Discussion Paper No. 753

SOCIALLY OPTIMAL LIABILITY RULESFOR FIRMS WITH
NATURAL MONOPOLY

Atsushi Tsuneki

August 2009

The Institute of Social and Economic ResearchOsaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Socially Optimal Liability Rules for Firms with Natural Monopoly
Atsushi Tsuneki
1. INTRODUCTION
It has been shown by Polinsky (1980) and Shavell (1980, 1987) that the strict liability
rule is socially superior to the negligence liability rule when firms are injurers,strangers are victims, and accidents have a unilateral nature. The analysis presupposesperfectly competitive behavior of firms.
We consider in this article the problem of socially efficient liability rules in a market
where natural monopoly prevails due to decreasing average cost. Intuitively, it can beconcluded that strict liability rules are more appropriate for firms with naturalmonopoly, for they are either allowed to enjoy the protection of the government or
exercise their market power. The present article shows that this intuition may notnecessarily be true from the view point of economic efficiency.
We consider a case where average cost pricing is achieved in a naturally monopolized
market either through well-organized government regulation or the weak invisible
hands of contestability. After the presentation of the model and its basic assumptions in
the next section, we discuss the effects of different liability regimes on resource
allocation in sections 3 and 4. Section 5 concludes our study with a summary of theimplications of this article, in comparison with those of previous contributions on thesubject.
2. THE

MODEL

Suppose that a monopolistic firm is selling its products to its customers. For theproduction of the good, the firm must bear a constant marginal cost c
the firm must bear a (probably very large) fixed cost F

0. In addition,

0 when it starts the business.

The production process involves some risk of accident, and there is a certain probability
per sale that an accident occurs and causes harm to strangers. The firm can partiallycontrol the probability and extent of harm that may occur in the event of accident by
paying attention to safety. We denote the cost of care x, expected amount of loss by theaccident δ

0 and suppose that δδ x , with δ x

0 and δ x

0.

The monopolistic firm discussed here minimizes its total cost either by proper
adaptation to government regulations or under the competitive pressure of thecontestable market. Therefore, analogous to the case of competitive firms shown byShavell (1980, 1987), this monopolistic firm chooses a socially optimal amount of care1x , either under a strict liability rule or a negligence rule with proper due carex

standard. Hereafter, we denote δ x

simply as δ.

Regarding the consumption side, we assume a partial equilibrium setting wheremarginal utility of income is constant for consumers. The market demand function
qD p , where D p0,p0, exists where q is the aggregate amount ofconsumption of the product of the monopolistic firm and p is a consumption price ofthe good.
Again, analogous to the case discussed by Shavell, the monopolistic firm must bear
the burden of liability when the legal rule is one of strict liability; the firm can escapethe burden of liability payment by appropriately choosing the due care level when thenegligence rule is adopted. This means that the average cost price adopted by themonopolistic firm will vary depending on kind of liability rules instated. Let us denotethe set of market price and quantity at the average cost pricing equilibrium (ACPEhereafter) as

under the strict liability regime, andp ,qp ,qunder thenegligence regime respectively. Then we have the average cost pricing formulae in bothregimes as follows:
(1 )

pc

δ

(2)

pc

FF,.

Suppose that all parties are risk neutral. Social welfare of the market is therefore
measured by the sum of the utilities of the customers minus the cost of production of thegood and the expected cost of accident. The maximum social surplus is obtained at themarginal cost pricing equilibrium where pc

δ, qD cδ , and the fixed cost F

is covered by the lump-sum tax.
Before starting on a detailed welfare analysis, let us summarize some additional
assumptions and preliminary results. We first assume that q0, which means that

this monopolistic firm provides positive welfare to the economy, at least when it isoptimally operated. We also assume that q(3)0, q

0. 1 We further assume thatFDis satisfied for the range of q between q and q . 2 Then, the proposition below follows.
In general, it is not possible to rule out the case q0, or even q0, even if we
assume that q0 . However, these cases are not particularly interesting andespecially irrelevant to the issue we discuss in this article.
2Assumption (3) means that the slope of the market demand curve is steeper than thatof the average cost curve at the ACPE.
12

Proposition 1(i) qq, qq.

(ii) the ACPE in the strict liability regime cannot be Pareto efficient.
The ACPE in the negligence regime cannot be Pareto efficient except for a coincidental
case that satisfies D c
Proof (i): As pc

F.

δδ is smaller than p in (1), the first result follows. We next

consider (1) as a function of δ, satisfying the demand condition as p δc

δF.

