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Liability rules and the selection of a socially optimal production technology
international Review of Law and Economics
(1987), 7(121-126)
NOTES LIABILITY RULES AND THE SELECTION J3)F A SOCIALLY OPTIMAL PRODUCTION TECHNOLOGY SIGMUND Texas southern
A. HORVITZ
University, Houston,
TX 71004, USA
AND
LOUIS H. STERN*
I. INTRODUCTION Many types of activities can lead to harmful effects upon second parties. The law has attempted to control these activities through the use of liability rules, i.e., rules which determine the conditions under which one party must pay for damages sustained by another party. In the case where the victim cannot avoid damages resulting from the injurer’s conduct-the unitateral case-the courts have applied two types of rules: strict liability and negligence. The injurer escapes liability under the negligence rule only if he was reasonably careful to avoid injury to others, while under the strict liability rule the injurer is absolutely liable. We confine our study to an analysis of liability rules within the unilateral context in which the victim who cannot protect himself from injury and is a “stranger” to the injurer is injured through the operation of the injurer’s production process.’ During the past few years economists have become interested in the question of the effect of imposing one or the other of these rules, or variants of each, upon the allocation of resources to potentially dangerous activities. Brown was the first to apply formal analysis to the problem.* He divides the injurer’s activity into two parts: a primary activity which implicitly assumes a fixed or constant production technology” used to produce a fixed quantity of output; and a secondary activity which involves the use of resources to avoid damages to others. He shows that in the unilateral case the negligence rule and the strict liability rule each leads the injurer to the same response and that each is socially efficient. Shave11 relaxes the assumption of fixed output and shows that in the unilateral case the strict liability rule but not the negligence rule is socially efficient. In this paper we restore the assumption of fixed output but relax the assumption of fixed *The authors are grateful to John G. Riley, Steven Shave& Ralph B. Turner, and an anonymous referee of this Journal for their helpful suggestions on earlier drafts of this paper. We are solely responsible for all remaining errors. 0144~8818/87/010121-06
$03.00
Q 1987 Butterworth
Publishers
122
Liabiiiry rules and production technology
production technology. We conclude that in the unilateral case if reasonable care is defined only in terms of the operation of a given production process the negligence rule (we refer to this version of the negligence rule as Negligence Rule I) and the strict liability rule lead, in the case of variable production technology, to different responses and that the strict liability rule but not the negligence rule is socially efficient. Posner correctly identifies the difference in the operation of the strict liability rule and Negligence Rule I. The strict liability rule, but not Negligence Rule I, compels the injurer to consider the residual damage which a production process generates despite its operation in an optimally careful manner.4 We also show that if reasonable care is defined in terms of the selection as well as the operation of a given production process the negligence rule and the strict liabiiity rule each leads within our analyticai framework to the same response and are socially optimal in the case of variable production technology. We refer to this version of the negligence rule as Negligence Rule II. In Section II we present Brown’s results. In Section III we modify the model5 and investigate the social efficiency of the conventional negligence rule and the strict liability rule within the context of variable production technology. In Section IV we consider the social efficiency of an alternative to the conventional negligence rule. Section V is a conclusion.
II. THE CASE OF CONSTANT
PRODUCTION
TECHNOLOGY
The injurer, a perfectly competitive producer of good Xt is constrained by constant production technology. This producer of good X is identical to every other producer of the good. The safety with which this good is produced is a function only of K, the care with which the injurer operates the production process. The net social gain obtained from the production of the good is given by
vs(K} = I O’p(Q)dQ - cz - C(K)Q - D(K)Q where Q is the fixed quantity of good X traded per unit of time; p(Q) is the demand function governing the good; C(K) is the carecost incurred per unit of &; D(K) is the residual damage generated per unit of Q; ca is the total cost of B producing D and is independent of K; o p(Q)dQ is the social value of and _iconsumption of the good. C(K)Q + D(K)g is the social cost of safety and C and D are deterministic functions of K with C’(K) > 0, c’(K) > 0 and D’(K) < 0, and D”(K) > 0. Net social gain is a maximum and the social cost of safety is a minimum at K” where dC(K)/d#
= - dD(K)/dK
d*C(K)/dK2 > - d’D(K)ldlYZ. The first condition states that marginal social cost and marginal social gain are just equal at K*, (marginal social costis the increase in C(K>Q while marginal social gain is the decrease in D(K)Q.) We denote minimum social cost by C*(K)Q + D*(K*)Q and maximum net social gain by n:. Negligence Rule I requires the injurer to pay the victim’s losses only if the injurer fails to exercise reasonable care in operating the given production process.
