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Producer Liability and Competition Policy When Firms Are Bound by a Common Industry Reputation

Andrzej Baniak, Peter Grajzl and A. Joseph Guse

Abstract

We contrast the laissez-faire regime with the regime of strict producer liability and draw the implications for competition policy in a setting where oligopolistic firms cannot differentiate themselves from rivals but rather are bound by a common industry reputation for product safety. We show that, first, unlike in the traditional products liability model, firms’ incentives to invest in precaution depend on market structure. Second, depending on the magnitude of expected damages awarded by the courts, laissez-faire can welfare dominate strict producer liability. Third, the relationship between social welfare and industry size, and hence the role for competition policy, depends on the institutional regime governing the industry. Under some circumstances, restricting industry size is unambiguously welfare-enhancing.

JEL Classification: K13; L13; D43

Acknowledgments

For helpful comments and discussion, we thank Yao-Yu Chih, Valentina Dimitrova-Grajzl, Brett Frischmann, Keisuke Hattori, Jeroen Hinloopen, Wolfgang Kerber, Elodie Rouviere, Tinni Sen, Raphael Soubeyran, Spencer Waller, participants at the annual meetings of the Midwest Economics Association, the International Society on New Institutional Economics, the European Association for Research in Industrial Economics, the European Association of Law and Economics, seminar participants at Washington and Lee University, and two anonymous referees.

Appendix A

Derivation of expressions [4] and [5]: Differentiating eq. [3] with respect to ai and qi for all i∈{1,…n} gives

[18]Wai=C(ai)qilqi=0,
[19]Wqi=αβQC(ai)ail=0

for all i∈{1,…n}. Letting qi>0 for all i∈{1,…n}, eq. [18] immediately implies eq. [4]. Using eq. [19] in turn implies eq. [5]. □

Derivation of expressions [8]–[10]: Taking aj and qj for all firms ji as given, firm i chooses ai and qi to maximize eq. [7]. The resulting first-order conditions (FOC) for an interior solution are

[20]Πiai=lqijqjqiC(ai)qi=0,
[21]Πiqi=βlaijqjjqjajjqj2qi+αβjqjljqjajjqjC(ai)=0.

After imposing symmetry (Q=nq), eq. [20] simplifies to

[22]ln+C(a)=0,

which implies expression [8]. Eq. [21] then simplifies to

[23]βq+αβnqlaC(a)=0.

Eq. [23] can be solved for q to obtain expression [9] and, when multiplying the resulting expression by n, expression [10]. □

Derivation of expressions [13]–[15]: Taking aj and qj for all firms ji as given, firm i chooses ai and qi to maximize eq. [12]. The resulting FOC for an interior solution are

[24]Πiai=(ld)qijqjqi(C(ai)+d)qi=0,
[25]Πiqi=β(ld)aijqjjqjajjqj2qi+αβjqj(ld)jqjajjqj(C(ai)+dai)=0.

Imposing symmetry (Q=nq), eq. [24] simplifies to

[26](ld)nC(a)d=0,

which implies expression [13]. Eq. [25] then simplifies to

[27]βq+αβnq(ld)aC(a)da=0.

Eq. [23] can be solved for q to obtain expression [14] and, when multiplying the resulting expression by n, expression [15]. □

Lemma A1: The following statements hold:

  1. The expression la+C(a) is decreasing for a<aFB, increasing for a>aFB, and attains a minimum at a=aFB.

  2. The expression –la–C(a) is increasing for a<aFB, decreasing for a>aFB, and attains a maximum at a=aFB.

  3. Let k1+C′(a1)=0 and k2 + C′(a2)=0. Then, a1>a2 if and only if k1<k2.

Proof: Follows from the fact that, first, la+C(a) is strictly convex in a and, second, l+C'(a)=0 at a=aFB; see eq. [4]. □

Lemma A2: When firms choose a common accident probability a and output level q, total industry output equals Q=nq, and social welfare equals

[28]W(a,Q)=0Q[αβy]dyQC(a)Qla=Q[α12βQC(a)la].
Then
[29]W(a,Q)a=Q[l+C(a)]
and
[30]W(a,Q)Q=αlaβQC(a).

