity for free-riding. The total effect will depend on the

relative strengths of these two effects. This is summarized in Proposition 5.



Proposition 5: Limited liability gives firms incentives to reduce their research since they face

lower expected liability but also to increase their research because the other firms reduce their

research in response to the decrease in their own expected liability.



        Limited liability has three different effects on each firm’s production decision. To find

the effect on the production decision, we need to estimate the effect of limited liability on the

maximum payoff from producing, or F = − ri + qp (r1 , r2 )[V − Di ] + (1 − q)V . In the appendix, we

find that the total effect on this value can be written as

                  7 8 −             4− 8 4−4 − −
              64 − 4 } 6 74 6 7 8 } } 6 74 6 7 8 } }                   4− 8 4−4 + −
        dF1 = − qp (r1 , r2 )dD1 + qp 2 (r1 , r2 )(V − D1 ) ∂D1 dD1 + qp 2 (r1 , r2 )(V − D1 ) ∂∂D22 dD2
                                                            ∂r2                                  r

                                                                                                           (2)
                  7 8 −            4− 8 4−4 − −
              64 − 4 } 6 74 6 7 8 } } 6 74 6 7 8 } }                 4− 8 4−4 + −
        dF2 = − qp(r1 , r2 )dD2 + qp1 (r1 , r2 )(V − D1 ) ∂D2 dD2 + qp1 (r1 , r2 )(V − D1 )∂∂D11 dD1
                                                           ∂r1                               r




The first term in this expression shows that limited liability clearly encourages production by

reducing the potential losses from producing. The second term shows an additional positive



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effect on production through the other firm’s research choice. When firm 1’s expected liability

falls, firm 2 actually has an incentive to increase its research in response, which lowers the risk

of a dangerous product and encourages production. The third term, however, shows that firm 2

also has an incentive to reduce its research when its own expected liability falls, which reduces

the payoff from producing by increasing the risk of inadvertently producing a dangerous product.

The total effect of limited liability on production decisions is unclear and will rest on the relative

strengths of these three effects. This is summarized in Proposition 6.



Proposition 6: Limited liability encourages production by reducing each firm’s expected

liability but the change in the research levels and thus the change in the risk of inadvertently

producing a dangerous product could discourage or encourage production.



       Determining the relative strengths of the multiple effects of limited liability on the

research and production decisions – and thus determining the net effects – is especially

complicated when we consider firms of varying asset levels. We show in the appendix that when

the firms are identical the net effect on research is negative although the effect on production is

still uncertain. However, the relative sizes of the conflicting effects will be heavily affected by

the relative sizes of the firms’ asset levels. In our model, firm 1 is assumed to be the higher asset

firm and so faces a higher expected liability even when liability is limited to its asset level. Firm

2, on the other hand is smaller and so faces a lower expected liability than firm 1 under a rule of

limited liability. In the expressions above, dD1 will be smaller (in absolute value terms) than

dD2 because firm 1’s expected liability does not fall as much when limited liability is used.




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Obviously, the relative magnitudes of dD1 and dD2 will determine the weights of the positive

and negative effects and thus the sign of the net effect.

       The net effect on research is more likely to be positive for a large firm in an industry with

a small firm. For this firm, its expected liability does not fall much with limited liability and so

it faces a rather weak incentive to reduce its research. However, the small firm will reduce its

research quite a bit and so the large firm will have a strong incentive to increase its research in

response. The net effect is more likely to be negative for a small firm in an industry with a large

firm. This firm faces a strong incentive to reduce its research as its expected liability falls

significantly with limited liability but a weak incentive to increase it since the large firm will not

reduce its research very much. This is summarized in Proposition 7.



Proposition 7: When firms have very different asset levels, limited liability can actually induce

more research from the larger firm although it will likely induce less from the smaller firm.



       The net effect of limited liability on the production decisions of firms of different sizes

will also depend on the relative magnitudes of dD1 and dD2 . When we consider firm 1, it is

clear that the reduction in its own expected liability ( dD1 ) introduces two positive effects on the

production payoff: it lowers the expected payout if the product is dangerous and it induces firm

2 to invest in more research as firm 1 reduces its own thus reducing the probability of an

undiscovered dangerous product. The reduction in the other firm’s expected liability ( dD2 ),

however, lowers the payoff from producing for firm 1 because it induces firm 2 to invest in less

research and thus increases the risk of inadvertently producing a dangerous product. We can

thus argue that it is more likely to see a firm produce less when it is a large firm in an industry



                                                                                                      13
with a small firm. The small firm will likely reduce its research in the face of limited liability

and so the large firm will face a higher risk of an undiscovered dangerous product. The small

firm likely produces more often since it can free-ride on the increased research of the large firm.

        It is worth noting at this point that only the payoff to the lower payoff firm matters in the

derivation of the WPBEs. This is explained in more detail in the appendix. In the case of

uniformly distributed exposure levels, we can show that firm 1 always has a lower payoff from

producing than firm 2. Therefore, we need only be concerned about the effects on firm 1. As

discussed, since firm 1 is the larger firm, it is likely that limited liability could actually induce

less production due to the large decrease in firm 2’s research levels. These results are

summarized in Proposition 8.



Proposition 8: When firms have very different asset levels, limited liability can actually induce

less production because of the increased risk of inadvertently producing a dangerous product.



The conclusions we can draw from this discussion is when limited liability is applied to an

industry with a small firm and a large firm, it is more likely to induce a positive net effect on

research for a large firm and a negative net effect on research for a small firm. The net effect on

production is likely to be negative for a large firm and positive for the small firm.



Spillover Liability

        The objective of this paper is to determine if and under what conditions allowing for

spillover liability would be welfare-increasing. As discussed above, in order for spillover

liability to be welfare-increasing, it must be true that the net effects of bankruptcy and spillover




                                                                                                        14
are to encourage more research from both firms and to induce the firms to produce under a wider

parameter range since the free-riding alone induced too little research and restricted production.

The effects of limited liability alone were discussed in the previous section. We have not

addressed whether limited liability will be welfare-increasing or welfare-reducing and we will

hold off any discussion of this and instead address it with spillover liability.

       When spillover liability is allowed, the most obvious impact is to increase the expected

liability of the more solvent firm, which in our model is firm 1. For an intermediate range of N,

firm 2 is bankrupted by its own liability and firm 1 is solvent in the face of its own liability.

