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Regulations to Supplement Weak Environmental Liability

Paul Calcott


Liability for environmental harm is often capped when precaution meets a minimum requirement and is often applied in conjunction with regulations. The correct setting for such regulations depends on the approach that courts take to evaluating the minimum requirement. It may also depend on which of the minimum requirement and capped liability is the more pressing consideration for the firm. Regulatory design is more straightforward when it is capped liability that is more pressing.

JEL Classification: K32; Q58


A Lemma 1

Proof. The argument proceeds in two steps. First, we establish that the firm’s choice of risk level is nonincreasing in θ. Let e=x+y and Z be the set of π,e for which π=π(x,y) for some x,yX×Y. Given the objective θπ+e, the marginal rate of substitution between π and e is θ. Therefore π is nonincreasing in θ by the Spence-Mirrlees condition.

The second step is to establish that reduced risk is achieved with more x. Assume the contrary. Then the change in precaution that delivers the reduction in risk can be decomposed into two movements. The first is an increase in y, holding x constant, which increases πx/πy by eq. 3. The second is a movement around the new isorisk curve, increasing y and decreasing x. This increases πx/πy by quasi-convexity. But problem 4 implies that πx/πy should be unchanged, so the assumption that x has reduced can be rejected.

B Three Proofs

B.1 Preliminaries

The impact on expected costs from a marginal increase in x along y=Y(x;θf+θl),l{c,d} is:


where j=h for social costs and j=c for private costs under capped liability. For points on y=Y(x;θf+θl), we can substitute in (θf+θl)πy+1=0 and rearrange to get:


where πx+πyY/x<0 by eq. 3.

B.2 Lemma 8

Proof. Consider the impact on social costs (j=h) of a movement along the firm’s best response under capped liability (l=c). Then θjθl>0 in eq. 26. Moreover, at the initial point the firm chooses x to minimize private costs, so (θf+θl)πx+1=0 and eq. 26 is locally negative by eq. 3. Consequently, welfare would be improved so long as the firm continues to comply with the regulation. Moreover, the firm would strictly prefer to comply with the duty when θd<θc, by Lemma 6. Consequently, it would also plan to comply with a regulation that only required incrementally more x than the cost minimizing way to reach the threshold, Xˆ(θf+θd).

B.3 Lemma 11

Proof. First, as θc<θd and costs are quasi-convex, the firm will want to keep y as low as is consistent with reaching the threshold, i. e., y=Y(xˉ;θf+θd). Now consider the impact on private costs (j=c) of increasing x above xˉ along y=Y(xˉ;θf+θd) (l=d). Then θjθl<0 in eq. 26. Moreover, [(θf+θl)πx+1]0 at the initial point so eq. 26 is locally positive.

B.4 Lemma 12

Proof. Consider the impact on social costs (j=h) of a movement along the threshold function (l=d). Then θjθl>0 in eq. 26. The initial point is the least-cost way to reach the threshold, so (θf+θd)πx+1=0. Then eq. 26 can be signed by Lemma 2 and assumption 8.

B.5 Heterogeneous Firms

Section 4.3 acknowledged heterogeneity of firms. In particular, θf was drawn from a continuous distribution with support [α,β]. Imagine that the initial setting for the regulation was at the lowest level of x that would be chosen without regulation, xˉ=Xˆ(α+θc). An incremental increase in xˉ would be socially beneficial for the firms that are induced to change precaution, by Lemma 8, and neither beneficial or harmful to the others. Consequently, it would be beneficial on balance.

C Example

C.1 Assumptions

Let π(x,y)=(ax)2+(ay)22, where X=Y=[0,a],a=1. Then Xˆ(θ)=aθ1 and Yˆ(θ)=Y(x;θ)=aθ1. Assume as in Section 3 and Section 4.1 that θc<θd<θh and so:


For simplicity, let θf1=0.

C.2 No Beneficial Regulation

Because x,y are independent, πxy=0, the regulator’s preferred point on y=Y(x;θc) is Xˆ(θh),Yˆ(θc). If this point violates the general criterion, πˉπ(x,y) i. e., if θd2(θh2+θc2)/2, then no regulation can be beneficial. The reason is that the regulator prefers the intersection of πˉ=π(x,y),y=Y(x;θc) to any other implementable point on y=Y(x;θc) by eq. 26 with l=c,j=h. Moreover, Xˆ(θd),Yˆ(θd) is the least cost way to attain πˉ=π(x,y), and this is attainable without regulation.

A similar argument holds for the biased general criterion of Section 4.1. If ϕˉ<ϕ(Xˆ(θh),Yˆ(θc)), then the best outcome attainable with regulation is on ϕˉ=ϕ(x,y), and so is dominated by the outcome without regulation.


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Published Online: 2016-11-12
Published in Print: 2016-10-1

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