S. A. HORVITZAND L. H. STERN
The strict liability rule requires the regard to the injurer’s conduct. A single injurer’s net gain function
injurer under
123
to pay the victim’s the strict liability
losses
without
rule is
where B is the price, given to the injurer, of good X, 4 is the injurer’s fixed output per unit of time; and c; is the injurer’s total cost of producing q and is independent of K. As long as the injurer employs reasonable (K*) or greater care, his net gain is measured under the negligence rule by npv(K) = 7jq - c; - C(K)?& otherwise,
the injurer’s
net gain is measured
by
The strict liability rule and Negligence Rule I each lead the injurer to K” (the socially optimal K) where net private gain (under these rules) is maximized and the private cost of safety is minimized. We refer to maximum net private gain obtained under the strict liability rule as T;,. The minimum private cost of safety is referred to by C*(K*)q + D*(K*)q. III. THE
CASE
OF VARIABLE
PRODUCTION
TECHNOLOGY
The production of a good or service can typically be accomplished through the employment of any one of several alternative production processes. This is the case of variable production technology. The removal of an obstruction in the way of a proposed road, for example, can be accomplished, for example, with a single blast, multiple blasts, chemical means, mechanical means, manual means or with certain combinations of these. We drop the assumption of constant production technology and assume that multiple production processes are available to the injurer for the production of good X. When we substitute the assumption of variable production technology for the assumption of constant production technology, the equivalence of the strict liability rule and Negligence Rule I disappears. Let Ti be the ith production technology for producing a given quantity of good X. This production process is operated subject to either the strict liability rule or the negligence rule with K = K*. T, belongs to a continuous set T of these production processes. We assume that T can be ordered according to increasing labor-capital ratios. Net social gain generated with the ith production process in the set T is
-
C*(T,)D
- D*(T,@
where p(Q) is the demand function governing good X and-c&T,) is the total cost, excluding care costs and residual damage, of producing Q units of good X with the ith production technology. Production technology T, has the smallest labor-capital ratio and is assumed to have the smallest total cost of production. The cost of production incurred
124
Liability rules and production technology
with T, is c&T,). The expression c&T,) - c,(T,) is the care cost incurred through the selection of the ith production technologyrather than T,. The expression [ca(TJ - ca(T,)] + C*(T,)Q is the care cost incurred through the selection and the operation of the ith production technology. (Recall that C*(T,)D is the care cost incurred through the operation of the ith production technology so as to yield maximum net social benefit.) The single injurer’s net gain obtained with the ith production process under the strict liability standard is 4r(TJ
= p4
-
c,(T,)
-
[c,(T,)
-
c,(T,)l
-
C*(T&
-
D*(T&
where c,(T,) is the total cost, exclusive of care costs and residual damage, of producing q units of good X with the ith production technology. The care cost incurred by the injurer through the selection and the operation of the ith production technology is [c;(T,) - c;(T,)] + C*(T&. If we simplify the above expression for n&( TJ we have $,(T,)
= j3q -
c;(T,)
-
C*(T,)q
In Figure lb we derive a n&(T) function C*(T)q and D*(T)q functions in Figure la. In Figure la we assume: 1. dc,(T)ldT
> 0, d2c,(T)/dT2
2. dC*(T)@dT 3. dc,(T)dT 4. dD*(T)qidT
D*(T,)q.
from
a set
of pq -
c;(T)
-
> 0
< 0, d2C*(T’)qldT2 > -dC*(T)qldT,
-
> 0
d2c,(T)ldT2
< 0, d2D*(7+)qldT2
> -d2C*(T)@dT2
> 0.