Proof: Straightforward, thus omitted. □

Proof of Proposition 1: If d=l, it follows from eqs. [4] and [13] that aL= aFB. The comparison of eqs. [4] and [8] reveals that aFB<aN. This proves part 1 of the result.

When d<l, the following holds

[31]l>ldn+d>ln.

Thus, from Lemma A1, part 3, aFB<a L<aN. This proves part 2 of the result.

Finally, when d>l, the following holds:

[32]ldn+d>l>ln.

Thus, from Lemma A1, part 3, aL<aFB<aN. This proves part 3 of the result. □

Proof of Proposition 2: From eqs. [10] and [15], note that for r∈{N,L} we have

[33]Qr=nβ(n+1)[αlarC(ar)].

Therefore, from Lemma A1, part 2,

[34]QFB=1β[αlaFBC(aFB)]>nβ(n+1)[αlaFBC(aFB)]nβ(n+1)[αlarC(ar)]=Qr.

Part 1 of the result then follows immediately from eq. [34] and Proposition 1, parts 1 and 2.

To prove part 2 of the result, note that aL<aFB<aN when d>l (see Proposition 1). Thus, from eq. [33], QN<QL if and only if

[35]αlaNC(aN)<αlaLC(aL).

When d=l, aL=aFB<aN (see Proposition 1) and, thus, by Lemma A1, part 2, expression [35] holds. Note that the right-hand side of expression [35] is a continuous function of d, whereas the left-hand side of expression [35] does not depend on d. Thus, for d>l and d close to l, the inequality in expression [35] holds as well and, therefore, QN<QL. When d increases, aL decreases (see eq. [13]) and, hence, C(aL) increases. Therefore, for d large enough, by continuity, the inequality sign in expression [35] is eventually reversed if αlaNC(aN)>αC(0), that is, if C(0)>laN+C(aN), or, equivalently, if [C(0)–C(aN)]/aN>l. □

Lemma A3: Let Wr and Qr, respectively, denote welfare and total industry output under regime r∈{N,L}. Then,

[36]Wr=β(n+2)2n(Qr)2.

Proof: Using eqs. [10] and [15] we can write

[37]Qr=nβ(n+1)[αlarC(ar)]

for r∈{N,L}. By eq. [28]

[38]W(ar,Qr)=Qr[αlarC(ar)12βQr].

Eq. [37] can be rewritten as

[39]αlarC(ar)=β(n+1)nQr.

Substituting eq. [39] into eq. [38] and collecting terms give eq. [36].

Proof of Proposition 3: Follows immediately from Lemma A3 and Proposition 2.

Proof of Proposition 4: From Lemma A3,

[40]Wr=β(n+2)2n(Qr)2

for r∈{N,L}. Then, the direct effect of n on Wr (viewing Wr as a continuous function of n and abusing notation somewhat) is

[41]Wrn=βn2(Qr)2

and

[42]WrQr=β(n+2)nQr.

From eq. [37], the direct effect of n on Qr (again abusing notation somewhat) equals

[43]Qrn=1β(n+1)2[αlarC(ar)]=1n(n+1)Qr,

where the last equality follows from eq. [39]. The competition effect thus equals

[44]Wrn+WrQrQrn=βn2(Qr)2+β(n+2)nQr1n(n+1)Qr=βn2(n+1)(Qr)2,

which is positive for all r∈{N,L}. This proves part 1 of the result.

To sign the industry reputation effect, first note that eq. [42] is positive for all r∈{N,L}. Thus, the sign of the industry reputation effect is determined by the sign of

[45]Qrararn.

Consider, first, the first derivative in the above product. We have

[46]Qrar=nβ(n+1)[l+C(ar)].

