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Optimal risk regulation of monopolists with subjective risk assessment

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Abstract

This study characterizes the optimal regulation of risky activities when the assessment of the probability of an accident is subjective. The optimism of stakeholders forms subjective risk perceptions, which substantially affect the optimal intervention. To explore this issue, we construct a moral hazard model with a limited liability constraint, where stakeholders have heterogeneous beliefs about the probability of an accident. First, we show that the optimism of a monopolist reduces the level of the preventive effort when regulatory instruments are fixed. We then analyze the case in which the regulator can set both a fine and a product price as regulatory instruments. The optimal product price increases with the monopolist’s degree of optimism, as the loose product market regulation encourages the preventive effort of the optimistic monopolist. Consequently, under such an optimal scheme, an increase in the optimism of the monopolist may increase the level of preventive effort.

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Notes

  1. Although we frame this study in the context of environmental risks, the scope is considerably wider. Further, the results can be applied to other types of risk regulation, such as bank regulation.

  2. For example, in a discussion regarding the nature of risk regulation, Hutter (2010) argues, “The nature of the evidence available for managing risks needs to be questioned. [...] Two clear messages of the volume are that we cannot expect to live in a risk-free society and that we have to be prepared for the unexpected” (p.261).

  3. Although our primary focus is on firms’ risk perceptions, we also analyze the effect of the regulator’s risk perceptions in Sect. 5.2.

  4. This is a subclass of Schmeidler (1989)’s Choquet expected utility theory, which is one of the most influential theories of decision-making under ambiguity. In the Choquet expected utility theory, people have a subjective belief called a capacity (not a probability), and they calculate the expected payoff based on the capacity using the Choquet integral.

  5. The literature on the optimal environmental policy under ambiguity is also related (e.g., Roseta-Palma and Xepapadeas 2004; Gonzalez 2008; Asano 2010; Athanassoglou and Xepapadeas 2012; Millner et al. 2013). Since environmental risks entail ambiguity, the optimal policy should be designed by taking ambiguity into account, which has been used to justify the precautionary principle. Previous studies focus on deriving the first-best instead of the optimal incentive design to achieve the second-best.

  6. In the literature of contracts under ambiguity, Weinschenk (2010) analyzes a moral hazard problem, although his work does not include the analysis of ambiguity -attitudes. Recently, Giraud and Thomas (2017) applied the Choquet expected utility with a neo-additive capacity to an adverse selection problem.

  7. In the literature of contracts, several studies have analyzed the effect of heterogeneous beliefs. la Rosa (2011) considers a moral hazard problem in which the agent overestimates his ability. In his setting, ability and efforts are complementary, whereas they are additive in our setting.

  8. Assuming non-monetary costs, limited-liability constraints are described in a tractable manner. Existing studies have extensively used this assumption since Laffont (1995). Note that similar results hold when K is a monetary cost.

  9. t includes an implicit and explicit subsidy in other dimensions of the firm’s activity. For example, the firm might also produce another product in a different market. Further, the profitability of this other market depends on economic regulation (e.g., entry restriction).

  10. If \(\delta =0\), \(\mathcal{P}_e\) contains only \(\pi _e\), whereas it contains any probability measures if \(\delta =1\).

  11. We assume that D is sufficiently large such that in our setting, the payoff in the case of an accident is worse than that without an accident, for both the regulator and the monopolist.

  12. From Proposition 1 (iii), \(\partial {{\hat{e}}}/\partial p <0\) only when \(p>p_M\). In this case, \(Q'(p)(p-C(e))+Q(p)<0\); this implies that \(p>C(e)\). Therefore, the first term in Eq. (8) is positive.

  13. The linear probability function has been widely adopted in the literature (e.g., Hiriart and Martimort (2006a, 2006b); Hiriart et al. (2011)).

  14. Using the smooth ambiguity theory, Alary et al. (2013) make a similar prediction in the analysis of self-protection. However, because their model does not contain the market-oriented profit of firms, the second term in our model does not exist; therefore, ambiguity aversion always reduces the level of effort. By contrast, we identify the effect on the business-as-usual profit and show that ambiguity-loving reduces the level of effort as long as the firm is almost risk-neutral.

  15. For example, the Kansai Electric Power Company (KEPCO) (the second largest electric power company in Japan) planned to spend 1950 billion yen (around 19.5 billion US dollar) from 2013–2015.