The first set of equations states that the cost of production increases as production processes with large labor-capital ratios are substituted for production processes with small labor-capital ratios. The second set of equations implies that the difficulty of controlling damages decreases as production processes with large labor-capital ratios are substituted for production processes with small laborcapital ratios. The third set of equations together with the first two sets of equations insures that the sum of the c;( T,) and the C*(T)if functions is a monotonically increasing function of T. The fourth set of equations implies that production
L
TI
T> TX
T
Tl
(la)
Tz (lb)
FIG.1
T3
T
125
S. A. HORVITZ AND L. H. STERN
processes with large labor-capital ratios are inherently less dangerous than production processes with small labor-capital ratios. The injurer obtains the maximum value of r&(7’) in the figure at T2 where d$,( d*$,(
T)ldT
= 0
T)/dT* < 0
The first condition implies that at T, marginal private cost and marginal private gain are just equal. (Marginal private cost is the decrease in fiq - c;(T) C*(T)q. The decrease in D*(T)if represents marginal private gain.) Since we have assumed that each of the firms producing good X is a representative firm, q:(T) as well as n;,_(T) is maximized at T,. Since the strict liability rule moves the injurer to T2 in Figure 1, the strict liability rule is socially efficient in the case of variable production technology. Since the injurer escapes liability for D*(T& under Negligence Rule I, net private gain under this rule is measured by BZJ - c;( T,) - C*( T,)q at the T, selected. Therefore, the injurer maximizes net private gain under this rule when it employs the T, for which this sum is a maximum. Hence, under Negligence Rule I, he prefers T, to T, in the figure. Although, privately, T, is more efficient than T2, T, is not socially efficient in the case of variable production technology. IV. REDEFINING
REASONABLE
CARE
The injurer behaves differently under the two rules in the example because Negligence Rule I does not force him to consider the residual damage which the operation of Tl generates since it is being carried out in an optimally careful manner. However, the courts can force the injurer under a negligence rule to take D*(T,)q into account by defining T, and not K* as reasonable care. (Then, in the figure T, < T2 constitutes negligence.) We refer to the negligence rule based upon the definition of reasonable care in terms of the selection, as well as the operation, of a given production process as Negligence Rule II. Negligence Rule II forces the injurer in Figure 1 to T2 where net private gain under this rule is maximized and is equal to Bq - c;(TJ - C*(T,)q. At T, net private gain under this rule is equal to pq - c;(T,) - C*(T,)q - D*(T,)q while at T3 net private gain under this rule is equal to pq - c;(T,) - C*(T,)q. Since n;(T) is a maximum at T,, Negligence Rule II, like the strict liability rule, insures that the privately efficient solution in the case of variable production technology coincides with the socially efficient solution. V. CONCLUSIONS We investigate the social efficiency of certain liability rules under the assumption that the injurer is not constrained by fixed production technology. This is done within a framework which is characterized by a number of restrictive assumptions. We assume within a partial equilibrium framework that the injurer is a perfectly competitive producer of a fixed quantity of output and that only he can avoid damage to the victim who is Shavell’s “stranger” to the injurer. We also assume that we can order a set of production processes with respect to the costs of, and the benefits from, the exercise of care. Finally, we assume that these costs and benefits are known with certainty. We show that the strict liability rule but not the conventional rule of negligence is socially optimal when the injurer can choose among competing production
126
Liability rules and production technology
technologies with which to produce output. We also show, however, that under this assumption of variable production technology, if negligence is redefined in terms of the selection as well as the operation of a given production process, the negligence rule as well as the strict liability rule is socially optimal. REFERENCES 1. We are considering 2. 3.
4. 5.
AND
NOTES
Shavell’s market case of “strangers”. S. Shavell, “Strict Liability Versus Negligence”, (1980) 71 J. Legal Stud. 1. J. P. Brown, “Toward an Economic Theory of Liability”, (1973), 2 J. Legal Stud. 232. In this paper we rely upon the distinction made by Frisch between constant and variable production techniques. The former term is used here to mean that there is only one technology which is available for producing a good or service, while the latter term is used to mean that there is a set of alternative technologies which are available for producing the good or service. R. Frisch, Theory of Production, Rand McNally (1965). R. A. Posner, “Theory of Negligence”, (1972) 1 J. Legal Stud. 29. The model of Section II is a deterministic version of a variation of the Hamada Model. K. Hamada, “Liability Rules and Income Distribution in Product Liability”, (1976) 66 Am. Econ. Rev. 228.
