Proposition 11 formally states the main result.
Proposition 11. Suppose the socially optimal action is positive, x∗ > 0. Then the actor’s chosen actionxis socially optimal if and only if the law optimizes net externalities.
Formally, suppose x∗ > 0. Then x = x∗ if and only if xdefined by condition (60) satisfies
δG0(x)= γC0(x)+(1−λ)H0(x). (64) Proof
Assume x∗ > 0. ThenS0(x∗)=0 follows from the first order condition (61) for social
welfare.
For one direction, assume x = x∗. This implies A0(x) = 0 and S0(x) = 0 (due
to first order conditions (60), (61) respectively). Equation (63) then impliesV0(x) = 0.
Condition (64) immediately follows from the definition ofV in equation (59).
For the other direction, assume x satisfies condition (64). The actor’s first order
condition (60) implies A0(x) ≤0. There are two possibilities:
1. SupposeA0(x)=0. Then equation (63) and condition (64) implyS0(x)= A0(x)+
V0(x) = 0. The first order condition (61) for social welfare is satisfied. The strict
concavity ofSthus implies the uniqueness of its optimizer, x= x∗.
2. Suppose, for a contradiction, A0(x) < 0. Then x = 0, and equation (63) and
condition (64) implyS0(0)= A0(0)+V0(0) <0. But x∗ >0 and 0= S0(x∗)> S0(0),
a contradiction toS00 < 0 (the strict concavity ofS).
The left-hand side of condition (64) sums the marginal positive externalities that the actor generates, while the right-hand side sums the marginal negative externalities. Proposition 11 proves that the actor has socially optimal incentives if positive and negative externalities are aligned at the margin — that is, net externalities are optimized. Moreover,
assuming the socially optimal action is positive, Proposition 11 proves that optimization of net externalities is also a necessary condition for social optimality. Thus, in this Model, condition (64) captures all allocation rules that generate socially optimal incentives.
Proposition 11 generalizes the externalities-internalization approach to inducing social optimality in the presence of externalities. In the present Model, complete internalization of externalities occurs under the triple (δ, γ, λ) = (0,0,1); in other words, the gain- and
cost-allocation rules allow the actor to keep all of her gain and cost, while the harm- allocation rule shifts to her all the harm that she imposes on the victim. As it is well understood (see section 5.1), this triple of allocation rules equates the actor’s utility with social welfare (I = S), leading to social optimality in her choice of action (x = x∗).
Proposition 11 explains this triple of allocation rules as a special case of a significantly more general condition of (net) externalities optimization: Because this triple induces zero net externalities (V = 0), the actor’s chosen action trivially optimizes net externalities.
Proposition 11 usually reveals uncountably many triples of gain-, cost- and harm- allocation rules that induce the socially optimal action. Many such triples do not com- pletely internalize externalities. The remainder of this subsection will describe some of these optimal triples.
Corollary 27 (Optimal restitution of net gain). Suppose the socially optimal action is positive (x∗ > 0). The actor’s chosen action xis socially optimal if the law shares some proportion of the actor’snetgain with the victim and shares thecomplementaryproportion of the victim’s harm with the actor.
Formally, x= x∗ > 0if
δ =γ = 1−λ <1. (65)
Proof
Assume x∗ > 0. The triple described by condition (65) gives the actor the following
utility:
A(x)=λG(x) −λC(x) −λH(x)= λS(x),
which implies she chooses x defined by A0(x) = λS0(x) = 0. Then λ > 0 implies
S0(x) = 0, which satisfies the first order condition (61) for social welfare. The strict
Corollary 27 reveals that restitution of (a proportion of) net gain — the law shifts to the victim the same proportion of the actor’s private gain and cost, δ = γ < 1 —
may induce social optimalitywithoutcomplete internalization of externalities.125 Social optimality arises if the actor also bears the complementary proportion of the victim’s harm,λ=1−δ. Intuitively, the law gives the actor a utility function that isproportionate to the social welfare function,A=λS; the law also gives rise to a net externalities function
that is proportionate to the social welfare function, V = (1−λ)S. Then the actor acts
to induce zero marginal utility if and only her action also induces zero marginal social welfare and zero marginal (net) externalities:
A0(x)=0 ⇔ S0(x)=0 ⇔ V0(x)=0.