Thus, by Lemma A1,

[47]signQrar=sign(aFBar).

Consider, next, the second derivative in eq. [45]. Accident probability ar is defined by γr(n) + C′(ar) = 0, where the function γr(n) varies across regimes r∈{N,L} as follows:

[48]γN(n)=ln,γL(n)=ldn+d.

Applying the implicit function theorem,

[49]arn=γr(n)C′′(ar)

and since C(·) is strictly convex, thus,

[50]signarn=sign[γr(n)].

Therefore,

[51]signQrararn=sign(aFBar)sign[γr(n)].

To prove part 2(a) of the result, note that from eq. [48], sign[–γ′N(n)]=1. Because aN>aFB (see Proposition 1), sign(aFBaN)=–1. Therefore, by eq. [51], the industry reputation effect under r= N is negative.

To prove part 2(b) of the result, observe that from eq. [48], sign[–γ′L(n)]=sign(ld). From Proposition 1, for d=l we have aL=aFB; for d<l we have aL>aFB; and for d>l we have aL<aFB. Thus, by eq. [51], the industry reputation effect under r=L equals zero for d=l and is negative for dl.

Parts 3 and 4 of Proposition 4 follow immediately from expression [17] and parts 1 and 2, which we prove above.

Proof of Proposition 5: From eq. [36], ∂Wr/∂n=–β(Qr)2n–2 and ∂Wr/∂Qr=β(n+2)Qrn–1. From eq. [37], ∂Qr/∂n=Qr[n(n+1)]–1. Then, the competition effect (see eq. [17]) equals

[52]Wrn+WrQrQrn=βn2(n+1)(Qr)2

and the industry reputation effect equals

[53]WrQrQrardardn=β(n+2)nQrQrardardn.

Thus, we can write the total effect (see eq. [17]), as

[54]dWrdn=βn2Qr1n+1Qr+n(n+2)Qrararn.

The right-hand side of eq. [54] can be expressed as T1r· [T2r + T3r], where

[55]T1rβn2Qr
[56]T2r1n+1Qr,
[57]T3rn(n+2)Qrardardn,

with T1r>0, T2r>0 for all r∈{N,L}, and T3r<0 for r= N and for r= L and dl (see Proposition 4).

Consider, first, eq. [55]. Using eq. [37], we can write

[58]T1r=1n(n+1)[αlarC(ar)].

Let n → ∞. Then, if r= N, αlaNC(aN) → αlC(1), where the limit of aN is 1. If r=L and dl, αlalC(aL) → αLC(κL), where κL∈(0,1) is the limit of aL defined by –C'(κL) = d. Thus, from eq. [58] it follows that, because the term [n(n + 1)]–1 → 0 as n → ∞, T1r → 0 for r=N and for r=L when dl.

Consider, next, eq. [56]. Using eq. [37] again, we can write

[59]T2r=nβ(n+1)2[αlarC(ar)].

From the argument above, the limit of αlarC(ar) as n → ∞ is finite for r=N and for r=L when dl. Because the term n[β(n + 1)]–2 → 0 as n → ∞, thus, T2r → 0 for r=N and for r=L when dl.

Finally, consider eq. [57]. Using eqs. [46] and [56], we can write eq. [57] as

[60]T3r=n2(n+2)β(n+1)[l+C(ar)]γr(n)C′′(ar).

Suppose, first, that r=N. Then, from eq. [48], it follows that

[61]T3N=n+2β(n+1)[l+C(aN)]lC′′(aN).

Thus, T3Nβ–1 · [l + C'(1)] · [(–l)/C"(1)] = β–1 · (–l2) · [C"(1)]–1τN <0 as n → ∞.

Suppose, second, that r= L and dl. Then,

[62]T3L=n+2β(n+1)[l+C(aL)](ld)C′′(aL)

and T3Lβ–1 · [ld] · [(–(ld))/C"(κL)] = β–1 · [(–(ld)2)/C"(κL)]≡τL as n → ∞. Note that τL<0 both when d>l and when d<l.

