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Equilibrium and Welfare in a Model of Torts with Industry Reputation Effects

Andrzej Baniak and Peter Grajzl EMAIL logo
From the journal Review of Law & Economics


We study the problem of torts in a framework where a firm’s accident adversely impacts all firms in the industry because of the presence of industry reputation effects. Industry reputation effects lead to interdependence among firms and give rise to strategic firm behavior. We characterize the industry equilibrium and the socially optimal industry configuration in such a setting. We then elucidate how the presence of industry reputation effects and the introduction of a liability regime in the form of a strict liability rule determine whether industry equilibrium is aligned or misaligned with the socially optimal industry configuration. Our results show that both the impact of industry reputation effects and the impact of the strict liability rule are in general contingent on the specifics of the tort problem at hand. In particular, we find that the presence of industry reputation effects can substitute for a suboptimal liability regime and that, in the presence of industry reputation effects, the introduction of the strict liability rule may be detrimental by steering the industry equilibrium away from the socially optimal industry configuration.

JEL Classifications: K13; L14; D60


Derivation of expression [4]:

Given [2], expression [3] becomes

where X is a random variable following a binomial distribution with mnk independent (Bernoulli) trials with success probability P. Drawing on the properties of binomially distributed random variables (see, e.g. Mood et al., 1974; Papoulis and Pillai, 2002), and . Using these facts and collecting terms,

which is expression [4]. □

Lemma A1:

Let, where. Then,

  1. ,

  2. ,

  3. .

Proof: Straightforward, thus omitted. □

Proof of Lemma 1:

To prove Lemma 1, we use Lemma A1 above. For R(m) in eq. [4], which is cubic in m, let , , , and d=0. Thus, from Lemma A1, part (i),

Because if and only if , which is in turn true if and only if , we also have

This proves part (i) of Lemma 1.

From Lemma A1, part (ii),

Then, if and only if or, equivalently, when . Thus, if and if . Note that if and only if or, equivalently, if and only if . and, clearly, as n→∞. This proves part (ii) of Lemma 1. Finally, from Lemma A1, part (iii),

which proves part (iii) of Lemma 1. □

Proof of Proposition 1:

Follows immediately from the discussion preceding Proposition 1 and is illustrated with Figure 2. □

Derivation of expression [13]:

With social harm per accident equal to H and with mn-k firms in the industry not investing in precaution, the expected social losses due to accidents equal

where X is a random variable following a binomial distribution with independent (Bernoulli) trials with success probability P, and thus E[X]=mP (see, e.g. Mood et al., 1974; Papoulis and Pillai, 2002). □

Proof of Lemma 2:

Rewrite expression [14] as

where R(m) is defined in eq. [4]. Then,

where (see Lemma 1). Thus, if and only if

which proves part (i) of Lemma 2.

Given the expression for ΔSC(m) above, we have

Because (see Lemma 1), and if . This proves part (ii) of Lemma 2. □

Proof of Proposition 2:

Follows immediately from the discussion preceding Proposition 2 and is illustrated with Figure 3. □

Proof of Lemma 3:

Trivial, thus omitted. □

Proof of Propositions 3–7:

Follows immediately from the discussion preceding each of the Propositions 3–7 and from the corresponding figures. □


For helpful comments and suggestions, we thank Atin Basu, Valentina Dimitrova-Grajzl, Joseph Guse, Anthony Heyes, Afshad Irani, Botond Kőszegi, Eric Linhart, Sergey Lychagin, Scott Masten, participants at the 2nd World Congress of the Public Choice Societies in Miami, the 10th Annual International Industrial Organization Conference at George Mason University School of Law, the 22nd Annual Meeting of the American Law and Economics Association at Stanford Law School, seminars at Central European University and Washington and Lee University, two anonymous reviewers, and the editor Francesco Parisi.


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  1. 1

    Related accounting literature provides evidence that accounting misstatements, which negatively affect shareholder wealth at the restating firm, induce share price declines among non-restating firms within the same industry (see, e.g. Xu et al., 2006; Gleason et al., 2008). Negative spillover (or “contagion”) effects have also been uncovered, for example, in the context of bankruptcy (Lang and Stulz, 1992), lawsuits (Hertzel and Kiholm Smith, 1993; Prince and Rubin, 2002), banking (see, e.g. Park, 1991), and the market for government debt (Landon and Smith, 2000).

  2. 2

    The standard law-and-economics model of torts is developed, for example, in Shavell (2004, 2007), Cooter and Ulen (2012), and Miceli (2008).

  3. 3

    Unlike in the context of torts where victims are third parties, models of strategic firm interaction are naturally more common in the literature on products liability and market structure (see, e.g. Polinsky and Rogerson, 1983).

  4. 4

    For early analyses and discussion of these issues in the context of products liability, see, e.g. Oi (1973) and Epple and Raviv (1978).

  5. 5

    Examining the impact of the extent of liability compensation, Baumann et al. (2011) study a model in which consumers cease to purchase products of firms which do not invest enough in accident prevention.

  6. 6

    Levin (2009) develops a stochastic version of Tirole’s (1996) model.

  7. 7

    Landon and Smith (1998) provide empirical evidence on the role of collective reputation for consumer behavior. Fatas et al. (2010) propose a mechanism that ensures adherence to high quality in an environment where firms are subject to collective reputation.