  16. https://www.nikkei.com/article/DGXNASFS1803C_Y3A110C1EE8000/.

  17. Therefore, it is always monotonic and never U-shaped.

  18. Another way is to suppose that there exists a threshold value \({\bar{T}}\) such that an accident does not happen if and only if \(e\ge {\bar{T}}\). \({\bar{T}}\) is a continuous variable, and people face ambiguity on the distribution of \({\bar{T}}\). This setting yields the same payoff functions.

  19. It is easy to see that \({\mathcal {N}}\) thus defined satisfies all the axioms the family of null events must satisfy according to Chateauneuf et al. (2007).

  20. Otherwise, the first-order condition is never satisfied because \(-Q(p)C'(e)[(1-\delta )\pi (e)+\delta \alpha ^M]-K'(e)<0\).

  21. \(F_p\) is obtained by substituting \(G=0\) into \(\frac{\partial F}{\partial p}\). That is, by calculating \(\frac{\partial F}{\partial p}\) from (14), we have

    $$\begin{aligned} \frac{\partial F}{\partial p}=&Q'(p)\left\{ -\gamma (1+2\lambda )[(1-\delta )\gamma e+\delta \alpha ^R]-\lambda \gamma (1-\delta )\delta \alpha ^M\right. \nonumber \\&\left. +\gamma ^2(1-\delta )^2[(1+\lambda )(p-c)-(1+2\lambda )e]\right\} \nonumber \\&+\gamma ^2(1-\delta )^2\lambda Q(p). \end{aligned}$$
    (14)

    Besides, from (14),

    $$\begin{aligned} \lambda Q(p)=-Q'(p)\left[ (1+\lambda )(p-c)-(1+2\lambda )e-\lambda \frac{\delta }{1-\delta }\frac{\alpha ^M}{\gamma }\right] . \end{aligned}$$

    By substituting this into (14), we have \(F_p\). Other derivatives are straightforwardly obtained.

  22. These are satisfied in the linear-quadratic model.

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Correspondence to Susumu Sato.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the anonymous referee, Akihiko Matsui, Takeshi Murooka, Yoshihiro Tamai, the participants of 2020 Autumn Meeting of the Japanese Economic Association, and the seminar participants at Hitotsubashi University for their helpful comments. This study is financially supported by JSPS KAKENHI Grant Numbers 20K22117 and 20K22131. All the remaining errors are ours.

Appendices

Appendices

1.1 Choquet expected utility with a neo-additive capacity

Let \(2^S\) be the power set of S and let \((S, 2^S)\) be a measurable space.Footnote 18 Players are assumed to be the Choquet expected utility maximizer (Schmeidler 1989). That is, given e, each player has a capacity over \(2^S\), and maximizes the Choquet expected utility based on the capacity. \(\theta ^i: {{\mathbb {R}}}_+\times S\rightarrow [0, 1]\) is a capacity kernel of player i. To be specific, \(\theta ^i_{e}\) represents the capacity player i has given e. Given this expression, the monopolist is assumed to evaluate the expected payoff given e as

$$\begin{aligned} \int _{S} U(s|e)d\theta ^M_e(s), \end{aligned}$$

while the regulator evaluates the expected payoff given e as

$$\begin{aligned} \int _{S} W(s|e)d\theta ^R_e(s). \end{aligned}$$

Note that each integral is the Choquet integral.

We further restrict our attention to a special class of capacities, so-called neo-additive capacity, which was introduced and axiomatized by Chateauneuf et al. (2007). In order to define the neo-additive capacity, we need to specify the family \({\mathcal {N}}\) of null events, each element of which is supposed to be an event that all players believe that it will never be realized. We assume that all players believe that any event is possible to occur. That is, we assume that \({\mathcal {N}} := \{\emptyset \}\). Recall that there are only finitely many events in the current model and this seems to be a reasonable specification of the null events in such a setting.Footnote 19

Under the current specification of \({\mathcal {N}}\), assuming the neo-additive capacity amounts to assuming the next:

Assumption 2

The neo-additive capacity we employ in this paper is defined by

$$\begin{aligned} (\forall i\in \{M, R\}) \ (\forall e\in {{\mathbb {R}}}_+) \ (\forall A\in 2^S\setminus (\emptyset \cup S)) \;\;\;\;\;\; \theta _e^i(A) := (1-\delta )\pi _e(A)+\delta \alpha ^i, \end{aligned}$$

where \(\pi _e\) is a probability measure such that \(\pi _e(\{0\})=\pi (e)\), \(\delta \in [0, 1]\), and \(\alpha \in [0, 1]\)

That is, the capacity for events except for \({\mathcal {N}}\) and S is given by a convex combination of a probability measure \(\pi \) and \(\alpha \). It is further known that the Choquet integral with respect to it is expressed as (1) (Chateauneuf et al. 2007). Hence, we obtain (1).