Thus, as n → ∞, T1r· [T2r + T3r] → 0 · [0 + τr] = 0, where τr < 0 for r= N and for r=L when dl. For r=N and for r=L when dl, by continuity, there, thus, exists an n0r<∞ such that dWr/dn<0 at n= n0r and, for n>n0r, dWr/dn → 0 as n → ∞. Hence, for r=N and for r=L when dl, Wr attains maximum at n < ∞.

Appendix B

This appendix demonstrates that, in instances when the industry reputation effect is negative, first, social welfare can decrease with the number of firms n and, second, the relationship between social welfare and the number of firms n can be either monotonic or non-monotonic.

Recall from Proposition 4 that the industry reputation effect is negative for r= N and for r= L when dl. Throughout this appendix, we assume that the per-unit cost of production equals

[63]C(ai)=12ρ(1ai)2,

where ρ>0. Thus, C′(ai)=–ρ(1 – ai)<0 for ai∈(0,1) and C"(ai)=ρ>0. Accordingly, from eq. [8]

[64]aN=1lρn

and from eq. [10]

[65]QN=nβ(n+1)αl+l2(2n1)2ρn2.

Similarly, from eq. [13]

[66]aL=1d(n1)+lρn

and from eq. [15]

[67]QL=nβ(n+1)αl+(n1)2d(2ld)+l2(2n1)2ρn2.

Social welfare under laissez-faire (WN) is then obtained by substituting in eq. [65] into eq. [26]. Social welfare under strict producer liability (WL) is calculated by substituting in eq. [67] into eq. [26].

Using thus-obtained expressions for WN and WL, we plot social welfare under a given regime against n for different values of the model’s parameters in Figures 16. Figures 16 use parameter values such that firm and industry output under any legal regime are strictly positive for all n>1. Moreover, Figures 36 use parameter values such that the equilibrium accident probability under strict producer liability, as defined in eq. [66], is between 0 and 1 for any n>1.

Figures 16 illustrate that the relationship between social welfare under a given regime for which the industry reputation effect is negative, and the number of firms n, varies with values of the model’s parameters. Specifically, in Figures 2, 4, and 6, the relationship between social welfare and n is monotonic. In contrast, in Figures 1, 3, and 5, the relationship between social welfare and n is non-monotonic. In all of the examples, however, social welfare is maximized for some n<∞ (see Proposition 5).

Figure 1 Plot of WN against n, when α = 4, β = 3, ρ = 1, and l = 0.75

Figure 1

Plot of WN against n, when α = 4, β = 3, ρ = 1, and l = 0.75

Figure 2 Plot of WN against n, when α = 4, β = 3, ρ = 1, and l = 1.5

Figure 2

Plot of WN against n, when α = 4, β = 3, ρ = 1, and l = 1.5

Figure 3 Plot of WL against n, when d<l and α = 5.5, β = 0.1, ρ = 8, l = 7.5, and d = 6.2

Figure 3

Plot of WL against n, when d<l and α = 5.5, β = 0.1, ρ = 8, l = 7.5, and d = 6.2

Figure 4 Plot of WL against n, when d<l and α = 5.5, β = 0.1, ρ = 8, l = 7.5, and d = 4.5

Figure 4

Plot of WL against n, when d<l and α = 5.5, β = 0.1, ρ = 8, l = 7.5, and d = 4.5

Figure 5 Plot of WL against n, when d>l and α = 5.5, β = 0.1, ρ = 10, l = 7.5, and d = 8.5

Figure 5

Plot of WL against n, when d>l and α = 5.5, β = 0.1, ρ = 10, l = 7.5, and d = 8.5

Figure 6 Plot of WL against n, when d>l and α = 5.5, β = 0.1, ρ = 11, l = 7.5, and d = 10.2

Figure 6

Plot of WL against n, when d>l and α = 5.5, β = 0.1, ρ = 11, l = 7.5, and d = 10.2

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  1. 1

    Fleckinger (2007, 1), for example, describes the environments in which firms share a common reputation as “an intermediate situation between the perfect information and the asymmetric information setting”.