Without spillover, firm 1 would only have faced its assigned liability. However, once spillover

is allowed, firm 1 faces its own assigned liability plus the unpaid liability of firm 2. Although

firm 1 is more likely to be bankrupted when spillover is allowed, its expected liability is still

always higher than without spillover. The traditional result that bankruptcy reduces expected

liability rests on the assumption that in those bankrupt cases the firm’s assets are less than the

liability and so the firm pays out less than its actual damages. In this case, the expanded range of

cases under which firm 2 is bankrupt are cases under which the firm would have been solvent

without spillover. Thus, under these cases the firm’s assets are actually greater than its damages

and so bankrupting the firm (taking its entire assets) actually increases the firm’s losses.

       Firm 2’s expected liability, however, remains unchanged when spillover is introduced.

Since firm 2 has a lower asset level than firm 1 but both face the same liability, liability can

never spill from firm 1 to firm 2. If firm 1 is bankrupted, then the damages must be high enough

that firm 2 is also bankrupted and so no liability can be transferred. Since firm 2 never faces

spillover liability, it faces the same expected liability as it did when spillover was not allowed.




                                                                                                      15
We denote the new expected liability levels as ELi,S , where it is easy to show

that EL2, S = EL2 < EL1 < EL1, S .

         Determining the effects of spillover on the research decisions of the firms is much more

straightforward than those of limited liability. The increased expected liability on firm 1 induces

firm 1 to invest in a higher research level and firm 2 to invest in a lower research level. With

firm 1 increasing its research, firm 2 can to free-ride on this higher research level and reduce its

own. We can see this by referring to (1), recognizing that dD2 is equal to 0 and dD1 is now

positive. These results are summarized in Proposition 9.



Proposition 9: Spillover liability induces more research from the more solvent firm but less

research from the less solvent firm.



         The effects of spillover on production are also quite clear. We show in the appendix that

spillover reduces the maximum payoff from producing to firm 1 but increases the maximum

payoff from producing to firm 2. Firm 1 faces higher expected liability if it produces and since

firm 2 reduces its research and free-rides, the benefits from producing fall further. Firm 2, on the

other hand, is able to reduce its research and free-ride on the increased research from firm 1. We

can see this by referring to (2) above and again recognizing that dD2 is equal to 0 and dD1 is

now positive. Since firm 1 is the lower payoff firm, this implies that production is overall less

likely after spillover. This is summarized in Proposition 10.



Proposition 10: Spillover liability restricts the parameter range over which production takes

place.



                                                                                                    16
Welfare Analysis

        The social desirability of spillover will depend on the effects of limited liability. The

effects of spillover are clear but those of limited liability depend on the parameters of the model.

We know that fully solvent firms engage in too little research and produce under too restricted a

parameter range and that spillover induces more research from firm 1, less from firm 2 and less

frequent production. Therefore, spillover will only definitely be welfare-increasing if (1) limited

liability induces so much more research from firm 2 that it is actually overinvesting prior to

spillover, (2) limited liability either induces less research from firm 1 or has such a minimal

positive effect that the firm is still underinvesting before spillover, and (3) limited liability

induces the firms to produce more than socially optimal. If only one or two of these conditions

are true, it does not necessarily mean that spillover is not welfare-increasing, only that it is not

clear whether or not it is. Certainly if none of these three conditions hold, spillover is welfare-

reducing. While we cannot determine the exact conditions under which each of these conditions

hold, we can draw some general conclusions about when it would be more likely to be true.

        In order for firm 1 to be investing in too little research prior to spillover, it must be true

that the limited liability induced either a reduction in research or an increase too small to

completely offset the free-riding problem. This is more likely when the firms are more similar in

size since then the research encouraging effect from firm 2’s reduction in research is less

pronounced. Because firm 1 is the larger firm, if the asset levels are very different, then firm 2

will reduce its research substantially in the face of limited liability, which could create a large

enough indirect research effect on firm 1 that it could end up actually increasing its research.




                                                                                                         17
       It is never possible that firm 2 will be investing in too much research prior to spillover,

since both free-riding and limited liability induce less research from firm 2.    Furthermore, the

more different the firms’ asset levels, the farther firm 2’s research choice will be from the

socially optimal level. When firm 2’s asset level is very low and firm 1’s relatively high, limited

liability induces a large decrease in firm 1’s research, both because its own liability is limited

and because firm 1 does not reduce its research very much and may actually even increase it.

       In order for the firms to be producing under too broad a parameter range prior to

spillover, limited liability must give a positive incentive to produce strong enough to outweigh

the production discouraging effects of limited liability as well as those presented by free-riding.

Although limited liability can encourage production by reducing the expected liability that each

firm faces, we showed above that when the firms have very different asset levels, limited liability

is likely to actually discourage production. In the WPBE, only the lower-payoff firm’s return

from producing matters, which in our model is firm 1. Although firm 1’s expected liability falls,

firm 2 reduces its research significantly and thus raises the risk of inadvertently producing a

dangerous product. When the firms’ asset levels are very different, it is more likely that this

latter effect will dominate.

       It is never possible for spillover to drive all three decisions (firm 1’s research, firm 2’s

research, and production) closer to the socially optimal because the effect on firm 2’s research is

always welfare-reducing. Furthermore, when asset levels are very different, it is likely that both

firm 1’s research decision and the production decision are also driven farther from the socially

optimal levels by spillover. Firm 1 is probably investing in too much research prior to spillover

and production is likely taking place too infrequently, making spillover undesirable. When asset

levels are more similar, it is possible that firm 1 might be underinvesting in research prior to




                                                                                                      18
spillover and production might be taking place too often, but similar asset levels present limited

opportunity for spillover to be allowed. When firms are similarly sized, there is narrow range of

parameter values over which one will be bankrupted and the other solvent. Furthermore, the

amount of liability that could spill over is small. These arguments are summarized in

Proposition 11.



Proposition 11: When firms are of very different asset sizes, spillover liability is more likely to

be welfare-reducing.



Conclusions

        At first glance, allowing for plaintiffs to shift their claims from bankrupt firms to solvent

firms might appear to be a simple way to overcome some of the incentive problems that limited

assets can create. The increased expected liability might induce more research and less

production, which would counteract the standard limited liability results of too little research and

too much production. However, we have shown that this intuition might be quite misleading for

two reasons. First, when firms’ research decisions are interdependent, limited liability does not

necessarily induce less research and more production. Second, although spillover does increase

expected liability, it only does so for the more solvent firms in the industry. These firms do

increase their research but the less solvent firms respond by reducing their research, limiting the

potential for an increase in welfare.

        With firms of different sizes, there may be frequent opportunity to use spillover, since it

is quite likely for one firm to be bankrupted and not the other. Furthermore, the magnitude of the

spillover liability could be quite large since the firms’ asset levels are so different.




                                                                                                   19
Unfortunately, these cases in which spillover might be easiest to apply might be just those cases

in which it is welfare-reducing.