Omitted proofs

1.1 Proof of Proposition 1

To simplify notations, we define the cross derivatives of \({{\mathcal {U}}}\) as follows: for \(x, y\in \{e, \alpha ^M, f, p\}\), \(\mathcal{U}_{xy}:=\frac{\partial ^2 {{\mathcal {U}}}}{\partial x \partial y}\).

We first prove that \({{\mathcal {U}}}_{ee}<0\) at \(e={\hat{e}}\).

$$\begin{aligned} \frac{\partial ^2 {{\mathcal {U}}}}{\partial e^2} \,=\,&(1-\delta )\pi ''(e)[{\Pi }(p, e)+f]-(1-\delta )\pi '(e)2Q(p)C'(e)\nonumber \\&-\,Q(p)C''(e)[(1-\delta )\pi (e)+\delta \alpha ^M]-K''(e). \end{aligned}$$

Here, from Eq. (2), \(\Pi (p, {\hat{e}})+f>0\).Footnote 20 Furthermore, from the assumption, \(\pi ''(e)\le 0\), \(\pi '(e)>0\), \(C'(e)>0\), \(C''(e)\), and \(K''(e)>0\). Hence, \({{\mathcal {U}}}_{ee}<0\) at \(e={\hat{e}}\).

  1. (i).

    By applying the implicit function theorem, \(\frac{\partial {\hat{e}}}{\partial f}=-{{\mathcal {U}}}_{ef}/{{\mathcal {U}}}_{ee},\) where

    $$\begin{aligned} \frac{\partial ^2 {{\mathcal {U}}}}{\partial e \partial f}=(1-\delta )\pi '(e)>0. \end{aligned}$$

    Hence, \(\frac{\partial {\hat{e}}}{\partial f}>0\).

  2. (ii).

    By applying the implicit function theorem, \(\frac{\partial {\hat{e}}}{\partial \alpha ^M}=-\mathcal{U}_{e\alpha ^M}/{{\mathcal {U}}}_{ee},\) where

    $$\begin{aligned} \frac{\partial ^2 {{\mathcal {U}}}}{\partial e \partial \alpha ^M}=-\delta Q(p)C'(e)<0. \end{aligned}$$

    Hence, \(\frac{\partial {\hat{e}}}{\partial \alpha ^M}<0\).

  3. (iii).

    By applying the implicit function theorem, \(\frac{\partial {\hat{e}}}{\partial p}=-{{\mathcal {U}}}_{ep}/{{\mathcal {U}}}_{ee},\) where

    $$\begin{aligned} \frac{\partial ^2 {{\mathcal {U}}}}{\partial e \partial p}=&-Q'(p)C'(e)[(1-\delta )\pi (e)\\&+\delta \alpha ^M]+(1-\delta )\pi '(e)[Q'(p)(p-C(e))+Q(p)]>0. \end{aligned}$$

    Notice that \(Q'(p)(p-C(e))+Q(p)>0\) because \(p<p_M\). Hence, \(\frac{\partial {\hat{e}}}{\partial p}>0\). \(\square \)

1.2 Stability and uniqueness in Example 1

By rearranging Eq.  (9), we obtain \(f={{\hat{f}}}(e,p)\):

$$\begin{aligned} {{\hat{f}}}(e,p):=\frac{[2\gamma (1-\delta )Q(p)+k]e+\delta \alpha ^MQ(p)}{\gamma (1-\delta )}-Q(p)(p-c). \end{aligned}$$
(12)

In addition, from Eq. (9),

$$\begin{aligned} \frac{\partial {\hat{e}}}{\partial f}=\frac{\gamma (1-\delta )}{2\gamma (1-\delta )Q(p)+k}. \end{aligned}$$
(13)