In other words, the law shares social welfare between the actor and the victim in a proportionate way, andanysuch sharing incentivizes the actor to take the socially optimal action.
Moreover, even restitution of gross gain may induce social optimality. Before Corollary 28 states the general result, Example 5 illustrates the conditions under which restitution of gross gain may be socially optimal.
Example 5. Suppose the wrongful actionxgenerates gainG(x)=4√xand costC(x)= x
to the actor and harmH(x)= xto the victim. A substitution exercise using the first order condition (61) for social welfare reveals that the socially optimal action is x∗ = 1.126
Let the actor bear all of her private cost (γ = 0).127 The pair of gain- and harm- allocation rules (δ, λ) = (0.25,0.5) is one of the many pairs that induce the actor to choose x= x∗ = 1. Under this pair, the choice x= 1satisfies her first order condition (60), that is, A0(1)= 0.75G0(1) −0.5H0(1) −C0(1) = 0.75×2×1−0.5−0.5×1−1 =0. Notice that the pair(δ, λ)= (0.25,0.5)doesnotinternalize all externalities; when x=1, net externalities sum toV(1)= 0.25G(1) −0.5H(1)= 0.5. This is the optimized value of
V becauseV0(1)= 0.25G0(1) −0.50
H(1)= 0.25×2×1−0.5−0.5=0.
Figure 8 depicts the social welfare functionS(the black dotted line), the actor’s utility function A(the green solid line) and the net externalities functionV (the red solid line) under the pair(δ, λ)=(0.25,0.5). As Figure 8 reveals, the actor’s choice x= 1satisfies
125Corollary 27 follows the same intuition as those underlying the socially optimal rules that Polinsky and
Rubinfeld (2003) and Huang (2016) proposed. See subsection 5.1.2 for a discussion of their proposals.
126More precisely,S0(x)=G0(x) −C0(x) −H0(x)=2x−0.5−1−1, andS0(x∗)=0 impliesx∗=1.
127See footnote 141 for a U.S. case recognizing that a securities law violator typically cannot offset her
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 Action (x) The actor's payoff A(x) given
δ=0.25, λ=0.5
Net externalities V(x) given δ=0.25, λ=0.5
Net externalities V(x) given δ=0.25, λ=0 The actor's payoff A(x) given δ=0.25, λ=0
Social welfare S(x)
The actor's choice given
δ=0.25, λ=0.5 Socially optimal The actor's choice given δ=0.25, λ=0
(x*)
Figure 8: The implications of two pairs of gain- and harm-allocation rules in Example 5, where the actor bears all of her private cost (γ = 0).
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Harm-allocation rule (γ) Gain-allocation rule (δ) (0,1) (0.25, 0.5)
Figure 9: Optimal pairs of gain- and harm-allocation rules for Example 5, where the actor bears her private cost (γ = 0).
the first order conditions ofS, AandV.
Pairs of gain- and harm-allocation rules that fail to optimize net externalities leads to a suboptimal action. One such suboptimal pair is (δ, λ) = (0.25,0). Under this pair, the actor’s first order condition (60) leads to the choice x =2.25,128which well exceeds the socially optimal action x∗ = 1. The action x = 2.25 also fails to optimize net externalities. Under the pair (δ, λ) = (0.25,0), the net-externalities function isV(x) =
0.25G(x) −H(x)=√x− x, the optimizer of which is0.25rather than2.25.
Figure 8 also depicts the actor’s utility function A(the green dashed line) and the net externalities functionV (the red dashed line) under the pair(δ, λ)=(0.25,0). In this case, the actor’s choice x= 2.25optimizes A, but it does not optimizeSorV.
The blue solid line in Figure 9 depicts the set of pairs of gain- and harm-allocation rules that induce the socially optimal action.129 This set contains the externalities- internalization pair (δ, λ) = (0,1), the pair (δ, λ) = (0.25,0.5) and uncountably many other pairs.