DDifferentiating the equation with respect to δ,(4) p δF1

D p1

follows. However, from assumption (3),1FD

0 is satisfied so that the secondbracket on the left-hand side of equation (4) is positive. Therefore, p δ

0 is impliedfor δ that corresponds to a value of q between q and q . This means that, startingfrom δ0, as we increase the value of δ while satisfying the average cost pricing

condition (1) and the market demand function, the market price increases constantly
and the level of consumption falls. Therefore, qq follows.

(ii): For the resource allocation to be Pareto efficient, the output level must be q . From(i), q is smaller than q and hence the ACPE in a strict liability regime cannot meetthis condition. To have q
Pareto efficient, pD cδFcqFD cc

δ , so that the ACPE in a negligence regime isδ must be satisfied. From these two equations,follows. With this relationship, it is easy to verify that q

q follows. //From the above proposition, we know that the ACPE can be at most second best,
barring the almost impossible exceptional cases. The rest of this article is devoted to theanalysis and comparison of the social welfare obtainable at the ACPE in the twoalternative liability regimes, in order to see which liability system should be second best
preferred at the ACPE. We distinguish the two cases q

q and qq . The formercase is discussed in the next section, and the latter case is discussed in section 4.
3. THE CASE WHEN q

q IS SATISFIED

The case discussed in this section offers us a clear-cut result, which is as follows.

3Proposition 2: When q

q is satisfied, the ACPE in the negligence regime is superior

to the ACPE in the strict liability regime in terms of social surplus.
Proof: Considering the above condition and Proposition 1(i), D cδqq

q issatisfied. Under this condition, the marginal benefit of consumption always exceedsc

δ so that the increase of consumption always improves welfare even if we consider

the expansion of harm that accompanies the increase of output by switching to thenegligence regime; the conclusion follows. //Diagrammatic illustration should help clarify the above result. The situation that is
discussed in this section can be illustrated as shown in Figure 1.
[Figure 1 about here]
The social surplus obtained at the Pareto efficient resource allocation W PE
characterized by the consumers’ surplus at the marginal cost price p
fixed cost F, i.e.,

cisδ, minus

F. By the same token, the social surplus at the ACPE inD p

the strict liability regime W S

and that in the negligence regime W N

can becharacterized. W S is identified with the consumers’ surplus at p , and W N is theconsumers’ surplus at p minus the amount of the expected harm δq , i.e., W S, and W N

D p

δq . From these representations, we canD psummarize the welfare cost formulae in the two liability regimes L S and L N asfollows:(5)

L SW PEW SD pF,(6)L NW PEW ND pF

δq .

In Figure 1, the first integrated part of the right-hand side of (5) is depicted as IAEK,
while Fpqc

δ q from (1) is depicted as IAHK. Therefore, L(S) coincides withthe triangle AEH. On the other hand, the integrated part of the right-hand side of (6) isJBEK. As Fpc qfrom (2), Fδqpc

δ q

is depicted as JBGK.

Therefore, L(N) can be depicted as the triangle, BEG. Comparing the two triangles AEH
and BEG, L(N) is definitely smaller than L(S) for the amount ABGH, which corresponds
to the net social value of additional consumption made possible by switching from thestrict liability regime to the negligence regime.
4. THE CASE WHERE q

q IS SATISFIED

Let us consider the alternative case q

q in this section. This case is illustrated inFigure 2.
4

[Figure 2 about here]
The welfare cost formulae (5) and (6) are still applicable to this case as well. Theintegrated part of the right-hand side of (5) is depicted as IAEJ in Figure 2, while
F coincides with IACJ. Therefore, L S is identified as the triangle AEC. The integrated
part in the right-hand side of (6) coincides with (minus)JEBK; F coincides with KBGLand δq equals JHGL. Therefore, L N equals the triangle EHB.
This diagrammatic illustration clarifies why the clear-cut result in the previous
section cannot be carried over to the present case. The welfare cost of the two liability
regimes L S and L N now depends on the relative distance of q and q from theoptimum consumption level q , and it is not a priori determinable. However, we havesome interesting comparative static results.
Proposition 3: As the level of the fixed cost F rises,(i)the welfare loss in the strict liability regime L S definitely increases,q.

(ii)the welfare loss in the negligence regime L N decreases iff q

Proof: (i) Differentiating L S in (5) with respect to F while considering p=p(F), we have
L S(7)
whereF1,evaluated at pp . By rewriting relation (1) as p F

and differentiating it with respect to F, again evaluating it at pF

1

is p FFFDD pF

FDD p

so that D p

1. However, from supposition (3), we have 0F

1 follows, and from (7), we have

FL S
F

L N

whereFF

1

δDD

D p

F

is p F evaluated at p

FD

D p

Fp , we haveFD

1

1,p F , we have

1,p . By rewriting relation (2) as p F

and differentiating it with respect to F, again evaluating it at p1

δ0.