By substituting Eqs. (12) and (13) into Eq. (5) and multiplying \(2\gamma (1-\delta )Q(p)+k\) on both sides, we have

$$\begin{aligned} F(e,p, \alpha ^R,\alpha ^M, k):=&-\lambda \left[ 2\gamma (1-\delta )Q(p) + k\right] \left[ (1-\delta )\gamma e + \delta \alpha ^R\right] \\&- \lambda \gamma (1-\delta )(ke + \delta \alpha ^MQ(p)) \\&+ \gamma ^2(1-\delta )^2\left\{ \int ^{\infty }_p Q(x) dx + Q(p)\right. \\&\left. \left[ (1+\lambda )(p-c) - (1+2\lambda )e\right] + D\right\} \\&-\,\gamma (1-\delta ) \left[ (1-\delta )\gamma e + \delta \alpha ^R\right] Q(p) - \gamma (1-\delta ) ke\\ =\,&0. \end{aligned}$$

Besides, by arranging Eq. (11), we have

$$\begin{aligned} G(e, p, \alpha ^R, \alpha ^M,k) := -p +c + \frac{1+2\lambda }{1+\lambda }e + \frac{\lambda }{1+\lambda }\frac{\delta }{1-\delta }\frac{\alpha ^M}{\gamma } - \frac{\lambda }{1+\lambda }\frac{Q(p)}{Q'(p)} = 0. \end{aligned}$$

Hence, the equilibrium condition for \(e^*\) and \(p^*\) is given by \(F(e^*,p^*, \alpha ^R,\alpha ^M, k)=0\) and \( G(e^*, p^*, \alpha ^R, \alpha ^M,k)=0\).

Note that, at \(e=e^*\) and \(p=p^*\), the following equations hold.Footnote 21

$$\begin{aligned} F_e := \frac{\partial F}{\partial e}&= -\gamma (1-\delta )(1+2\lambda )\left[ k + 2\gamma (1-\delta )Q(p)\right]<0; \\ F_p := \frac{\partial F}{\partial p}&= -\gamma (1-\delta )\left[ (1-\delta )\gamma e + \delta \alpha ^R\right] Q'(p)(1+2\lambda )>0; \\ G_e :=\frac{\partial G}{\partial e}&= \frac{1+2\lambda }{1+\lambda } >0; \\ G_p := \frac{\partial G}{\partial p}&= -1 - \frac{\lambda }{1+\lambda }\frac{d}{dp}\left( \frac{Q(p)}{Q'(p)}\right) <0. \end{aligned}$$

The solution \((e^*, p^*)\) is unique if \(\det J>0\), where

$$\begin{aligned} J:= \left( \begin{array}{cc} F_e &{} F_p \\ G_e &{} G_p \end{array}\right) . \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned} \det J =&F_eG_p - F_pG_e \\ =\,&\gamma (1-\delta )(1+2\lambda )\left\{ \left[ k + 2\gamma (1-\delta )Q(p)\right] \left[ 1+\frac{\lambda }{1+\lambda }\frac{d}{dp}\left( \frac{Q(p)}{Q'(p)}\right) \right] \right\} \\&+ \gamma (1-\delta )\left[ (1-\delta )\gamma e + \delta \alpha ^R\right] Q'(p)\frac{1+2\lambda }{1+\lambda } \end{aligned} \end{aligned}$$

If \(Q(p) = a-p\), \(\det J >0\) if and only if

$$\begin{aligned} (1+2\lambda ) k + 2\gamma (1-\delta ) (1+2\lambda ) (a-p) - \left[ (1-\delta )\gamma e + \delta \alpha ^R\right] > 0. \end{aligned}$$

Substituting Eq. (11) into this, \(\det J > 0\) is less likely to hold when \(e^*\) is large. If \(\det J >0\) holds at \(e= 1/\gamma \), \(\det J >0\) holds for any equilibrium value of \(e^* \in (0,1/\gamma )\). A calculation shows that a \(\det J >0\) at \(e=1/\gamma \) holds if and only if

$$\begin{aligned} k > {{\hat{k}}} \equiv \frac{1-\delta + \delta \alpha ^R-2(1-\delta )\left[ \gamma (1+\lambda )(a-c) - 1 - 2\lambda - \lambda \frac{\delta }{1-\delta }\alpha ^M\right] }{1+2\lambda }. \end{aligned}$$