                                                                                                20
                                   MATHEMATICAL APPENDIX

1: General model
         In this appendix, we write firm i’s expected payout when the product is dangerous as Di .
We use this general term to allow us to more simply consider changes in this level due to limited
liability and spillover. We are assuming that V < Di for all versions of the model, or that the
expected payout levels are so high that they exceed the benefits of producing, even when the
firms face limited liability. We have briefly described the methodology of a WPBE in the text.
The main requirement for a WPBE is sequential rationality. Here we consider each of the
possible WPBEs, identify the conditions when each will exist and determine the research choices
under each WPBE.
         The assumption that V < Di implies that no strategy under which the firm would produce
when the research reveals danger could ever be part of a WPBE. If a firm knows that the product
is dangerous, it recognizes that it will always face liability and because the expected liability
always exceeds the value of producing, it would never be optimal for the firm to produce. There
are therefore, only four possible strategy combinations that might be a WPBE: [(r1; DNP, P), (r2;
DNP, P)], [(r1; DNP, P), (r2 DNP, DNP)], [(r1; DNP, DNP), (r2 DNP, P)], and [(r1; DNP,
DNP), (r2 DNP, DNP)]. We will investigate each of these combinations and identify the
conditions under which it is a WPBE.
2.1: [(r1; DNP, P), (r2; DNP, P)]
         Under this strategy combination, both firms produce if no danger is revealed and neither
produces if danger is revealed. The risks of accidentally producing a dangerous product are
outweighed by the benefits of producing a potentially safe product when the firm does not know
if the product is dangerous. In order for this strategy to be optimal, the research level must be
high enough to drive the conditional expected liability (the conditional probability that the
product is dangerous when the research does not reveal that it is times the expected liability)
below the value of producing but not so high that the firm can do better by not researching or
producing at all. Because we are looking for a WPBE and thus require sequential rationality, we
will examine the decisions in each stage, beginning with the last stage.
         In order for this strategy to be sequentially rational at the production stage, it must be true
that the conditional expected liability is low enough that it is outweighed by the value of
producing. This requirement is formalized in Lemma A1.

Lemma A1: If [(r1; DNP, P), (r2; DNP, P)] is a WPBE, then V >
                                                                               qp ( r1 , r2 )
                                                                          qp ( r1 , r2 ) + (1− q )   Di for i = 1, 2.

Only when the conditions of Lemma A1 hold would it would be optimal for both firms to
produce when there is no evidence of danger but to not produce when there is evidence of
danger. The benefit of producing (V) exceeds the conditional expected liability.
        Lemma A1 describes the conditions necessary for the second stage production decisions
to be sequentially rational. However, sequential rationality also extends to the first stage
research decisions. The research levels chosen must be those that maximize the payoff to each
firm. The expected payoff of playing (ri; DNP, P) when the other firm does the same is given
by Ai − ri + qp(r1 , r2 )[V − Di ] + (1 − q )V . With probability qp(r1 , r2 ) , the product is dangerous but
the research will not reveal that it is, in which case the firm will produce, gain the benefit V, but



                                                                                                                        21
also pay out the expected liability. The firm also receives the net benefit Ai − ri . With
probability (1 − q) , the product is not dangerous (and so of course the research does not reveal

as Ai − ri . Finally, with probability q[1 − p(r1 , r2 )] , the product is dangerous and the research
that it is). In this case, the firm produces and receives the benefit of producing V, as well

reveals that it is. The firm does not produce and simply receives the net benefit Ai − ri .
Weighing each of these payoffs by the associated probability gives the expected payoff function

− 1 + qp1 (r1 , r2 )[V − D1 ] = 0 and − 1 + qp 2 (r1 , r2 )[V − D2 ] = 0 . We denote these research levels as
above. The research levels that maximize the expected payoff to each firm are given by

r1* and r2* . These choices are restated in Lemma 2.

                                                                           *             *
Lemma A2: If [(r1; DNP, P), (r2; DNP, P)] is a WPBE, then r1 = r1 and r2 = r2 .

           Note that because the expected liability levels are not equal, these research levels are not
equal. If the firms had identical asset levels (or were always solvent), then the research levels
would be the same for both firms. The second order conditions are
 qp11 (r1 , r2 )[V − D1 ] qp12 (r1 , r2 )[V − D1 ]
 qp 21 (r1 , r2 )[V − D2 ] qp 22 (r1 , r2 )[V − D2 ]
                                                     > 0 . We can simplify this to

 q 2 [ p11 p 22 − p 21 p12 ][V − D1 ][V − D2 ] > 0 . Since we know that [V − D1 ] < 0 and [V − D2 ] < 0 ,
this condition can be simplified to p11 p 22 − p 21 p12 > 0 . We assume that p11 < p 21 and that
 p 22 < p12 . So a firm’s research has a stronger marginal effect on its own research productivity
than the other firm’s. Under this assumption, our second order conditions hold.
            The final requirement for sequential rationality is that neither firm can realize a higher
payoff by choosing a different strategy. Specifically, the payoff to each firm of choosing ( ri* ;
DNP, P) exceeds the payoff of deviating and choosing (ri; DNP, DNP), given that the other firm
will choose ( ri* ; DNP, P). It is easy to see that the highest payoff that a firm could realize under
(ri; DNP, DNP) would be A. Since the firm would never produce under (ri; DNP, DNP), there is
no benefit from research and so the optimal choice of r would be 0. Therefore, if [(r1; DNP, P),
(r2; DNP, P)] is sequentially rational, it must be true that the payoff to each firm --
 Ai − ri + qp(r1 , r2 )[V − Di ] + (1 − q )V -- exceeds Ai . This is formalized in Lemma A3.
          *          *    *




Lemma A3: If [(r1; DNP, P), (r2; DNP, P)] is a WPBE, then
Ai − ri + qp(r1 , r2 )[V − Di ] + (1 − q )V > Ai for i = 1, 2.
       *       *    *




        Lemmas A1 – A3 describe the conditions necessary for [(r1; DNP, P), (r2; DNP, P)] to
be sequentially rational. If the conditions of Lemmas A1 – A3 all hold, then [(r1; DNP, P), (r2;
DNP, P)] is a WPBE. However, notice that if the inequality in Lemma A3 holds for firm i, then
the inequality in Lemma A1 will also hold at the optimal research choices. We can therefore
combine the necessary conditions in Proposition A1.




                                                                                                           22
Proposition A1: If Ai − ri + qp(r1 , r2 )[V − Di ] + (1 − q )V > Ai for i = 1, 2, then [( r1 ; DNP, P),
                              *        *    *                                             *

   *
( r2 ; DNP, P)] is a WPBE.

