Choosing Whether to Comply with a General Duty or with a Specification Standard

Lemma 8

Assume that x is a normal input in producing safety. Then Xˆ(θ) is nondecreasing in θ and X˜ is nonincreasing in π˜.

proof

Let MRT=πx/πy. Then interior values of X˜,Y˜ are characterized by the tangency:

Application of the Implicit Function Theorem reveals that:

where Δ1 is the determinant. This expression is signed by eq. (1). Now consider Xˆ(θf),Yˆ(θf). Interior values are characterized by the FOCs:

This time the Implicit Function Theorem reveals that:

which is also signed by eq. (1). As [0,a]×[0,a] is a lattice, the presence of boundaries will not reverse these directions.

Lemma 9

Consider a firm faced with choice of compliance, (x,y)∈Sd∪Ss. If it prefers to comply with the standard rather than the duty, then the latter option is safer, π(xˉ,Y∗(xˉ,θf))>π˜.

proof

Imagine instead that π(xˉ,Y∗(xˉ,θf))≤π˜. Then there would be a weakly convex combination of Xˆ(θf),Yˆ(θf) and xˉ,Y∗(xˉ,θf), that would exactly meet the duty. The firm would weakly prefer this convex combination to xˉ,Y∗(xˉ,θf) by convexity. But the firm prefers X˜,Y˜ to other precaution combinations that also have risk equal to π˜, by Lemma 3. Consequently it would prefer X˜,Y˜ to xˉ,Y∗(xˉ,θf) by transitivity.

Lemma 10

Increasing xˉ results in more safety

proof

The impact of an incremental increase in xˉ on the probability of an incident at a firm that remains compliant with the standard is:

which is negative by eq. (1).

proof

First, we establish that precaution taken by noncompliers, Xˆ(θf),Yˆ(θf), would be more risky than choices taken by compliers. This is immediate if compliance would be with the duty, by eq. (5) and Lemma 1. It also holds for those that comply with a standard, as the impact on risk of a firm increasing x to comply with a standard is negative by Lemma 10. Moreover firms with high values of θf are less enthusiastic about safe options than firms with low values, by Lemma 1. This establishes part (i).

According to Lemma 9, if any firms switch to comply only with the standard, then this results in less safety than X˜,Y˜ does. But then Lemma 1 implies that X˜,Y˜ would be preferred by firms with the higher values of θf. This establishes part (ii).

proof

The only consequence of introducing choice of compliance is to remove penalties when x,y∈Ss∖Sd. Therefore, there is no motive for any change in precaution, unless it is to Ss∖Sd. But as (X˜,Y˜)∉Ss∖Sd, any such change will be to xˉ,Y∗(xˉ,θf) by Lemma 4. This establishes part (i). Furthermore, risk will increase according to Lemma 9. This establishes part (ii) regarding compliers. Finally, initial noncompliers will increase x along y=Y∗(x,θf) and so will reduce risk by Lemma 10.

proof

The duty is too permissive, π(Xˆ(θs),Yˆ(θs))<π˜. But π(xˉ,Y∗(xˉ,θf))≥π˜ by Lemma 6. Continuity ensures that there is a point x,y such that π(x,y)=π˜ which is a convex combination of Xˆ(θs),Yˆ(θs) and xˉ,Y∗(xˉ,θf), and which would be socially preferred to xˉ,Y∗(xˉ,θf) by quasi-convexity. However X˜,Y˜ is socially preferred to this x,y by Lemma 3. Transitivity gives us the result that X˜,Y˜ has lower social costs than xˉ,Y∗(xˉ,θf). This establishes part (i). Now imagine that the firm would not comply under DO. Then any change would be from Xˆ(θf),Yˆ(θf) to xˉ,Y∗(xˉ,θf), which results in a decrease in risk by Lemma 10. But then this decision improves social efficiency by Lemma 1.