Thus, for \(k \ge {{\hat{k}}}\), the equilibrium is stable. \(\square \)

1.3 Proof of Proposition 2

Using the implicit function theorem, we have

$$\begin{aligned} J \left( \begin{array}{c} \frac{d e^*}{d\omega } \\ \frac{d p^*}{d\omega } \end{array}\right) = \left( \begin{array}{c} -\frac{\partial F}{\partial \omega } \\ -\frac{\partial G}{\partial \omega } \end{array}\right) \end{aligned}$$

for \(\omega \in \{\alpha ^R, \alpha ^M\}\). By using the Cramer’s rule, we have

$$\begin{aligned} \frac{d e^*}{d\omega } = \frac{1}{\det J}\det \left( \begin{array}{cc} - \frac{\partial F}{\partial \omega }&{} F_p \\ - \frac{\partial G}{\partial \omega } &{} G_p \end{array}\right) ; \ \ \frac{d p^*}{d\omega } = \frac{1}{\det J}\det \left( \begin{array}{cc} F_e &{}- \frac{\partial F}{\partial \omega } \\ G_e &{} - \frac{\partial G}{\partial \omega } \end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} \frac{\partial F}{\partial \alpha ^R}&= -\delta \left[ \lambda k + \gamma (1-\delta )(1+2\lambda )Q(p)\right] < 0; \\ \frac{\partial G}{\partial \alpha ^R}&= 0; \\ \frac{\partial F}{\partial \alpha ^M}&= -\lambda \gamma (1-\delta )\delta Q(p); \\ \frac{\partial G}{\partial \alpha ^M}&= \frac{\lambda }{1+\lambda }\frac{\delta }{1-\delta }\frac{1}{\gamma }. \end{aligned}$$

By Assumption 1, we have \(\det J >0\), and thus

$$\begin{aligned} \text {sign}\left( \frac{de^*}{d\alpha ^M}\right) =&\text {sign}\left( -\frac{\partial F}{\partial \alpha ^M}G_p + \frac{\partial G}{\partial \alpha ^M}F_p\right) ; \\ \text {sign}\left( \frac{dp^*}{d\alpha ^R}\right) =&\text {sign}\left( \frac{\partial F}{\partial \alpha ^M}G_e - \frac{\partial G}{\partial \alpha ^M}F_e \right) <0. \end{aligned}$$
  1. (i).

    Hence, We have \(dp^*/d\alpha ^M>0\) if and only if

    $$\begin{aligned} \begin{aligned} \frac{\partial F}{\partial \alpha ^M}G_e - \frac{\partial G}{\partial \alpha ^M}F_e =&\frac{\lambda }{1+\lambda }\delta \left[ (1+2\lambda )k + \gamma (1-\delta )(1+2\lambda )Q(p)\right] > 0. \end{aligned} \end{aligned}$$
  2. (ii).

    \(de^*/d\alpha ^M > 0\) if and only if

    $$\begin{aligned} \begin{aligned} -\frac{\partial F}{\partial \alpha ^M}G_p + \frac{\partial G}{\partial \alpha ^M}F_p =&-\lambda \gamma (1-\delta )\delta Q(p)\left[ 1 +\frac{\lambda }{1+\lambda } \frac{d}{dp}\left( \frac{Q(p)}{Q'(p)}\right) \right] \\&-\frac{\lambda }{1+\lambda }\delta \left[ (1-\delta )\gamma e + \delta \alpha ^R\right] Q'(p)(1+2\lambda ), \end{aligned} \end{aligned}$$

    which may be either positive or negative.

  3. (iii).

    When \(Q(p)= a-p\), from (11),

    $$\begin{aligned} p=e+\frac{1+\lambda }{1+2\lambda }c+\frac{\lambda }{1+2\lambda }a+\frac{\lambda }{1+2\lambda }\frac{\delta }{1-\delta }\frac{\alpha ^M}{\gamma }. \end{aligned}$$
    (15)