2.2: [(r1; DNP, DNP), (r2, DNP, P)], [(r1; DNP, P), (r2; DNP, DNP)]
         Proposition A1 lays out the conditions under which a WPBE exists in which both firms
produce when no evidence of danger is found. However, it is also possible that when one firm
chooses to produce when no evidence of danger is found, the other firm might find it optimal to
not produce, even when no danger is discovered. This asymmetry is possible because of the
different expected liability levels. Even if the conditional expected liability for one firm is
driven low enough that the benefits of producing exceed it, this will not always be true for the
other firm who may face higher expected liability due to higher assets.
         As listed, there are two possible asymmetric equilibriums (where asymmetric refers to the
strategy choice). We will first address the strategy combination [(r1; DNP, DNP), (r2, DNP, P)].
In this case, firm 1 never produces. As we discuss this combination – and the conditions under
which it will be a WPBE – it will be obvious that [(r1; DNP, P), (r2; DNP, DNP)] can never be a
WPBE. Since firm 2 is the lower asset firm, it always faces the lower expected liability, and so it
would never be optimal for firm 1 to produce and for firm 2 not to produce. We again examine
the conditions necessary for sequential rationality at each stage, beginning with the production
stage.
         In order for [(r1; DNP, DNP), (r2, DNP, P)] to be sequentially rational at the production
stage, it must be true that the value of producing is less than the conditional expected liability for
firm 1 but greater than the conditional expected liability facing firm 2. Only then would it be
optimal for firm 1 to not produce and firm 2 to produce when there is no evidence of danger.
This is formalized in Lemma A4.

Lemma A4: If [(r; DNP, DNP), (r; DNP, P)] is a WPBE,
then qp ( r1 , r2 1)+2(1− q ) D1 > V > qp( r1 , r21) +2(1− q ) D2 .
          qp ( r , r )                     qp ( r , r )




Since firm 1 faces a higher expected liability than firm 2 does, it is possible for the inequality in
Lemma A4 to hold. Notice that in order for the other asymmetric strategy combination -- [(r1;
DNP, P), (r2; DNP, DNP)] – to be sequentially rational, this inequality would have to be
reversed. Since D1 > D2 , this can never be true and so it is obvious that [(r1; DNP, P), (r2; DNP,
DNP)] can never be a WPBE. This is given in Corollary A1.

Corollary A1: Because D1 > D2 , [(r1; DNP, P), (r2; DNP, DNP)] is never sequentially rational
at the production stage and so can never be a WPBE.

        Sequential rationality at the research stage requires that the research levels be chosen
such that they maximize the firm’s expected payoff. Firm 1’s expected payoff under [(r1; DNP,
DNP), (r2; DNP, P)] is simply A1 − r1 . The firm never produces and so never receives the benefit

obviously 0. Firm 2’s expected payoff is A2 − r2 + qp(0, r2 )[V − D2 ] + (1 − q)V . The research
of producing but also never faces any liability. The research level that maximizes this function is

level that maximizes this expression is given by − 1 + qp 2 (0, r2 )[V − D2 ] = 0 . We denote this
                      M
research level by r2 . We summarize these results in Lemma A5.


                                                                                                      23
                                                                                        M
Lemma A5: If [(r; DNP, DNP), (r; DNP, P)] is a WPBE, then r1 = 0 and r2 = r2 .

It is easy to see that r2 > r2* . Since firm 1 performs no research if it does not produce, firm 2
                         M


can no longer free-ride on any of the research of firm 1 and so must increase its own above the
level it would choose when both firms produce.
         Finally, if this strategy combination is sequentially rational, it must be true that the
expected payoff of playing the strategy for each firm exceed the expected payoff of playing any
other strategy. For firm 1, this means that its payoff from not producing or researching must be
greater than the payoff it could realize by choosing (r; DNP, P). The highest possible payoff that
firm 1 could receive by choosing (r; DNP, P) is A1 − r1M + qp (r1M , r2M )[V − D1 ] + (1 − q)V , where
 r1M is given by − 1 + qp1 (r1M , r2M )[V − D1 ] = 0 . Notice that this research level is less than the
level chosen in the first WPBE because firm 2 is investing in a very high level of research if it
produces alone. For firm 2, the payoff from researching and producing alone must exceed the
payoff from not producing, or (r; DNP, DNP). The best firm 2 could do when never producing
would obviously be to choose not to research all and earn a net benefit of A2 . These conditions
are summarized in Lemma A6.
                                        M
Lemma A6: If [(0; DNP, DNP), ( r2 ; DNP, P)] is a WPBE, then
− r1M + qp(r1M , r2M )[V − D1 ] + (1 − q )V < 0 and − r2M + qp(0, r2M )[V − D2 ] + (1 − q )V > 0 .

We can see that r1M < r1* because r2 > r2* . Since firm 2 invests in more research, firm 1 can
                                        M


reduce its research and free-ride.
                                                                                 M
       If the conditions of Lemmas A4 – A6 all hold, then [(0; DNP, DNP), ( r2 ; DNP, P) is a
WPBE. However, if the inequalities in Lemma A6 hold, then those in Lemma A4 will also hold.
Therefore, we can express the conditions for [(0; DNP, DNP), ( r2 M ; DNP, P) to be a WPBE as
follows.

Proposition A2: If − r1M + qp(r1M , r2M )[V − D1 ] + (1 − q )V < 0 and
− r2M + qp(0, r2M )[V − D2 ] + (1 − q )V > 0 , then [(0; DNP, DNP), ( r2 M ; DNP, P)] is a WPBE.

        Finally, it is easy to derive the conditions for [(r1; DNP, DNP), (r2; DNP, DNP)] to be a
WPBE by following the general procedure of the preceding analysis. In this WPBE, neither firm
ever produces. Even if the firms engaged in research levels high enough that the conditional
expected liability was driven below the benefit of producing, the firms would still be better off
not researching or producing at all.

Proposition A3: If − r2       + qp(0, r2 )[V − D2 ] + (1 − q)V < 0 , then [(0; DNP, DNP), (0; DNP,
                          M            M


DNP)] is a WPBE.