(ii) Differentiating L N in (6) with respect to F while considering p(8)ccFDF

p , we have

1 . Substituting the relation into (8), we have

L NF

15

δ

D1DFDFFD

1. Since 0, which is in turn identical to q
p , which is equivalent to q

FD1q

1, we have

L N
F

0 iff δF. This inequality is equivalent to cDDδδ because of the consistency with theD cdemand condition. //The implication of the above proposition is clear enough. As for the case underdiscussion in this section, as F becomes larger, the negligence regime, socially, becomesincreasingly preferable to the strict liability regime.
In contrast, the next proposition makes a case for the adoption of the strict liabilityregime.
Proposition 4: As the level of the unit expected harm δ increases,(i)the welfare loss at the negligence regime L N increases iff qq.

(ii)the welfare loss at the strict liability regime L S decreases iffF

where θF,

D p .

and ε

Proof: (i) Considering pricing condition (2), p and hence q cannot be affected by thechange of δ. Therefore, from (6),L N

D c

δqqq , and the result follows.

(ii) Differentiating L S in (5) with respect to δ while considering p
L SD pTherefore,D c
L S

equivalent to0 iff1FDδqq , whereis p δp(δ), we haveevaluated at p. Substituting (4), evaluated at pp.

p , this is1. Rearranging terms, and substituting the definitionsof θ and ε and (1), result (ii) follows. //This shows that an increase in the expected external harm makes the choice of thenegligence scheme socially less preferable. In contrast, the effect on the desirability ofthe strict liability regime is ambiguous; however, it is suspected that in many cases, thestrict liability regime becomes more desirable socially as expected harm increases.
65. CONCLUSION
The conclusion of this article seems clear enough. The rigorous proofs of Shavell (1980)and Polinsky (1980) indicate that, the strict liability regime is socially more preferable
to the negligence regime for controlling the external harm caused by firms, if themarket is perfectly competitive. Once imperfect competition is introduced, however, theresult can be easily overturned.
In this article, we introduced imperfect competition by considering the case of anatural monopoly; however, the environment remained quasi-competitive, either as aresult of the second-best regulation of the government or the competitive pressure ofpotential entry created by the contestability of the market. Even in thisquasi-competitive case, it is expected that in most cases, the negligence regime issocially more desirable than the strict liability regime. This is due to the fact that underconditions of natural monopoly, the strict liability scheme requires firms to set very highprices in order to cover both their fixed costs and high damage payments. On the otherhand, the effect of excessive provision under the negligence scheme that is responsible
for economic inefficiency in perfectly competitive market conditions is weakened when
there is a natural monopoly. This is because in a natural monopoly firms can chargehigher prices than in perfect competition in order to cover their fixed costs.
If we were to consider further cases of a monopolistic economy, it may be expectedthat the negligence scheme would have emerged as the more preferable liability rule forsociety than the case discussed in this article. This is because, as the monopoly power ofa firm becomes stronger, its supply of products becomes smaller than the economy
analyzed in this article, so that the case of the firm discussed in section 3 is more likely
to apply, and the negligence regime becomes definitely more efficient.3 We hope that thepresent article has shed some light on the relationship between the market structure
and the socially desirable liability system to firms.

The desirability of the negligence regime with monopoly is analyzed by Polinsky andRogerson (1983) for the case of product liability.
37Reference
Baumol, W., J. Panzar, and R. Willig. 1982. Contestable Markets and the Theory ofIndustrial Structure. Harcourt Brace & Jovanovich.
Polinsky, A.M. 1980. Strict Liability vs. Negligence in a Market Setting. American

Economic Review 70:363-367.
Polinsky, A.M., and W.P. Rogerson. 1983. Product Liability, Consumer Misperceptions,and Market Power. Bell Journal of Economics 14:581-589.
Shavell, S. 1980. Strict Liability versus Negligence. Journal of Legal Studies 9:1-25.
Shavell, S. 1987. Economic Analysis of Accident Law. Harvard University Press.

8p
q=D(p)

pIApJCc+δKHpcp

BFδc

GEqq*FcOqq

Figure 19p

p=D(q)

IAppJcCEF+c+δpFHδKpcBL+cG0qsqqq

Figure 210