Figure 8 also suggests that, when the actor fails to bear all private cost and all social harm (λ < 1 or γ > 0), it is socially optimality to have an intermediate gain-allocation rule, δ ∈ (0,1). Under such an intermediate gain-allocation rule, both the actor and the
victim receive a positive proportion of the wrongful gain. Corollary 28 formalizes this observation.130
Corollary 28 (Optimal restitution). Suppose the actor’s chosen action is positive and socially optimal (x= x∗ > 0). Then:
1. The gain-allocation rule takes an intermediate form (δ ∈ (0,1)) if and only if the cost- and harm-allocation rules fail to hold the actor liable for all of her private cost and all of the social harm (γ >0orλ <1).
In particular, if the actor bears all of her private cost and none of the social harm (γ = λ=0), then the gain-allocation rule takes the following intermediate form:
δ = H 0( x) G0(x) = 1− C0(x) G0(x) = H0(x) H0(x)+C0(x). (66)
128More precisely,A0(x)=0.75G0(x) −C0(x)=1.5x−0.5−1=0, andA0(x)=0 impliesx=2.25.
129Formally, this optimal set is{(δ, λ) ∈ [0,1] × [0,1]|λ=1−2δ}.
2. The gain-allocation rule allows the actor to keep all of her private gain (δ = 0)if and only if the cost- and harm-allocation rules hold her liable for all of her private cost and all of the social harm (γ =0andλ=1).
Proof
Assume x = x∗ > 0. Then S0(x) = 0 due to condition (61), and Proposition 11
impliesxis induced by a triple of rules(δ, γ, λ)that satisfies condition (64). These imply
δG0(x) −γC0(x) − (1−λ)H0(x)= 0= G0(x) −H0(x) −C0(x),
a rearrangement of which gives gives
δ = G
0(x) − (1−γ)C0(x) −λH0(x)
G0(x) .
For one direction, consider two cases:
1. Supposeλ <1 orγ > 0. ThenS0(x)= 0,C0 > 0, H0 > 0,λ ≥ 0 andγ ≤ 1 imply
0 = G0(x) −C0(x) −H0(x) < G0(x) − (1−γ)C0(x) −λH0(x) < G0(x). Then δ ∈ (0,1).
Condition (66) follows from assumingλ=γ = 0 and usingS0(x)=0.
2. Supposeλ= 1 andγ =0. Then condition (64) impliesδ= 0.
For the other direction, supposeδ ∈ (0,1)(respectively,δ =0) and, for a contradiction, λ =1 andγ = 0 (respectively,λ < 1 orγ > 0). Then case 2 (respectively, case 1) above
leads to a contradiction.
The intuition underlying part 1 of Corollary 28 is most apparent in a simple scenario in which the actor retains all of her private cost (γ = 0) and does not bear any of the
social harm (λ= 0). If the gain-allocation rule allows her to keep all of her gain (δ =0),
then she has perverse incentives to over-act without regard to the resulting harm. By comparison, if the gain-allocation rule disgorges all of her gain (δ = 1), then she has no
incentives to take positive action. Thus, to incentivize her to take a positive action that is socially optimal (x= x∗ > 0), the gain-allocation rule has to give her an expectation of
receiving some, but not all, of the wrongful gain. The same intuition explains the social optimality of intermediate restitution when the actor bears some, but not all, of the social harm (λ∈ (0,1)).
However, if the cost- and harm-allocation rules hold the actor liable for all of her private cost (γ = 0) and social harm (λ = 1), then part 2 of Corollary 28 reveals that the gain-
allocation rule must allow her to keep all of her private gain (δ= 0) in order to incentivize
a positive, socially optimal action. This result is the externalities-internalization theory; the triple(δ, γ, λ)= (0,0,1)removes all externalities. Thus social optimality requires the
imposition of restitutionary liability only in cases where complete compensation of the social harm is unattainable; as soon as the law can shift all wrongful harm to the actor, social optimality stops requiring any disgorgement of her private gain.
Corollary 28 doesnotcover cases in which the socially optimal action is zero,x∗ =0.
In these cases, any onerous liability that disincentivizes the actor from acting positively would be socially optimal. As subsection 5.3.2 will elaborate, this intuition explains the scope and limitations of externalities optimization more generally.