(1987), 7(121-126)
NOTES LIABILITY RULES AND THE SELECTION J3)F A SOCIALLY OPTIMAL PRODUCTION TECHNOLOGY SIGMUND Texas southern
A. HORVITZ
University, Houston,
TX 71004, USA
AND
LOUIS H. STERN*
I. INTRODUCTION Many types of activities can lead to harmful effects upon second parties. The law has attempted to control these activities through the use of liability rules, i.e., rules which determine the conditions under which one party must pay for damages sustained by another party. In the case where the victim cannot avoid damages resulting from the injurer’s conduct-the unitateral case-the courts have applied two types of rules: strict liability and negligence. The injurer escapes liability under the negligence rule only if he was reasonably careful to avoid injury to others, while under the strict liability rule the injurer is absolutely liable. We confine our study to an analysis of liability rules within the unilateral context in which the victim who cannot protect himself from injury and is a “stranger” to the injurer is injured through the operation of the injurer’s production process.’ During the past few years economists have become interested in the question of the effect of imposing one or the other of these rules, or variants of each, upon the allocation of resources to potentially dangerous activities. Brown was the first to apply formal analysis to the problem.* He divides the injurer’s activity into two parts: a primary activity which implicitly assumes a fixed or constant production technology” used to produce a fixed quantity of output; and a secondary activity which involves the use of resources to avoid damages to others. He shows that in the unilateral case the negligence rule and the strict liability rule each leads the injurer to the same response and that each is socially efficient. Shave11 relaxes the assumption of fixed output and shows that in the unilateral case the strict liability rule but not the negligence rule is socially efficient. In this paper we restore the assumption of fixed output but relax the assumption of fixed *The authors are grateful to John G. Riley, Steven Shave& Ralph B. Turner, and an anonymous referee of this Journal for their helpful suggestions on earlier drafts of this paper. We are solely responsible for all remaining errors. 0144~8818/87/010121-06
$03.00
Q 1987 Butterworth
Publishers
122
Liabiiiry rules and production technology
production technology. We conclude that in the unilateral case if reasonable care is defined only in terms of the operation of a given production process the negligence rule (we refer to this version of the negligence rule as Negligence Rule I) and the strict liability rule lead, in the case of variable production technology, to different responses and that the strict liability rule but not the negligence rule is socially efficient. Posner correctly identifies the difference in the operation of the strict liability rule and Negligence Rule I. The strict liability rule, but not Negligence Rule I, compels the injurer to consider the residual damage which a production process generates despite its operation in an optimally careful manner.4 We also show that if reasonable care is defined in terms of the selection as well as the operation of a given production process the negligence rule and the strict liabiiity rule each leads within our analyticai framework to the same response and are socially optimal in the case of variable production technology. We refer to this version of the negligence rule as Negligence Rule II. In Section II we present Brown’s results. In Section III we modify the model5 and investigate the social efficiency of the conventional negligence rule and the strict liability rule within the context of variable production technology. In Section IV we consider the social efficiency of an alternative to the conventional negligence rule. Section V is a conclusion.
II. THE CASE OF CONSTANT
PRODUCTION
TECHNOLOGY
The injurer, a perfectly competitive producer of good Xt is constrained by constant production technology. This producer of good X is identical to every other producer of the good. The safety with which this good is produced is a function only of K, the care with which the injurer operates the production process. The net social gain obtained from the production of the good is given by
vs(K} = I O’p(Q)dQ - cz - C(K)Q - D(K)Q where Q is the fixed quantity of good X traded per unit of time; p(Q) is the demand function governing the good; C(K) is the carecost incurred per unit of &; D(K) is the residual damage generated per unit of Q; ca is the total cost of B producing D and is independent of K; o p(Q)dQ is the social value of and _iconsumption of the good. C(K)Q + D(K)g is the social cost of safety and C and D are deterministic functions of K with C’(K) > 0, c’(K) > 0 and D’(K) < 0, and D”(K) > 0. Net social gain is a maximum and the social cost of safety is a minimum at K” where dC(K)/d#
= - dD(K)/dK
d*C(K)/dK2 > - d’D(K)ldlYZ. The first condition states that marginal social cost and marginal social gain are just equal at K*, (marginal social costis the increase in C(K>Q while marginal social gain is the decrease in D(K)Q.) We denote minimum social cost by C*(K)Q + D*(K*)Q and maximum net social gain by n:. Negligence Rule I requires the injurer to pay the victim’s losses only if the injurer fails to exercise reasonable care in operating the given production process.