  8. 8

    For models of industry populated by ex ante identical firms, see, for example, d’Aspremont etal. (1983), Donsimoni et al. (1986), Polinsky and Rogerson (1983), and Shaffer (1995).

  9. 9

    Allowing for an investment in precaution to merely reduce the likelihood of an accident (rather than entirely eliminate the risk of an accident taking place) complicates the algebra, but should not alter the main conclusions of our analysis, as long as industry-wide reputational losses occur only when accidents are caused by firms not investing in precaution.

  10. 10

    The assumption that a firm can eliminate the risk of an accident by investing in precaution renders the strict liability rule equivalent to the negligence rule with a due standard of care set at .

  11. 11

    That is, we could write , where p is the probability that the firm is sued (conditional on experiencing an accident), q the probability that the firm is found liable, l the amount of damages determined by the court, and r firm-specific reputation loss. We assume that r is small.

  12. 12

    We thank two anonymous reviewers for the suggestion to explore an S-shaped specification of .

  13. 13

    The social psychology literature has identified escalated behavioral responses in a variety of contexts (see, e.g. Mikolic et al., 1997;Shukla-Mehta and Albin, 2002;Levine et al., 2011).

  14. 14

    Expression [2] implies that for a given number of industry accidents j, increases with industry size n. Given that we take industry size as exogenously fixed throughout the article, this implicit assumption comes without any loss of generality. Moreover, while the question of the relationship between industry size and industry-wide reputational sanctions incurred by each firm for a given number of accidents is ultimately an empirical one, it is plausible that larger industries are under closer public scrutiny than smaller industries and, therefore, that industry-wide reputational sanctions for a given number of accidents are stronger in the case of larger industries than they are in the case of smaller industries.

  15. 15

    From part (ii) of Lemma 1, the inflection point of R(m) is at .

  16. 16

    Letting , therefore, leads to qualitatively identical outcomes as when assuming that the ex post loss incurred by each firm in the industry due to diminished industry reputation (ρ(j)) is strictly increasing and strictly convex in the number of accidents j – a scenario we worked out in an early version of the article. A full proof of this point is available upon request.

  17. 17

    Industry configuration m∈{1,…,n–1} simultaneously satisfying conditions [9] and [11], or, equivalently, expressions [10] and [12], is evidently not unique when it comes to the identity of specific firms that invest in precaution and those that do not invest in precaution. In fact, there are strict Nash equilibria for any given m∈{1,…,n–1} that simultaneously satisfy conditions [9] and [11]. Because firms are identical, however, all of the equilibria for a given m∈{1,…,n–1} imply qualitatively the same behavior and have the same welfare repercussions (see Proposition 2 and Section 5). We, thus, do not refer to this situation as one of “true” multiple equilibria (using the standard meaning of the phrase).

  18. 18

    While is defined (only) on the set {1,2,…,n}, for graphical convenience, we illustrate in all of the figures as if were a continuous function defined on the entire interval [0,n].

  19. 19

    Note that an m such that is not an equilibrium. (In Figure 2, this corresponds to the situation where the line cuts the curve in the decreasing portion of .) Intuitively, with or, equivalently, , when m-1 firms are not investing in precaution, the mth firm currently also not investing in precaution is better off deviating and instead investing in precaution. Similarly, with or, equivalently, , with firms investing in precaution, the kth firm currently investing in precaution is better off deviating and instead not investing in precaution.

  20. 20

    None of our results change if we instead assume that the next expected social costs due to industry reputation effects are proportional to R(m), rather than to nR(m).

  21. 21

    We note that when , for a specific value of , SC(m) attains two minima so that and , with .

  22. 22

    If ρ is small enough that , then possibly (see Proposition 1, part (iii)) – the outcome that would prevail if there existed no industry reputation effects (ρ = 0).

  23. 23

    If before policy intervention (L = 0) and policy intervention increases L to L > 0, but L > 0 is small, then the new industry equilibrium may still be mIND = n (see Proposition 1, part (iii)), rendering the introduction of strict liability inconsequential.

  24. 24

    Since all firms in the industry are (ex ante) identical, industry equilibrium and socially optimal industry configuration in the absence of industry reputation effects can be characterized by focusing on a representative firm.

  25. 25

    The exception occurs either when or when . In the former case, all firms take precaution in the equilibrium and such an industry configuration is also socially optimal (). In the latter case, no firm takes precaution in the equilibrium and such an industry configuration is also socially optimal ().

  26. 26

    All of the qualitative conclusions of the analysis continues to hold if we instead let before the introduction of strict liability to allow for existence of positive (but limited) firm-specific reputation losses when a firm causes an accident.

  27. 27

    In a similar vein, the literature on law and agent-specific informal (reputational) sanctions allows for the possibility that passage and enforcement of a law shapes informal sanctions; see for example Cooter and Porat (2001:413), Deffains and Fluet (2013:949), and Ganuza et al. (2013:Sec. 7).

  28. 28

    In a typical multiple-tortfeasor setting without industry reputation effects, the negligence rule has been found to be superior to the strict liability rule (see, e.g. Landes and Posner, 1980; Shavell, 2007:167–168).

Published Online: 2013-10-29

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