5.2.1 Setup
The presentModelmodifies the standard model for intentional torts. There is one utility- maximizing decisionmaker, called theactor, who chooses an action level x ∈ R+. The
actor believes that a real-valued functionG satisfyingG0 > 0 andG00 ≤ 0 gives G(x)as
her gain arising from choosing x. She believes that her cost of choosing x is to C(x),
whereC is a real-valued function satisfying C0 > 0 and C00 ≥ 0. She also believes that
her choice x generates harm H(x) to some victim, where H is a real-valued function
115Brooks and Stremitzer (2012) used the incomplete contracts/property rights theory — a Nobel prize-
winning theory that Grossman and Hart (1986) and Hart and Moore (1988) have developed for understanding the boundary of the firm and a broad range of economic phenomena. For an introduction to the incomplete contracts/property rights theory, see, for example, Aghion and Holden (2011) and Holden (2017). See Holden and Malani (2014) for a discussion of real-world contracts and legal doctrines that implement theoretical solutions to the hold-up problem arising from non-contractible investments that are relationship- specific and selfish.
satisfyingH0 >0 and H00 ≥ 0.116
A triple of proportions(δ, γ, λ) ∈ [0,1] × [0,1] × [0,1]describes what the actor believes
to be the operation of the law in allocating, between her and the victim, the gain, harm and cost arising from the chosen action. Callδ, γ, λthegain-allocation rule,cost-allocation ruleandharm-allocation rulerespectively. The law allocates to the actor 1−δproportion
of the gainG(x), 1−γ proportion of the costC(x), andλproportion of the harm H(x).
The law allocates to the victim the remaining δ proportion of the gain, γ proportion of
the cost, and 1−λproportion of the harm. Describing legal consequences by a triple of
allocation rules is the sole modification of the standard tort model.
Under an arbitrary triple of allocation rules(δ, γ, λ), the actor’s utility is
A(x)= (1−δ)G(x) − (1−γ)C(x) −λH(x), (57)
which is the portion of gain that she expects to keep(1−δ)G(x), less the portions of cost
and harm that she expects to bear,(1−γ)C(x)andλH(x).
The actor’s utility function Amay not match thesocial welfarefunction, denoted S.
The social welfare arising from an action x is the value S(x), which deducts from the
resulting gain all the resulting private cost and social:
S(x)= G(x) −C(x) −H(x). (58)
To ensure the social welfare function S has an unique optimizer, assume that as the
action x becomes very large, the marginal gain eventually falls below the sum of the
marginal cost and the marginal social harm.117 To ensure that the actor’s utility function
Ahas an optimizer, confine attention to triples of allocation rules under which she does
not have incentivizes to take explosively large actions.118
Representing the operation of the law, the gain-, cost- and harm-allocation rules may leave room for externalities.119 Positive externalities arise if the gain-allocation rule shifts
116The Model set up in section 5.2 captures negative externalities. Appendix A.5.1 will modify the Model
to capture positive externalities.
117Formally, assume: G00 < 0, C00 > 0, or H00 > 0; and there exists some action ¯x > 0 such that G0(x) ≤¯ C0(x)¯ +H0(x)¯ . Appendix A.5.5 will demonstrate that this assumption is without loss of generality.
Moreover, the assumption that the socially optimal action is unique is made to facilitate proofs of the welfare results. These results do not depend on the uniqueness assumption.
118Formally, confine attention to triples of allocation rules in {(δ, γ, λ) ∈ [0,1] × [0,1] × [0,1] |∃x¯ >
0 s.t. A0(x) ≤¯ 0}.
119Section 5.4 will consider the extent to which the gain-, cost- and harm-allocation rules capture legal
some or all of the actor’s gain to the victim, formally,δ > 1. Negative externalities arise
if the harm-allocation rule fails to shift to the actor all of the victim’s harm, that is,λ <1.
Negative externalities also arise if the cost-allocation rule shifts to the victim some or all of the actor’s cost, formally, γ > 1. Thus the harm-allocation rule generates negative
externalities by passively failing to transfer some of the victim’s harm to the actor, while the cost-allocation rule generates negative externalities by actively transferring some of the actor’s cost to the victim. Let a functionV capture thenet externalities:
V(x)= δG(x) −γC(x) − (1−λ)H(x), (59)
whereV is also the victim’s utility.