    Hence, by substituting (15) into (15), we have

    $$\begin{aligned} de^*/d\alpha ^M>0 \Leftrightarrow \frac{\lambda (1+2\lambda )}{1+\lambda }\delta \left[ - \gamma (1-\delta )(a-p^*) + \gamma (1-\delta ) e^* + \delta \alpha ^R\right] > 0, \end{aligned}$$

    which is equivalent to

    $$\begin{aligned} \begin{aligned} \Omega (e, \delta , \alpha ^M, \alpha ^R) :=&-\gamma (1-\delta ) \frac{1+\lambda }{1 + 2\lambda } (a-c) + 2\gamma (1-\delta )e\\&+ \delta \left( \frac{\lambda }{1+2\lambda }\alpha ^M + \alpha ^R\right) >0 \end{aligned} \end{aligned}$$
    (16)

    holds for \(e=e^*\). If either \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)\le 0\) or \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)\ge 0\) holds for all \(\alpha ^M\), the proposition straightforwardly holds. Hence, it suffices to focus on the case where for some \(\alpha ^M\), \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)< 0\), while for other \(\alpha ^M\), \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)>0\). We claim that there exists a unique \({\bar{\alpha }}^M\) such that \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)<0\) if and only if \(\alpha ^M<{\bar{\alpha }}^M\). To see this, let \(\alpha ^M_T\) be the smallest value of \(\alpha ^M\) such that \(\Omega (e^*, \delta , \alpha ^M_T, \alpha ^R) = 0\). Then, it is shown that for all \(\alpha ^M > \alpha ^M_T\), \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)\ge 0\). Prove by contradiction. Suppose that there exists \(\alpha ^M > \alpha ^M_T\) such that \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)< 0\). Then, for some of such \(\alpha ^M\), we must have \(\alpha ^M \ge \alpha ^M_T\) and \(e^*(\alpha ^M) \ge e^*(\alpha ^M_T)\). However, because (16) is increasing in \(\alpha ^M\) and e, this leads to \(\Omega (e^*, \delta , \alpha ^M, \alpha ^R)>0\), a contradiction. Thus, there exists \(\alpha ^M_T\) such that \(\Omega (e^* \delta , \alpha ^M_T, \alpha ^R) = 0\), and, it is unique. Letting \({\bar{\alpha }}^M := \alpha ^M_T\), we have the result. \(\square \)

1.4 Proof of Proposition 4

Similarly to the proof of Proposition 2, we have

$$\begin{aligned} \text {sign}\left( \frac{de^*}{d\alpha ^R}\right) = \text {sign}\left( -\frac{\partial F}{\partial \alpha ^R}G_p\right)<0; \ \ \text {sign}\left( \frac{dp^*}{d\alpha ^R}\right) = \text {sign}\left( \frac{\partial F}{\partial \alpha ^R}G_e\right) <0. \end{aligned}$$

This shows the proposition. \(\square \)

1.5 Regulation with fixed price

In this section, we show that when p is fixed, the equilibrium effort level is decreasing in the monopolist’s optimism \(\alpha ^M\), which is shown by analyzing the solution to Eq. (7). As a regularity condition, we assume that there exists a unique solution to (7) and the second-order condition is satisfied.

Assumption 3

There is a unique solution to (7). Furthermore, for such f, \(\frac{\partial ^2 \bar{{\mathcal {W}}}(f, {{\hat{e}}}(f), p)}{\partial f^2}\le 0\).

Note that this assumption is satisfied in the linear-quadratic model. Under this assumption, by solving (7), we obtain the regulator’s optimal fine level \(f^{**}\). Note that this assumption is satisfied in the linear quadratic model because \(\frac{\partial ^2 \bar{{\mathcal {W}}}(f, {{\hat{e}}}(f), p)}{\partial f^2}< 0\) for any f. Given this fine level, the equilibrium effort level \(e^{**}:={\hat{e}}(f^{**})\) is determined.

Now we analyze the impacts of the monopolist’s optimism. For this, we obtain the following results.

Proposition 5

The following statements hold:

  1. (i)

    \(e^{**}\) is decreasing in \(\alpha ^M\).

  2. (ii)

    Suppose that \(\pi '''(e)\le 0\), \(C'''(e)\ge 0\), and \(K'''(e)\ge 0\) holds for all e. If \(K'''(e)\ge -(1-\delta )3Q(p)\pi ''(e)C'(e)\) for all e,Footnote 22\(f^{**}\) is increasing in \(\alpha ^M\).

