It has been shown by Polinsky and Shavell 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 if prefect competition among firms prevails. This article considers the problem of socially efficient liability rules in a market where natural monopoly prevails due to decreasing average cost. We especially consider a quasi-competitive 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. In contrast to the perfectly competitive economy, the present article shows that in most cases, the negligence regime is socially more desirable than the strict liability regime from the view point of economic efficiency.

Atsushi Tsuneki

English

Research Papers in Economics

Discussion Paper No. 753 
 
 
 
 
 
 
 
 
 
SOCIALLY OPTIMAL LIABILITY RULES 
FOR FIRMS WITH 
NATURAL MONOPOLY 
 
 
Atsushi Tsuneki 
 
 
 
 
 
 
 
 
 
 
 
 
August 2009 
 
 
The Institute of Social and Economic Research 
Osaka University 
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan 
 1 
 
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 presupposes 
perfectly 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 be 
concluded that strict liability rules are more appropriate for firms with natural 
monopoly, 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 not 
necessarily 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 the 
implications of this article, in comparison with those of previous contributions on the 
subject. 
 
2.  THE  MODEL 
Suppose that a monopolistic firm is selling its products to its customers. For the 
production of the good, the firm must bear a constant marginal cost c 0 . In addition, 
the firm must bear a (probably very large) fixed cost  F 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 partially 
control 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 the 
accident  δ 0   and suppose that  δ δ  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 the 
contestable market. Therefore, analogous to the case of competitive firms shown by 
Shavell (1980, 1987), this monopolistic firm chooses a socially optimal amount of care   2 
 
x x  , either under a strict liability rule or a negligence rule with proper due care 
standard. Hereafter, we denote  δ x   simply  as δ. 
Regarding the consumption side, we assume a partial equilibrium setting where 
marginal utility of income is constant for consumers. The market demand function 
q D   p  , where D  p   0 ,       p   0, exists where q is the aggregate amount of 
consumption of the product of the monopolistic firm and  p is a consumption price of 
the 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 escape 
the burden of liability payment by appropriately choosing the due care level when the 
negligence rule is adopted. This means that the average cost price adopted by the 
monopolistic firm will vary depending on kind of liability rules instated. Let us denote 
the set of market price and quantity at the average cost pricing equilibrium (ACPE 
hereafter) as  p ,q      under the strict liability regime, and  p ,q     under  the 
negligence regime respectively. Then we have the average cost pricing formulae in both 
regimes as follows: 
(1  )   p   c δ 
F
   , 
(2)   p   c 
F
   . 
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 the 
good and the expected cost of accident. The maximum social surplus is obtained at the 
marginal cost pricing equilibrium where p   c δ ,  q   D  c δ  , 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 q   0 , which means that 
this monopolistic firm provides positive welfare to the economy, at least when it is 
optimally operated. We also assume that  q   0 ,q    0 .1  We further assume that 
(3)   
 
D      
F
   
is satisfied for the range of  q between q  and q .  2  Then, the proposition below follows. 
                                                  
1 In general, it is not possible to rule out the case q   0 , or even  q   0 , even if we 
assume that q   0. However, these cases are not particularly interesting and 
especially irrelevant to the issue we discuss in this article. 
2Assumption (3) means that the slope of the market demand curve is steeper than that 
of the average cost curve at the ACPE.   3 
 
 
Proposition 1(i)  q   q  ,q    q  . 
(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    δ  
F
 . 
Proof (i): As p   c δ  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 δ    c   δ  
F
D      . 
Differentiating the equation with respect to  δ, 
(4)  p  δ  1  
F
  D  p   1  
follows. However, from assumption (3),  1  
FD    
    0  is satisfied so that the second 
bracket on the left-hand side of equation (4) is positive. Therefore, p  δ   0  is implied 
for  δ that corresponds to a value of q between q  and  q . This means that, starting 
from  δ 0 , 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,  q   q   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 meet 
this condition. To have q   q    D   c   δ , so that the ACPE in a negligence regime is 
Pareto efficient, p   c 
F
    c δ must be satisfied. From these two equations, 
D c    δ  
F
   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 the 
analysis and comparison of the social welfare obtainable at the ACPE in the two 
alternative 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 q   q  . The former 
case 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. 
 4 
 
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 c    δ  q    q    q   is 
satisfied. Under this condition, the marginal benefit of consumption always exceeds 
c 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 the 
negligence 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   is 
characterized by the consumers’ surplus at the marginal cost price p   c δ ,  minus 
fixed cost F, i.e.,   D p      F
 
    . By the same token, the social surplus at the ACPE in 
the strict liability regime W S  and that in the negligence regime W N  can be 
characterized.  W S  is identified with the consumers’ surplus at p , and W N  is the 
consumers’ surplus at p  minus the amount of the expected harm δq , i.e., W S   
  D p   
 
   , and W N      D p     
 
   δq  . From these representations, we can 
summarize the welfare cost formulae in the two liability regimes L S  and L N  as 
follows: 
(5)  L S   W  PE   W  S       D p      F
  
    , 
(6)  L N   W  PE   W  N      D p      F
  
     δ q  . 
In Figure 1, the first integrated part of the right-hand side of (5) is depicted as IAEK, 
while  F p  q     c δ   q   from (1) is depicted as IAHK. Therefore, L(S) coincides with 
the triangle AEH. On the other hand, the integrated part of the right-hand side of (6) is 
JBEK. As F   p    c   q   from (2), F δ q      p      c   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 the 
strict 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 in 
Figure 2. 5 
 
[Figure 2 about here] 
The welfare cost formulae (5) and (6) are still applicable to this case as well. The 
integrated part of the right-hand side of (5) i s depicted as IAEJ in Figure 2, w hile 
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 KBGL 
and  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 the 
optimum consumption level q , and it is not a priori determinable. However, we have 
some 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, 
(ii)the welfare loss in the negligence regime  L N  decreases  iff q   q  . 
 
