A Appendix

In this paper, it is assumed that the per-firm damage D is linked to the production, but is not directly proportional to the output produced by this firm. It corresponds to a configuration under which an important accident occurs during the production or consumption process. Figure 1 represents the link between the production and the damage, with the per-firm output q_i represented on the X-axis, and the expected damage D represented on the Y-axis.

On Figure 1, the expected damage that is not proportional to the output is represented by full lines with a drastic change for the production level qˉ. This expected damage is

where qˉ is a production threshold above which risks appears. In the paper, it is assumed that qˉ→0, which means that the damage may occur as soon as the firm enters the business with an output qi>0.Figure 1 represents the damage D happening if no effort is made (λi= 0), and the expected damage (1−λi)D linked to an effort λi.

Alternatively, the dashed lines represent the “classical case” with the expected damage, (1−λi)dqi, proportional to the output q_i, and with d denoting the per-unit damage (see Polinsky and Rogerson 1983). Figure 6 represents the damage dqi happening if no effort is made (λi= 0), and the expected damage (1−λi)dqi linked to an effort λi. Note that the dashed curve continuously increases with the output qi, while the continuous curve discretely increases for the output qˉ.

Figure 6:

The expected damage.

As seen in point 2 of Section 6, an expected damage (1−λi)D not directly proportional to the output leads to results that are close to the ones when the damage is proportional to the output, but not internalized in the price by lack of liability or information.

B Proof of Proposition 1

The proof starts with the configuration for which all firm may enter whatever the standard. This is the case when the effort compatible with the duopoly and given by eq. (2) is such that λˉ2≥1, leading to the constraint f≤f2 with f2=2π2 represented on Figure 1. With f≤f2, both firms enter the market in areas A and B whatever the standard and the effort. In area A, the standard λ∗ is defined by eq. (6), with λ∗≤1 equivalent D≤D1 with D1=f represented on Figure 1. By using eq. (5), the optimal level of welfare is

In area B, the standard λ∗ is equal to 1 because D>D1.

When the constraint f>f2 is satisfied one firm may be impeded from entering the market. If λ∗<λ‾2+ϵ, the standard λ∗ defined by eq. (6) is compatible with both firms entering, which is the case with D<D3 with D3=2fπ2 represented on Figure 1. However for this case with λ∗<λ‾2+ϵ, the regulator may distort the number of firms entering by selecting a standard λˉ2+ϵ, with only one firm entering the market. By using eq. (7) with one firm, the welfare is

The regulator selects the standard λ*=D/f with both firms entering the market in area G, because W2λ∗>W1λ‾2+ϵ corresponding to D<D4 on Figure 1 with

In area F, the standard λˉ2+ϵ is selected and one firm enters the market, because D>D4 corresponding to W2λ∗<W1λ‾2+ϵ and D<D3 corresponding to λ‾2+ϵ>λ∗.

When D3<D<D1 corresponding to λ‾2+ϵ<λ∗<1, the standard maximizing the welfare is λ*=D/f and one firm enters the market, corresponding to area E on Figure 1. In area C, the standard is λ*=1 because D>D1 and one firm enters the market. These optimal levels of effort in areas E and C are possible, when the effort compatible with the monopoly and given by eq. (4) is such that λˉ1≥1, leading to the constraint f≤f1 with f1=2π1 represented on Figure 1. With f>f1, the optimal levels λ*=MinD/f,1 are not always compatible with the monopoly. If λ∗=D/f>λ‾1+ϵ, the standard λ∗ defined by eq. (6) is not compatible with one firms entering, which is the case with D>D2 with D2=2fπ1 represented on Figure 1. In area D, defined by f>f1 and D>D2, the standard is defined by λˉ1 given by eq. (2), and one firm enters the market. For simplifying the analysis we assume that W1λˉ1 is always positive.

QED

C Proof of Proposition 2

As the maximization of eqs. (7), (8) and (9) leads to Minλ∗,1 with λ∗ defined by eq. (6), when the international welfare is maximized, many constraints of Figure 1 applies to Figure 2. However, the decision to restrict the number of firms via the standard and to favor the domestic firm differs. This decision by the domestic regulator comes from the comparison of the domestic welfare with the standard λ∗ and the domestic welfare with one firm entering with a standard λ‾2+ϵ. When both domestic and foreign firms enter the market, the welfare for the domestic country defined by eq. (8) with the standard λ∗ is

When only the domestic firm enters the market because of the domestic standard λ‾2+ϵ, the welfare is given by W1λ‾2+ϵ defined by eq. (16). It is assumed that the domestic regulator maximizing the domestic welfare is able to protect his firm, because of the proximity of the consumers, only present in the domestic country. Because of this assumption, we do not study the possibility of overreaction by the foreign regulator for favoring the foreign firm that could enter.