Proof

To simplify notations, we define the cross derivatives of \(\bar{\mathcal{W}}\) as follows: for \(x, y\in \{\alpha ^R, \alpha ^M, f\}\), \(\bar{\mathcal{W}}_{xy}:=\frac{\partial ^2 \bar{{\mathcal {W}}}(f, {{\hat{e}}}(f), p)}{\partial x \partial y}\).

(i): \(e^{**}\).:

Step 1. While we have regarded the regulator’s problem as a problem of choosing f, we can also regard the regulator’s problem as a problem of choosing e. Take the inverse of \({\hat{e}}(f)\) and define the inverse function by \({\hat{f}}(e):= {\hat{e}}(f)^{-1}\). Similarly, define \({\hat{t}}(e)\) by \({\hat{t}}({\hat{f}}(e))\). Then, the regulator’s problem can be rewritten as

$$\begin{aligned} \max _{e} \bar{{\mathcal {W}}}({\hat{f}}(e), e, p). \end{aligned}$$

The first-order condition of this problem can be rewritten as

$$\begin{aligned}&\frac{\partial \bar{{\mathcal {W}}}({\hat{f}}(e), e, p)}{\partial e}=0 \nonumber \\ \Leftrightarrow&-\lambda \frac{\partial {\hat{f}}(e)}{\partial e}\nonumber \\&\left[ (1-\delta )\pi (e)+\delta \alpha ^R\right] +(1-\delta )\pi '(e)\left[ CS(p)+\Pi (p, e)-\lambda {\hat{f}}(e)+D\right] \nonumber \\&-Q(p)C'(e)\left[ (1-\delta )\pi (e)+\delta \alpha ^R\right] -K'(e)=0. \end{aligned}$$
(17)

Here, from the inverse function theorem, \({\partial {\hat{f}}(e)}/{\partial e}={1}/\left( {\partial {\hat{e}}(f)}/{\partial f}\right) .\) Hence, (7)=0 \(\Leftrightarrow \) (17)=0. Therefore, from Assumption 3, there exists a unique f satisfying (17)=0 and for such e, the second-order condition is also satisfied. Step 2. By the implicit function theorem,

$$\begin{aligned} \frac{\partial e^{**}}{\alpha ^M}=-{\frac{\partial ^2 \bar{{\mathcal {W}}}({\hat{f}}(e), e, p)}{\partial \alpha ^M \partial e}}\left( {\frac{\partial ^2 \bar{{\mathcal {W}}}({\hat{f}}(e), e, p)}{\partial e^2}}\right) ^{-1}, \end{aligned}$$

where

$$\begin{aligned} \frac{\partial ^2 \bar{{\mathcal {W}}}({\hat{f}}(e), e, p)}{\partial e^2}<0; \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 \bar{{\mathcal {W}}}({\hat{f}}(e), e, p)}{\partial \alpha ^M \partial e}=-\lambda \left[ \frac{\partial ^2 {\hat{f}}(e)}{\partial \alpha ^M \partial e}+(1-\delta )\pi '(e)\frac{\partial {\hat{f}}(e)}{\partial \alpha ^M}\right] . \end{aligned}$$
(18)

Here, from Proposition 1, \(\frac{\partial f}{\partial \alpha ^M}>0\). Furthermore,

$$\begin{aligned} \frac{\partial ^2 {\hat{f}}(e)}{\partial \alpha ^M \partial e}=-\frac{\partial }{\partial \alpha ^M}\left( \frac{{{\mathcal {U}}}_{ee}}{{{\mathcal {U}}}_{fe}}\right) =-\frac{\pi ''(e)}{\pi '(e)}\frac{\partial f}{\partial \alpha ^M}\ge 0. \end{aligned}$$

Hence, (18)\(<0\), implying that \(\frac{\partial e^{**}}{\alpha ^M}<0\).