Proof: (i) Differentiating  L S   in (5) with respect to F while considering p=p(F), we have 
(7)    
 L S 
 F  D  p  
   
 F  1 ,  
where 
   
 F  is p  F  evaluated at p p  . By rewriting relation (1) as  p F    c   δ  
F
D   F   and differentiating it with respect to F, again evaluating it at p p  , we have 
 1 
FD     
       D p  
   
 F  1 . However, from supposition (3), we have 0  1 
FD     
         1 ,  
so that  D p  
   
 F  1   follows, and from (7), we have 
 L S 
 F  0 .  
(ii) Differentiating L N  in (6) with respect to F while considering p p  F , we have    
(8)   
 L N 
 F    1 δ
D     
D     D p  
   
 F  1 ,  
where 
   
 F  is p  F  evaluated at p p  . By rewriting relation (2) as  p F   c 
F
D   F   
and differentiating it with respect to F, again evaluating it at p p  , we have 
 1 
FD     
       D p  
   
 F  1. Substituting the relation into (8), we have 
 L N 
 F    1 6 
 
δ
D     
D      1 
FD     
       
  
 1. Since 0  1 
FD     
         1 , we have 
 L N 
 F  0  iff δ
D     
D      
FD     
      , which is in turn identical to  q   
F
 . This inequality is equivalent to  c δ   
F
    p  , which is equivalent to q   q    D  c δ   because of the consistency with the  
demand condition. // 
 
The implication of the above proposition is clear enough. As for the case under 
discussion in this section, as  F  becomes larger, the negligence regime, socially, becomes 
increasingly preferable to the strict liability regime. 
In contrast, the next proposition makes a case for the adoption of the strict liability 
regime. 
 
Proposition 4: As the level of the unit expected harm  δ increases, 
(i)the welfare loss at the negligence regime  L N  increases  iff q   q  . 
(ii)the welfare loss at the strict liability regime L S  decreases iff     
     
   , 
where  θ 
F
        F and ε  
  
   D  p  . 
Proof: (i) Considering pricing condition (2), p  and hence q  cannot be affected by the 
change of  δ. Therefore, from (6), 
 L N 
      D  c δ    q    q    q  ,  and the result follows. 
(ii) Differentiating L S  in (5) with respect to δ while considering p p (δ), we have 
 
 L S 
    D  p  
   
    D  c δ    q      
    q  ,  where 
   
    is  p  δ   evaluated  at  p p  . 
Therefore, 
 L S 
    0 iff 
  
    
   
  .   Substituting (4), evaluated at p p   , this is 
equivalent to 
  
   1 
FD     
         1 . Rearranging terms, and substituting the definitions 
of  θ and ε  and (1), result (ii) follows. // 
 
This shows that an increase in the expected external harm makes the choice of the 
negligence scheme socially less preferable. In contrast, the effect on the desirability of 
the strict liability regime is ambiguous; however, it is suspected that in many cases, the 
strict liability regime becomes more desirable socially as expected harm increases. 7 
 
 
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 the 
market is perfectly competitive. Once imperfect competition is introduced, however, the 
result can be easily overturned. 
In this article, we introduced imperfect competition by considering the case of a 
natural monopoly; however, the environment remained quasi-competitive, either as a 
result of the second-best regulation of the government or the competitive pressure of 
potential entry created by the contestability of the market. Even in this 
quasi-competitive case, it is expected that in most cases, the negligence regime is 
socially more desirable than the strict liability regime. This is due to the fact that under 
conditions of natural monopoly, the strict liability scheme requires firms to set very high 
prices in order to cover both their fixed costs and high damage payments. On the other 
hand, 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 charge 
higher 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 expected 
that the negligence scheme would have emerged as the more preferable liability rule for 
society than the case discussed in this article. This is because, as the monopoly power of 
a 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 the 
present article has shed some light on the relationship between the market structure 
and the socially desirable liability system to firms. 
  
                                                  
3  The desirability of the negligence regime with monopoly is analyzed by Polinsky and 
Rogerson (1983) for the case of product liability.   8 
 
Reference 
Baumol, W., J. Panzar, and R. Willig. 1982. Contestable Markets and the Theory of 
Industrial 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. 
  9 
 
p 
 q=D(p) 
 
 
      
 
 
 
 
 
 
 
 
 
 
p  I  A  p  c δ 
F
  
 
 
  p  J  C  B  p  c 
F
  
  
 
 
c+δK  H  G  E 
 
 
c 
 
O    q   q   q*  q 
 
Figure 1 
  10 
 
p p=D(q) 
 
       
 
 
 
 
 
 
 
 
 
 
 
 I  A 
  p    
p 
F
 +c+δ 
 J  C  E H 
c δ  
 
  p      K                                                               B    
p 
F
 +c 
 c  L  G 
 
   0                                    q
s   q        q         q 
 
 
 
Figure 2 
 
 

Socially Optimal Liability Rules for Firms with Natural Monopoly

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