S. A. HORVITZAND L. H. STERN
The strict liability rule requires the regard to the injurer’s conduct. A single injurer’s net gain function
injurer under
123
to pay the victim’s the strict liability
losses
without
rule is
where B is the price, given to the injurer, of good X, 4 is the injurer’s fixed output per unit of time; and c; is the injurer’s total cost of producing q and is independent of K. As long as the injurer employs reasonable (K*) or greater care, his net gain is measured under the negligence rule by npv(K) = 7jq - c; - C(K)?& otherwise,
the injurer’s
net gain is measured
by
The strict liability rule and Negligence Rule I each lead the injurer to K” (the socially optimal K) where net private gain (under these rules) is maximized and the private cost of safety is minimized. We refer to maximum net private gain obtained under the strict liability rule as T;,. The minimum private cost of safety is referred to by C*(K*)q + D*(K*)q. III. THE
CASE
OF VARIABLE
PRODUCTION
TECHNOLOGY
The production of a good or service can typically be accomplished through the employment of any one of several alternative production processes. This is the case of variable production technology. The removal of an obstruction in the way of a proposed road, for example, can be accomplished, for example, with a single blast, multiple blasts, chemical means, mechanical means, manual means or with certain combinations of these. We drop the assumption of constant production technology and assume that multiple production processes are available to the injurer for the production of good X. When we substitute the assumption of variable production technology for the assumption of constant production technology, the equivalence of the strict liability rule and Negligence Rule I disappears. Let Ti be the ith production technology for producing a given quantity of good X. This production process is operated subject to either the strict liability rule or the negligence rule with K = K*. T, belongs to a continuous set T of these production processes. We assume that T can be ordered according to increasing labor-capital ratios. Net social gain generated with the ith production process in the set T is
-
C*(T,)D
- D*(T,@
where p(Q) is the demand function governing good X and-c&T,) is the total cost, excluding care costs and residual damage, of producing Q units of good X with the ith production technology. Production technology T, has the smallest labor-capital ratio and is assumed to have the smallest total cost of production. The cost of production incurred
124
Liability rules and production technology
with T, is c&T,). The expression c&T,) - c,(T,) is the care cost incurred through the selection of the ith production technologyrather than T,. The expression [ca(TJ - ca(T,)] + C*(T,)Q is the care cost incurred through the selection and the operation of the ith production technology. (Recall that C*(T,)D is the care cost incurred through the operation of the ith production technology so as to yield maximum net social benefit.) The single injurer’s net gain obtained with the ith production process under the strict liability standard is 4r(TJ
= p4
-
c,(T,)
-
[c,(T,)
-
c,(T,)l
-
C*(T&
-
D*(T&
where c,(T,) is the total cost, exclusive of care costs and residual damage, of producing q units of good X with the ith production technology. The care cost incurred by the injurer through the selection and the operation of the ith production technology is [c;(T,) - c;(T,)] + C*(T&. If we simplify the above expression for n&( TJ we have $,(T,)
= j3q -
c;(T,)
-
C*(T,)q
In Figure lb we derive a n&(T) function C*(T)q and D*(T)q functions in Figure la. In Figure la we assume: 1. dc,(T)ldT
> 0, d2c,(T)/dT2
2. dC*(T)@dT 3. dc,(T)dT 4. dD*(T)qidT
D*(T,)q.
from
a set
of pq -
c;(T)
-
> 0
< 0, d2C*(T’)qldT2 > -dC*(T)qldT,
-
> 0
d2c,(T)ldT2
< 0, d2D*(7+)qldT2
> -d2C*(T)@dT2
> 0.