        We can rewrite the conditions for each WPBE in terms of the parameter q. Proposition 1
states that the WPBE in which both firms produce when no evidence of danger is found exists



                                                                                                      24
when − ri + qp (r1 , r2 )[V − Di ] + (1 − q)V > 0 . A simple application of the envelope theorem
         *                   *       *


shows that − ri + qp (r1 , r2 )[V − Di ] + (1 − q)V is decreasing in the expected liability and so the
                     *                   *        *


expression is lower for firm 1. We can therefore rewrite the conditions for this WPBE as
− r1 + qp (r1 , r2 )[V − D1 ] + (1 − q)V > 0 . Focusing on the parameter q, we refer to the q level
    *        *    *


that solves − r1 + qp (r1 , r2 )[V − D1 ] + (1 − q)V = 0 as q a . Because
                     *                   *        *


− r1 + qp (r1 , r2 )[V − Di ] + (1 − q)V is also decreasing in q, all q levels such that q < q a will
    *            *       *


satisfy − r1 + qp (r1 , r2 )[V − D1 ] + (1 − q)V > 0 . We summarize this in Corollary A2.
             *                   *       *




Corollary A2: If q < q a , then [( r1 ; DNP, P), ( r2 ; DNP, P)] is a WPBE.
                                                          *              *



                                                                                    M
       The conditions under which [(0; DNP, DNP), ( r2 ; DNP, P)] is a WPBE are given in
Proposition 2. We refer to the q level such that − r1M + qp(r1M , r2M )[V − D1 ] + (1 − q )V = 0 as qb
and the q level such that − r2M + qp(0, r2M )[V − D2 ] + (1 − q)V = 0 as qc . When qb < q < q c ,
then the conditions of Proposition 2 are satisfied. This is summarized in Corollary A3.

Corollary A3: If qb < q < q c , then [(0; DNP, DNP), ( r2 ; DNP, P)] is a WPBE.
                                                                                   M




       Finally, we can see from Proposition A3 that if − r2 + qp(0, r2 )[V − D2 ] + (1 − q)V < 0 ,
                                                                                        M          M


[(0; DNP, DNP), (0; DNP, DNP)] is a WPBE. We have already solved for the q level that
solves − r2 + qp(0, r2 )[V − D2 ] + (1 − q)V = 0 as qc . Therefore, when q > qc , the condition of
           M            M


Proposition 3 is satisfied. We present this result in Corollary 3:

Corollary A4: If q > qc , then [(0; DNP, DNP), (0; DNP, DNP)] is a WPBE.

       We can use Corollaries A1 – A3 to map the existence ranges for the three possible
WPBEs. We can show that the ranges for the asymmetric WPBE and the WPBE in which both
firms produce cannot overlap. 11 PUT THIS NOTE INTO THE TEXT Figure 1 depicts the
ranges of q over which each WPBE exists.

             Figure A1: Existence Ranges for the three possible WPBEs in terms of q
                                                                             Firm 1 never
             Both produce                                                    produces. Firm 2
             if no evidence                               No WPBE            produces if no
                                                                                                            Neither firm
             of danger.                                   exists.            evidence of danger.
                                             qa                     qb                                 qc   ever produces.
                                                      *


                                                                                                                             q
q=0

        As figure A1 shows, when the probability of danger (q) is very low, the only WPBE is
that in which both firms produce when there is no evidence of danger. Since a low value of q


                                                                                                                             25
implies a low level of expected liability, the value of producing exceeds the conditional expected
liability and both firms produce as long as there is no evidence of danger. When the probability
of danger is very high, the only WPBE is that in which neither firm ever produces. In this case,
the expected liability is so high that the value of producing never exceeds it and so the firms
never produce. When q falls in an intermediate range, the only WPBE is that in which firm 1
does not produce but firm 2 does. The q level is high enough that the expected liability
outweighs the benefits of producing for firm 1 but low enough that the benefits exceed the
expected liability for firm 2.
         Notice that there is also an intermediate range in which no WPBE exists. In this range
( q a < q < qb ), firm 1 would prefer to not research or produce instead of choosing r1 and
                                                                                       *


producing. Firm 2, however, prefers to research and produce and would even prefer to choose
   M                                                                           M
 r2 and produce over not producing. However, if firm 2 were to choose r2 and produce alone,
firm 1 would want to enter again. Firm 2 is researching at a much higher level and firm 1 can
                                                                                                   *
free-ride on this. Of course, if firm 1 were to enter, firm 2 would reduce its research back to r2
and firm 1 would exit.

1.1: Comparing Firm Choices Under Full Solvency to Socially Optimal Choices

        It is easy to show that even fully solvent firms invest in suboptimal levels of research.
The firms take into account only their private net cost of failing to discover a dangerous product,
or V − ND . However, the socially optimal decisions require that the firms consider the full
socially cost of failing to discover, or 2V − 2 ND . We can easily see the effect of this on welfare.
If we rewrite the first order conditions of Lemma A2 as −1+ pF p1 [NC ] = 0,−1+ pF p2 [NC ] = 0 ,
where NC is higher (less negative) under the private decisions, we can see that
               − p F p1   p F p12 [ NB ]                         ?

                          p F p 22 [ NB ]        − p F 2 (NB )[ p1 p 22 − p 2 p12 ]
                                                 64 444744448
 ∂r1           − p F p2
∂NB
       =   p F p11 [ NB ] p F p12 [ NB ]
                                             =   p F 2 (NB )2 [ p11 p 22 − p p12 ]
                                                                                      . The sign of this depends on the sign of
           p F p 21 [ NB ] p F p 22 [ NB ]
                                                 14444 244421 4
                                                              4             43
                                                                 +


 p1 p 22 − p 2 p12 . In equilibrium, we know that the research choices will be identical and, relying
on the symmetry of the probability function, we can see that p1 = p2 . We also know that the
probability function is convex, or that p11 p 22 − p12 p 21 > 0 . However, we can use the symmetry
and the equality of the research levels in equilibrium to claim that
 p11 p 22 − p12 p 21 = 2( p 22 − p12 ) > 0 . The sign of p1 p 22 − p 2 p12 = p1 ( p 22 − p12 ) is then clearly
negative, meaning that the firms will engage in less research than is socially optimal.
          We also claim that our fully solvent firms actually produce under too narrow a parameter
range. To see this result, first recognize that rSO and rSO were the values that maximized
                                                                                                (       )
A1 + A2 − r1 − r2 + [pF p(r1, r2 ) + (1 − pF )]2VP − pF p(r1,r2 ) N + N d . Because of this, it must be true
that A1 + A2 − rSO − rSO + [ p F p (rSO , rSO ) + (1 − p F )]2V P − p F p(rSO , rSO ) N + N d >                  (      )
                                   [                                          ]                     (
A1 + A2 − r * − r * + pF p(r * , r * ) + (1 − p F ) 2VP − p F p(r * , r * ) N + N d . Since our firms are   )
symmetric, we can rewrite this as − rSO + [ p F p(rSO , rSO ) + (1 − p F )]VP − p F p (rSO , rSO )                                ( )d >
                                                                                                                                  N+N


           [                                         ]                                 ( )d by dividing each side of the inequality in
                                                                                                                                   2
                                                                                        N +N
− r * + p F p (r * , r * ) + (1 − pF ) VP − p F p (r * , r * )                            2

half.