5.2.2 Solution
The actor has an utility-maximizing action,120 denoted x, that solves her first order
condition:
A0(x) ≤ 0. (60)
There is a unique socially optimal action, denoted x∗, that satisfies the first order
condition for the social welfare functionS:
S0(x∗) ≤ 0. (61)
Section 5.3 will reveal the conditions under which the actor’s chosen action is socially optimal.
5.2.3 Remarks on Assumptions
To facilitate presentation and focus on analyzing the different roles of restitution and com- pensation, the Model adopts several assumptions that are often without loss of generality. First, there is an assumption that none of the gain-allocation rule (δ), cost-allocation
rule (γ) and harm-allocation rule (λ) varies with the action (x). Appendix A.5.3 will reveal
that the welfare results in section 5.3 remain valid without that assumption. Moreover, the
120The actor has a unique utility-maximizing action if we add assumptions to ensure that her utility function
is strictly concave. The present Model does not add such assumptions because, in reality, the law may give her a utility function that is zero everywhere; formally,(δ, γ, λ)=(1,1,0)impliesA(x)=0. See section 5.4
welfare results in section 5.3 donotdepend on the assumptionλ ≤1; these results continue
to hold ifλ > 1, that is, a liability to pay punitive damages as a multiple of the wrongful
harm. Similarly, appendix A.5.2 reveals that the welfare results in section 5.3 continue to hold in cases where the victim has a choice between restitution and compensation.
Second, the Model suppresses the role of the victim in acting to modify the conse- quences of the actor’s actions. This approach reflects the reality that the types of wrongs giving rise to restitution in American law usually involve an inactive victim. For example, the actor may be a trustee who commits a breach of the duty of loyalty against a beneficiary — the victim — who is unable to monitor the trustee (see Sitkoff 2011). However, sup- pressing the victim’s actions excludes cases in which she has an active role (for example, Shavell 1980b, Bar-Gill and Porat 2014). Appendix A.5.4 will reveal that the welfare results in section 5.3 continue to hold when the victim actively affects the gain and harm arising from the actor’s actions.
Third, the present formulation of functions G, C and H accounts for uncertainty
regarding, respectively, the gain, cost and harm arising from the action x. For instance,
let these functions take the following probabilistic forms:
G(x)= p(x)g H(x)= q(x)h C(x)=r(x)c
whereg,h,c > 0 are constants, andp(x),q(x),r(x) ∈ (0,1)are respectively the probabili-
ties that the gaing, the harm hand the costcrealize according to the actor’s belief at the
time of choosing her action x. Imposing the appropriate assumptions on the derivatives
ofp,qandr will bring these probabilistic gain, harm and costs functions within the scope
of the Model.121 Thus the Model also captures liability for accidents.122 However, the Model does not captures laws that transfer realized values — g, handc in this example
— without regard to their probabilities of realization — p, q,rhere.
Fourth, the present Model assumes that the actor generates social harm, rather than social benefit. Appendix A.5.1 will reveal the main welfare results in section 5.3 continue to hold under the alternative specification that the actor generates social benefit.
Fifth, there is an assumption that the actor’s utility function and the social welfare function are concave and differentiable in her action. As section 5.3 will make apparent,
121More precisely, assumep0,q0,r0 >0;p00 ≤0,q00≥0,r00 ≥0, with at least one holding strictly; and p0(x)¯ g ≤q0(x)h¯ +r0(x)c¯ for some ¯x>0.
these technical assumptions are made to simplify the proofs of the welfare results.123 Appendix A.5.5 will show that the underlying intuition of the proposed externalities- optimization theory does not depend on these assumptions.
However, there is a loss of generality in excluding administrative costs and the pos- sibility of contracting to vary or avoid the law. Administrative costs are assumed to be independent or sunk costs that do not affect the actor’s incentives at the time of choosing her action. Such an assumption also rules out cases where litigation costs vary with the allocation rules.124 Subsection 5.3.3 will consider information costs. The possibility of contracting is left for future research.
Moreover, neither the Model nor its modifications in the appendices accounts for the law’s expressiveness.