The first set of equations states that the cost of production increases as production processes with large labor-capital ratios are substituted for production processes with small labor-capital ratios. The second set of equations implies that the difficulty of controlling damages decreases as production processes with large labor-capital ratios are substituted for production processes with small laborcapital ratios. The third set of equations together with the first two sets of equations insures that the sum of the c;( T,) and the C*(T)if functions is a monotonically increasing function of T. The fourth set of equations implies that production
L
TI
T> TX
T
Tl
(la)
Tz (lb)
FIG.1
T3
T
125
S. A. HORVITZ AND L. H. STERN
processes with large labor-capital ratios are inherently less dangerous than production processes with small labor-capital ratios. The injurer obtains the maximum value of r&(7’) in the figure at T2 where d$,( d*$,(
T)ldT
= 0
T)/dT* < 0
The first condition implies that at T, marginal private cost and marginal private gain are just equal. (Marginal private cost is the decrease in fiq - c;(T) C*(T)q. The decrease in D*(T)if represents marginal private gain.) Since we have assumed that each of the firms producing good X is a representative firm, q:(T) as well as n;,_(T) is maximized at T,. Since the strict liability rule moves the injurer to T2 in Figure 1, the strict liability rule is socially efficient in the case of variable production technology. Since the injurer escapes liability for D*(T& under Negligence Rule I, net private gain under this rule is measured by BZJ - c;( T,) - C*( T,)q at the T, selected. Therefore, the injurer maximizes net private gain under this rule when it employs the T, for which this sum is a maximum. Hence, under Negligence Rule I, he prefers T, to T, in the figure. Although, privately, T, is more efficient than T2, T, is not socially efficient in the case of variable production technology. IV. REDEFINING
REASONABLE
CARE
The injurer behaves differently under the two rules in the example because Negligence Rule I does not force him to consider the residual damage which the operation of Tl generates since it is being carried out in an optimally careful manner. However, the courts can force the injurer under a negligence rule to take D*(T,)q into account by defining T, and not K* as reasonable care. (Then, in the figure T, < T2 constitutes negligence.) We refer to the negligence rule based upon the definition of reasonable care in terms of the selection, as well as the operation, of a given production process as Negligence Rule II. Negligence Rule II forces the injurer in Figure 1 to T2 where net private gain under this rule is maximized and is equal to Bq - c;(TJ - C*(T,)q. At T, net private gain under this rule is equal to pq - c;(T,) - C*(T,)q - D*(T,)q while at T3 net private gain under this rule is equal to pq - c;(T,) - C*(T,)q. Since n;(T) is a maximum at T,, Negligence Rule II, like the strict liability rule, insures that the privately efficient solution in the case of variable production technology coincides with the socially efficient solution. V. CONCLUSIONS We investigate the social efficiency of certain liability rules under the assumption that the injurer is not constrained by fixed production technology. This is done within a framework which is characterized by a number of restrictive assumptions. We assume within a partial equilibrium framework that the injurer is a perfectly competitive producer of a fixed quantity of output and that only he can avoid damage to the victim who is Shavell’s “stranger” to the injurer. We also assume that we can order a set of production processes with respect to the costs of, and the benefits from, the exercise of care. Finally, we assume that these costs and benefits are known with certainty. We show that the strict liability rule but not the conventional rule of negligence is socially optimal when the injurer can choose among competing production
126
Liability rules and production technology
technologies with which to produce output. We also show, however, that under this assumption of variable production technology, if negligence is redefined in terms of the selection as well as the operation of a given production process, the negligence rule as well as the strict liability rule is socially optimal. REFERENCES 1. We are considering 2. 3.
4. 5.
AND
NOTES
Shavell’s market case of “strangers”. S. Shavell, “Strict Liability Versus Negligence”, (1980) 71 J. Legal Stud. 1. J. P. Brown, “Toward an Economic Theory of Liability”, (1973), 2 J. Legal Stud. 232. In this paper we rely upon the distinction made by Frisch between constant and variable production techniques. The former term is used here to mean that there is only one technology which is available for producing a good or service, while the latter term is used to mean that there is a set of alternative technologies which are available for producing the good or service. R. Frisch, Theory of Production, Rand McNally (1965). R. A. Posner, “Theory of Negligence”, (1972) 1 J. Legal Stud. 29. The model of Section II is a deterministic version of a variation of the Hamada Model. K. Hamada, “Liability Rules and Income Distribution in Product Liability”, (1976) 66 Am. Econ. Rev. 228.