                                                                                                                                           26
2: Limited Liability
       When a limited liability rule is used, the most obvious effect is a reduction in both firms’
expected liability levels. The firm’s expected liability under limited liability is given by
ELi = p(ND < Ai )DE[N | ND < Ai ] + p(ND > Ai )Ai . To see that this is less than its expected
damages, note first that we can rewrite p(ND < Ai )DE[N | ND < Ai ] + p(ND > Ai )Ai as
( )[−
   1
 N −N
         Ai 2
         2D
                −   N 2D
                      2               ]
                           + NAi . It is then easy to show that ∂ELi i =
                                                                 ∂A                                                      ( )[−
                                                                                                                             1
                                                                                                                            N−N
                                                                                                                                           Ai
                                                                                                                                           D               ]
                                                                                                                                                + N , which is obviously
positive. This implies both that the expected payout falls under limited liability and also the firm
1 faces a higher expected liability than firm 2 since firm 1 has a higher asset level.
        We can see the effects of limited liability on the research levels and existence conditions
for the WPBEs by adjusting the above analysis to account for a reduction in both D1 and D2 .
We will first address the situation in which the firms are assumed to be of equal assets. This
implies that D1 = D2 whether or not limited liability is imposed. We then turn to the case of
unequal assets, which is more relevant for our objectives.
        When firms have identical asset levels and face limited liability, the first order conditions

evidence of danger become − 1 + qp1 (r1 , r2 )[V − D ] = 0 and − 1 + qp 2 (r1 , r2 )[V − D ] = 0 . We know
that define the research levels under the WPBE in which both firms produce when there is no

that when limited liability is imposed, D will fall below the expected damages and so the effects
of limited liability can be seen from:
                                                           qp1 ( r1 ,r2 ) qp12 ( r1 , r2 ) [V − D1 ]
                                          ∂r1              qp2 ( r1 ,r2 ) qp22 ( r1 , r2 ) [V − D2 ]                   q 2 [ p1 p 22 − p2 p12 ][V − D1 ]
                                          ∂D    =     qp11 ( r1 , r2 ) [V − D1 ]   qp12 ( r1 , r2 ) [V − D1 ]
                                                                                                                  =                    +
                                                      qp21 ( r1 , r2 ) [V − D2 ] qp22 ( r1 ,r2 ) [V − D2 ]

Because our firms are identical, p1 = p 2 in equilibrium. We know that [V − D ] < 0 and that
                                 ∂r1
 p 22 < p12 , which implies that ∂D > 0 . Deriving the same result for firm 2 is straightforward.
         When the firms are not identical, we cannot treat limited liability as an equal decrease in
Di for each firm. Instead, we must consider each firm separately. We find the following partial
derivatives:
                        qp1    qp12 [V − D1 ]                                                                          0    qp12 [V − D1 ]
        ∂r1                    qp22 [V − D2 ]              q 2 [ p1 p22 ][V − D2 ]              ∂r1                         qp22 [V − D2 ]                     2
                                                                                                                                                                   [ p2 p12 ][V − D1 ]
                =                                      =                              >0               =                                            =−q                                  <0
                         0                                                                                            qp2
        ∂D1         qp11 [V − D1 ]   qp12 [V − D1 ]                   +                        ∂D2              qp11 [V − D1 ]    qp12 [V − D1 ]                         +
                    qp21 [V − D2 ] qp22 [V − D2 ]                                                               qp21 [V − D2 ] qp22 [V − D2 ]
                        qp11 [V − D1 ] 0                                                                              qp11 [V − D1 ] qp1
        ∂r2             qp21 [V − D2 ] qp2                 q 2 [ p2 p11 ][V − D1 ]             ∂r2                    qp21 [V − D2 ] 0                         2
                                                                                                                                                                   [ p1 p21 ][V − D2 ]
        ∂D2     =   qp11 [V − D1 ]   qp12 [V − D1 ]
                                                       =              +              >0        ∂D1    =         qp11 [V − D1 ]    qp12 [V − D1 ]
                                                                                                                                                    =−q                  +               <0
                    qp21 [V − D2 ] qp22 [V − D2 ]                                                           qp21 [V − D2 ] qp22 [V − D2 ]

It is easy to so that simultaneous decreases in D1 and D2 have counteracting effects on each
firm’s research choice. The reduction in a firm’s own liability induces it to choose a lower
research level ( ∂∂D11 > 0 and ∂∂D22 > 0 ) but the reduction in the other firm’s liability induces it to
                   r             r


choose a higher output level because the other firm will reduce its own ( ∂∂D12 < 0 and
                                                                                                                                                                                    ∂r2
                                                                            r
                                                                                                                                                                                    ∂D1   < 0 ).
The net effect on each firm’s research level will depend on the degree to which limited liability
reduces each firm’s expected payout. The total effects of a reduction in both D1 and D2 on the
research choices are given by:




                                                                                                                                                                                                   27
                                            64 + 4 } 64 + 4 }
                                                      7 8 −                        7 8 −
                                            q 2 [ p1 p22 ][V − D2 ]       q 2 [ p2 p12 ][V − D1 ]
                                     dr1 =             +            dD1 −            +            dD2
                                                       +                             +
                                                      7 8 −                         7 8 −
                                            q 2 [ p2 p11 ][V − D1 ]       q 2 [ p1 p21 ][V − D2 ]
                                            64 4 } 64 4 }
                                     dr2 =             +            dD2 −            +            dD1
Obviously the relative sizes of dD1 and dD2 will determine the net effects.
         We can also find the effects on the production decisions (or the existence conditions for
the WPBE in which both firms produce when no evidence of danger is found) by analyzing how
simultaneous changes in both firms’ expected liability levels affect the maximum value of
 F = − ri + qp (r1 , r2 )[V − Di ] + (1 − q)V for i = 1, 2. The direct effect of a reduction in a firm’s
own expected liability (ignoring any change in the other firm’s research level) is clearly positive
by the envelope theorem. We can write ∂∂D1 = − qp(r1 , r2 ) which is positive when considering a
                                                           F                    *


reduction in expected payout. However, since the other firm also adjusts its research in response
to its lower expected liability, there is a second effect. Finally, since the other firm’s expected
liability falls simultaneously, there is a third effect in that each firm responds to the expected
liability change of the other firm.