  2. 2

    For recent surveys of the voluminous literature on products liability, see Daughety and Reinganum (2014), Geistfeld (2009), and Shavell (2004, 2007).

  3. 3

    Baniak and Grajzl (2013) study the interaction of strict producer liability and industry reputation effects in a model of torts where harm occurs in a non-market setting.

  4. 4

    See Hattori and Yoshikawa (2013) for a model with endogenous firm investments in a common property resource that could be interpreted as representing firms’ common reputation.

  5. 5

    For discussion about the relationship between tort law and antitrust law, see also “Note: Deception as an Antitrust Violation” (2012).

  6. 6

    The traditional products liability model summarized by Daughety and Reinganum (2014) in fact implies that profit-maximizing firms choose the socially optimal level of precaution regardless of the liability rule in place. Firms choose socially suboptimal precaution when, for example, consumers systematically misperceive the risk of harm (Spence 1977; Polinsky and Rogerson 1983; Marino 1988a). However, the level of precaution selected by firms in models where consumers misperceive risk, but at the same time all other assumptions of the traditional products liability model continue to hold, is still independent of market structure.

  7. 7

    For analyses allowing for “scale effects” in expected harm, see Marino (1988a, 1988b) and Spulber (1988). For models featuring fixed costs of safety, see Daughety and Reinganum (2006) and Baumann and Friehe (2012).

  8. 8

    Chiang and Masson (1988) do not provide a fully fledged welfare analysis. However, they do show that in the presence of a common industry reputation, when firms have no market power, labor is the only input, and returns to scale are constant, firm consolidation leads to higher wages.

  9. 9

    The literature, of course, suggests reasons other than the existence of a common industry reputation for why greater intra-industry competition, as captured by the number of firms in an industry, need not increase social efficiency. One prominent reason is the existence of fixed set-up costs that firms must incur upon entry; see, e.g. Mankiw and Whinston (1986) and references therein. Another is the application of the strict market share liability rule; see Marino (1991). Stucke (2013) provides an illuminating discussion of when competition leads to suboptimal results.

  10. 10

    Chen and Hua (2012) do not study common industry reputation effects. Instead, they focus on the impact of full producer liability, partial producer liability, and punitive damages on monopolist’s incentives to increase product safety through ex ante investment when the firm can also take ex post (i.e. after sales) remedial measures concerning product quality.

  11. 11

    Assuming risk aversion rather than risk neutrality would render the model intractable, but would, we anticipate, not change the key qualitative insights.

  12. 12

    In the related products liability literature that does not study implications of firms sharing a common industry reputation, linear demand is directly assumed (as opposed to derived from an underlying utility function) for example in Polinsky and Rogerson (1983) and Daughety and Reinganum (1995). In contrast, in the industrial organization literature on common industry reputation, Fleckinger (2007), for example, starts from a setup with heterogeneous consumers and develops a multiplicative inverse demand.

  13. 13

    Another scenario that gives rise to free-riding in firms’ choice of precaution is the application of the market share liability rule, under which firms are held strictly liable for their market share of the total damages caused by the industry; see Marino (1991).

  14. 14

    When consumers’ knowledge of risk of failure is perfect, the precise allocation of liability for losses from defective products does not matter for firms’ investments in precaution under the assumptions of the traditional products liability model (see, e.g. Shavell 1980; Landes and Posner 1985). Laissez-faire performs just as well as strict producer liability. Given administrative and litigation costs associated with products liability, laissez-faire is in fact the preferred regime.

Published Online: 2014-5-3
Published in Print: 2014-10-1

©2014 by De Gruyter

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