The domestic regulator selects the standard λ*=D/f with both firms entering the market in area G₁, because Wˉ2dλ∗>W1λ‾2+ϵ corresponding to D<D5 on Figure 2 with

In area G₂, the standard λˉ2+ϵ is selected and the domestic firm enters the market, because D>D5 corresponding to Wˉ2dλ∗<W1λ‾2+ϵ . The area G₂ exists since the frontier D5<D4 in Figure 2. In area G₂, the standard is λˉ2+ϵ and only one domestic firm enters the market. This standard is protectionist since the choice in Figure 1 was λ*=D/f with both firms entering the market.

QED

D Proof of Proposition 3

The maximization of eqs. (10) and (11) leads to Minλ∗,1 with λ∗ defined by eq. (6), which explains many areas and frontiers of Figure 3 similar to Figure 1.

This decision by the domestic regulator comes from the comparison of the domestic welfare with the standard λ∗ and the welfare with one firm entering with a standard λ‾2+ϵ. When both domestic and foreign firms enter the market, the welfare for the domestic country defined by eq. (10) with the standard λ∗ is

When one firm enters the market, the welfare (11) with the standard is

The domestic regulator selects the standard λ*=D/f with both firms entering the market in area G₃, because Wˆ2fλ∗>Wˆ1fλ‾2+ϵ corresponding to D<D6 on Figure 3 with

In area G₄, the standard λˉ2+ϵ is selected and one foreign firm enters the market, because D>D6 corresponding to Wˆ2fλ∗<Wˆ1fλ‾2+ϵ. The area G₄ exists since the frontier D6<D4 in Figure 3. In area G₄, the standard is λˉ2+ϵ and only one foreign firm enters the market. This standard is protectionist, since the choice in Figure 1 was λ*=D/f with both firms entering the market. As D6<D5, the area G₄ in Figure 3 is larger than area G₂ in Figure 2.

QED

E Proof of Proposition 4

The domestic regulator will impose the highest effort maximizing eqs. (12) or (13). Under duopoly the welfare is maximized for a standard equal to 1 when f < f₂with f2=2π2, meaning the entry of both firms (in areas A and B of Figure 4).

When f2<f<f1 with f1=2π1, both firms may enter the market if the standard λˉ2 given by eq. (2) is selected. In this case by using eq. (12) the welfare is

With f2<f<f1 and one firm entering market, the highest level of effort is equal to one. In this case, the welfare defined by eq. (13) is CS1 The comparison between this value with one firm CS1 and W˜2dλˉ2  with two firms given by eq. (23) leads to

For D<D7 on Figure 4, W˜2dλˉ2 >CS1, the regulator chooses the standard λˉ2 and 2 foreign firms enter the market (namely in area G₅ and G₆). For D>D7 on Figure 4, W˜2dλˉ2 <CS1, the regulator chooses the standard equal to 1 and 1 foreign firm enters the market (namely in area F₁ and E₁).

With f > f₁ and one firm entering market, the highest level of effort is equal to λˉ1 given by eq. (2). In this case, the welfare defined by eq. (13) is

The comparison between this value W˜1dλˉ1  with one firm and W˜2dλˉ2  with two firms given by eq. (23) leads to

For D<D8 on Figure 4, W˜2dλˉ2 >W˜1dλˉ1 , the regulator chooses the standard λˉ2 , and 2 foreign firms enter the market (namely in area G₅ and G₆). For D>D8 on Figure 4, W˜2dλˉ2 <W˜1dλˉ1 , the regulator chooses the standard equal to λˉ1 and 1 foreign firm enters the market (namely in area F₂ and E₂).

Proposition 4 comes from the comparison between Figure 1 and Figure 4.

QED

F Proof of Proposition 5

When the constraint f<f2 is satisfied, both firms may enter the market (see proposition 1). In this case, the effort λ⃗=Min2D/f,1 maximizes the welfare defined by eq. (14). For D<D10= f/2, namely in area A₁, the effort is λ⃗=2D/f that is higher than the international standard λ*=D/f. For D10= f/2<D<D1, namely in area A₂, the effort is λ⃗=1 that is higher than the international standard λ*=D/f.

For the optimal effort λ⃗=2D/f, the welfare is

The domestic regulator chooses this effort λ⃗=2D/f with both firms entering when the welfare W⃗2dλ⃗ is greater than the welfare with only one firm entering because of a standard λ‾2+ϵ . By using W1λ‾2+ϵ with one domestic firm defined by eq. (16), the inequality W⃗2dλ⃗≥W1λ‾2+ϵ is satisfied for D≤D9, namely in area G₈, with

For D4<D≤D9, namely in area G₇, the standard λ‾2+ϵ. is selected leading to the entry of the domestic firm.

The frontier D4 is given by eq. (17) and the rest of frontiers of Figure 5 is equivalent to frontiers delineating the top of areas in Figure 1 and Figure 2.

QED