                                  [                                                                         ]
         To be more explicit, we can write the derivative of F with respect to a reduction in firm
i’s expected payout as ∂∂D1 = − qp (r1, r2 ) + qp1 (r1, r2 )[V − D1 ] ∂D11 + qp2 (r1 , r2 )[V − D1 ] ∂D21 − ∂D11 dD1 .
                          F                                           ∂r                             ∂r     ∂r


We can use our first order conditions to eliminate the second and last terms in the braces because

                                      [                                                          ]
we know they will add to 0. We then write the derivative of F with respect to a reduction in firm
j’s expected payout as ∂∂D2 = − ∂D12 + qp1 (r1, r2 )[V − D1 ] ∂D12 + qp2 (r1 , r2 )[V − D1 ] ∂D22 dD2 . We can
                         F       ∂r                            ∂r                            ∂r


again eliminate the first and second terms in the braces by using the first order conditions. This
leaves us with a total effect of
 dF = − qp(r1 , r2 )dD1 + qp 2 (r1 , r2 )(V − D1 ) ∂∂D21 dD1 + qp 2 (r1 , r2 )(V − D1 ) ∂∂D22 dD2 .
                                                     r                                    r


         The first term in this above expression is the direct effect of a reduction in a firm’s own
expected payout. The second is the effect of a decrease in firm 1’s expected payout on the
research of firm 2 and the resulting effect on the risk of inadvertently producing a dangerous
product. Finally, the third term is the effect of the reduction in the second firm’s expected
payout on firm 1’s research level and thus on the equilibrium probability level. We can sign the
component terms of the total derivative but not the entire expression. For reductions in D1 and
 D2 , the first and third terms are positive while the second is negative.
         In the WPBE in which both firms produce when there is no evidence of danger, the only
payoff that matters is the lower one. We can easily show that when D1 > D2 , the difference
between the payoffs from producing and not producing, or − ri + qp (r1 , r2 )[V − Di ] + (1 − q)V is
                                                                            *         *    *


greater for firm 2. If we compare − r1 + qp (r1 , r2 )[V − D1 ] + (1 − q)V to
                                               *           *   *


− r2 + qp (r1 , r2 )[V − D2 ] + (1 − q )V , we can see that because D1 > D2 and r1* > r2* it is also true
    *          *   *


that − r1 + qp (r1 , r2 )[V − D1 ] + (1 − q)V < − r2 + qp(r1 , r2 )[V − D2 ] + (1 − q)V . Therefore, when
         *         *    *                              *           *   *


we analyze the effects of these changes in Di on the firm’s production decisions, the only
production decision that will affect the existence of the WPBE in which both firms produce is
that of firm 1.



                                                                                                                   28
OLD PROOF: The derivative of −ri + qp(r1 ,r2 )[V − ELi ] + (1− q)V with respect to Ai is equal
                                                                                                         *                 *        *


                              * ∂EL
                                                  (
to −qp(r1 , r2 ) ∂A i i + −1 + qpi (r1 , r2 )[V − ELi ]                                                                    )   ∂ri *
                                                                                                                                        + qp j ( r1 , r2 )[V − ELi ] ∂Aj i . We can appeal to the
                                                                                                                                                                                       ∂r
                                                                                                                                                                                            *
                  *                                                                *        *                                                         *      *
                                                                                                                               ∂ Ai

first order conditions to eliminate the middle term, giving us
−qp(r1 ,r2 ) ∂ELi i + qp j (r1 ,r2 )[V − ELi ] ∂Aj i . If we investigate the sign of
          *       *                                        *        *                                    ∂r      *
                                                                                                                                                                                         ∂r j *
              ∂A                                                                                                                                                                         ∂A i     , we can see that our
first order conditions ( −1 + qpi (r1 , r2 )[V − ELi ] = 0 for i = 1, 2) give us the result that
                                                                                        *            *

              qp11 ( r1* ,r2* )[ V −EL1 ]
                                                                          ∂EL
                                                           qp1 (r1* ,r2* ) ∂A 1                                       +
                                                                                                               64748 6 74 64 − 4 }
                                                                                                                                 4− 8          7 8 +
                                                                                                         − q 2 p 21 (r1* ,r2* )[V − EL2 ] p1 (r1* ,r2* ) ∂A 1
                                                                                                                                                        ∂EL
              qp 21 ( r1* ,r2* )[ V −EL 2 ]
                                                                             1
∂r2
  *
      =                                                                                         =                                                                    < 0 . We can therefore sign the
                                                                        0                                                                                        1
∂A1       qp11 (r1* ,r2* )[V −EL1 ] qp12 (r1* ,r2* )[ V − EL1 ]                                                                         +
          qp 21 (r1* ,r2* )[V −EL 2 ] qp 22 (r1* ,r2* )[ V − EL2 ]

             4+ 8 +               4 _ 8 4− 8 −
derivative −qp(r1 ,r2 ) ∂A i + q p j (r1 ,r2 )[V − ELi ]∂Aj i < 0
            6 74 }              6 74 6 74 }*
                 *   * ∂ELi             *   *           ∂r




Spillover
       The main effect of spillover liability is to increase firm 1’s expected liability while

                                                                                                                                            (                                                            )
leaving that of firm 2 unchanged. We can write firm 1’s expected liability under spillover as
                                  A +A                        A +A                                                                                                                                                          A1 + A 2
 p(ND < A2 )DE[N | ND < A2 ] + p( 1 2 2 > ND > A2 ) 2E[N | 1 2 2 > ND > A2 ]D − A2 + p(ND >                                                                                                                                            )A1
                                                                                                                                        (                                                           )
                                                                                                                                                                                                                               2
                                                                                                A1 + A 2                                               A1 + A 2                                               A1 + A 2
or as p(N <                   A2
                                      )DE[N | N <                       A2
                                                                             ] + p(                               >N>          A2
                                                                                                                                    ) 2E[N |                         >N>        A2
                                                                                                                                                                                     ]D − A2 + p(N >                     )A1 .
                                                                                                               A12+A2 − AD2   A12+A2 + AD2  
This can be simplified to  N − N  D                                                                        +  D  2 D 2  D − A2  +  N − 2 D  A1 or further to
                              D                                         D                         2D                           D                         2D                     D                               2D


                           −N                                                                                                                             
                                                                                                                                                       A1 +A2


                                                                                                               N − N                       N−N 
                                                                                                     +N
                                                                        A2                      A2
                                                                        D                       D




( )[                                                                             ]
                                                                                                     2


                   − N 2D − (
           A22                2
                                              A1 + A 2 )2
  1
                                                               + NA1 . We can easily show that this is greater than

( )[−                                                       ]
 N−N       2D                                    4D


                          −                  + NAi , or that EL1,S > EL1 . We have already shown that EL1 > EL2 and we
                      2               2
   1              Ai              N D
 N −N             2D               2

know that firm 2’s expected liability is unaffected by spillover. We then have the result that
EL1,S > EL1 > EL2 = EL2,S .
       The effects of this change in the expected liability on the firms’ research choices in the
WPBE in which both firms produce when no evidence of danger is found is simple to see. We
                                                                                                                                 qp1        qp12 [V − D1 ]
                                                                                                                 ∂r1                        qp22 [V − D2 ]               q 2 [ p1 p22 ][V − D2 ]
                                                                                                                       =                                             =                             > 0 and
                                                                                                                                  0
                                                                                                                           qp11 [V − D1 ]        qp12 [V − D1 ]
know from the above derivatives that                                                                             ∂D1                                                                +
                                                                                                                           qp21 [V − D2 ] qp22 [V − D2 ]
                  qp11 [V − D1 ] qp1
∂r2               qp21 [V − D2 ] 0                                           2
                                                                                 [ p1 p21 ][V − D2 ]
∂D1   =    qp11 [V − D1 ]                 qp12 [V − D1 ]
                                                               =−q                     +                         < 0 . We can therefore conclude that when spillover is
           qp21 [V − D2 ] qp22 [V − D2 ]

allowed, firm 1 increases it’s research while firm 2 reduces its research.
       The effects of spillover liability on production will in part depend on the research effects.
We can again find the effects on production decisions by analyzing the effect on the maximum
value of − ri + qp (r1 , r2 )[V − Di ] + (1 − q)V > 0 for i = 1, 2. For firm 1, we need evaluate the
             *        *    *


effect on − r1 + qp (r1 , r2 )[V − D1 ] + (1 − q)V of an increase in D1 , considering both the direct
                                  *                   *         *


effect as well as the indirect effect through firm 2’s research. We can write the derivative as
− qp (r1 , r2 ) + qp 2 (r1 , r2 )[V − ELi ] ∂r21 . We know from the preceding paragraph that firm 2 will
                                                                                                             *
              *           *                           *         *
                                            ∂D

reduce its research in response to spillover liability so we can see that the sign of this derivative


                                                                                                                                                                                                                                 29
is negative. Therefore, firm 1 faces a lower payoff from producing under spillover liability.
Firm 2, however, sees a higher payoff from producing. If we take the derivative of
− r2 + qp (r1 , r2 )[V − D2 ] + (1 − q )V with respect to D1 , we find qp1 (r1 , r2 )[V − D2 ] ∂D1 . Since
    *        *   *                                                             *   *           ∂r *
                                                                                                1



we know that firm 1 increases its research, the sign of this derivative is clearly positive.
1
  White 2002
2
  See Shavell (), Beard (), Larson ().
3
  Hensler et al 2001, p 9
4
  White 2002
5
  “USG Files for Bankruptcy, Blames Lawsuits” June 25, 2001
6
  see Shavell (1983), Schwartz (1985) for a dangerous products context and Klepper (2002),
Kamien, Muller, and Zang (1992), and Katz (1986) for a cost-reducing R&D context
7
  The Food and Drug Administration Modernization Act of 1997 established a public databank of
clinical trials results for certain types of drugs and some have suggested expanding the bank’s
coverage in light of the recent controversy over childhood depression medications.
8
  Notice that due to the assumptions made on the probability function, it is always better for the
social planner to distribute the research between two firms and not simply have one firm
research. For a given level of research expenditures R = r1 + r2 , we can show that allocating the
research between the two firms will always produce a lower probability of producing a
dangerous product than assigning all of the research to one firm, or that p(r1,r2 ) < p(r1 + r2,0).

 p(αv + (1− α )w ) < αp(v ) + (1− α ) p(w ). If we set v = (R,0) and w = (0,R), then we have
Since the probability function is assumed convex, it must be true that
     r           r        r             r              r             r

p(αR, (1− α )R) < αp(R,0) + (1 − α ) p(0, R). However, due to the symmetry of the probability
function, this is identical to p(αR, (1− α )R) < p(R,0). Setting α = r1 gives us the desired result.
                                                                             R
9
  We are using the word frequently loosely here. When we say that the firms produce less
frequently, we mean that they produce under a more limited parameter range.
10
   The model can certainly be adjusted to address the case when this inequality does not hold.
However, the model becomes rather uninteresting since if a firm will produce even if it knows
the product is dangerous, it certainly will never invest in any research.
11
   In order for the ranges to overlap, it must be true that the conditions for the existence of the
asymmetric WPBE and the WPBE in which both firms produce when there is no evidence of
danger can be satisfied simultaneously. The conditions for the asymmetric equilibrium are
− r1M + qp(r1M , r2M )[V − EL1] + (1− q)V < 0 and − r2M + qp(0, r2M )[V − EL2 ] + (1− q)V > 0 . The
conditions for the WPBE in which both firms produce when no evidence of danger exists are
given by Ai − ri* + qp( r1*, r2* )[V − ELi ] + (1− q)V > Ai . It is easy to show that it is never possible
for − r1M + qp(r1M , r2M )[V − EL1] + (1− q)V < 0 and − r1 + qp (r1 , r2 )[V − EL1 ] + (1 − q)V > 0 to
                                                           *         *   *


hold simultaneously. Since r2M > r2 , it must be true that
                                        *


− r1M + qp(r1M , r2M )[V − EL1 ] + (1 − q )V > − r1 + qp (r1 , r2 )[V − EL1 ] + (1 − q )V . r1M is the
                                                  *        *    *


research level that maximizes − r1M + qp(r1M , r2M )[V − EL1 ] + (1 − q)V and r1* is the research level
that maximizes − ri + qp (r1 , r2 )[V − ELi ] + (1 − q)Vi . A simple application of the envelope
                     *         *   *


theorem gives the result that
 − r1M + qp(r1M , r2M )[V − EL1 ] + (1 − q )V > − r1 + qp (r1 , r2 )[V − EL1 ] + (1 − q )V . From this, we
                                                    *        *    *




                                                                                                             30
can conclude that if − r1 + qp (r1 , r2 )[V − EL1 ] + (1 − q)V > 0 , then
                                       *        *   *


− r1M + qp(r1M , r2M )[V − EL1] + (1− q)V < 0 . We can rephrase this as if                               p ( r1M
                                                                                                                    − r1M +V
                                                                                                                   , r2M ) [EL1 −V ]+V
                                                                                                                                               <q,
                 − ri   *
                            +V                                                          − r1M +V                               − ri   *
                                                                                                                                          +V
then                [ELi −V ]+V
       p ( r1* ,r2* )
                                  < q . It is therefore never possible for   p ( r1M   , r2M   ) [EL1 −V ]+V
                                                                                                               <                  [ELi −V ]+V
                                                                                                                     p ( r1* ,r2* )
                                                                                                                                                 . MUST
BE A BETTER WAY TO SHOW THIS RESULT.




                                                                                                                                                      31

								
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