(ii): \(f^{**}\).:

We next prove that \(f^{**}\) is increasing in \(\alpha ^M\) under a mild condition. By applying the implicit function theorem, \(\frac{\partial f^{**}}{\partial \alpha ^M}=-\bar{{\mathcal {W}}}_{f\alpha ^M}/\bar{{\mathcal {W}}}_{ff},\) where \(\bar{{\mathcal {W}}}_{ff}<0.\) Hence, it suffices to prove that \(\bar{\mathcal{W}}_{f\alpha ^M}>0\). Step 1. First, observe that

$$\begin{aligned} \bar{{\mathcal {W}}}_{f\alpha ^M}=\frac{\partial {\hat{e}}}{\partial \alpha ^M}\frac{\partial \bar{{\mathcal {W}}}_f}{\partial {\hat{e}}}+\frac{\partial ^2 {\hat{e}}}{\partial \alpha ^M \partial f}A, \end{aligned}$$

where

$$\begin{aligned} A:=&(1-\delta )\pi '({\hat{e}}(f))\left[ CS(p)+\Pi (p, {\hat{e}}(f))-\lambda f+D\right] \\&-Q(p)C'({\hat{e}}(f)) \left[ (1-\delta )\pi ({\hat{e}}(f)))+\delta \alpha ^R\right] -K'({\hat{e}}(f)). \end{aligned}$$

From Proposition 1, \(\frac{\partial {\hat{e}}}{\partial \alpha ^M}<0\). Furthermore, it is easy to show that \(\frac{\partial \bar{{\mathcal {W}}}_f}{\partial {\hat{e}}}<0\). In addition, at the optimum (i.e., \(f=f^{**}\)), the first-order condition directly implies that

$$\begin{aligned} \frac{\partial {\hat{e}}}{\partial f}A=\lambda \left[ (1-\delta )\pi ({\hat{e}})+\delta \alpha ^R\right] >0. \end{aligned}$$

This further implies that \(A>0\) because \(\frac{\partial {\hat{e}}}{\partial f}>0\) from Proposition 1. Therefore, \(\bar{{\mathcal {W}}}_{f\alpha ^M}>0\) as long as \(\frac{\partial ^2 {\hat{e}}}{\partial \alpha ^M \partial f}\ge 0\). Step 2. The next step is to prove that \(\frac{\partial ^2 {\hat{e}}}{\partial \alpha ^M \partial f}\ge 0\). Here,

$$\begin{aligned} \frac{\partial ^2 {\hat{e}}}{\partial \alpha ^M \partial f}=&\frac{\partial }{\partial \alpha ^M}\left( \frac{-(1-\delta )\pi '(e)}{{{\mathcal {U}}}_{ee}}\right) \nonumber \\ =&-(1-\delta )\frac{1}{{{\mathcal {U}}}_{ee}^2}\left[ \frac{\partial {\hat{e}}}{\partial \alpha ^M}\pi ''({\hat{e}})\mathcal{U}_{ee}-\frac{\partial {\hat{e}}}{\partial \alpha ^M}\pi '({\hat{e}})\frac{\partial {{\mathcal {U}}}_{ee}}{\partial {\hat{e}}}\right] . \end{aligned}$$
(19)

The first inequality comes from the implicit function theorem. Here, \(\frac{\partial {\hat{e}}}{\partial \alpha ^M}<0\) and \({{\mathcal {U}}}_{ee}<0\) from Proposition 1. Hence, if \(\frac{\partial \mathcal{U}_{ee}}{\partial {\hat{e}}}\le 0\), (19)>0. Step 3. The last step is to prove that \(\frac{\partial {{\mathcal {U}}}_{ee}}{\partial {\hat{e}}}\le 0\).

$$\begin{aligned} \frac{\partial {{\mathcal {U}}}_{ee}}{\partial e} =&(1-\delta )\pi '''(e)[\Pi (p, e)+f]-(1-\delta )3Q(p)\pi '(e)C''(e)\nonumber \\&-Q(p)C'''(e)[(1-\delta )\pi (e)+\alpha ^M]\nonumber \\&-(1-\delta )3Q(p)\pi ''(e)C'(e) -K'''(e). \end{aligned}$$
(20)

Here, the first three terms in (20) are non-positive as long as \(\pi '''(e)\le 0\) and \(C'''(e)\ge 0\). Hence, a sufficient condition for (20)\(\ge 0\) is that

$$\begin{aligned}&-(1-\delta )3Q(p)\pi ''(e)C'(e)-K'''(e)\le 0 \Leftrightarrow K'''(e)\\&\quad \ge -(1-\delta )3Q(p)\pi ''(e)C'(e). \end{aligned}$$

\(\square \)

1.6 Price-setting monopolist

Consider the case where the monopolist can choose p as well as e. To make analysis as tractable as possible, we assume the linear-quadratic specification where \(C(e) = c+e\), \(K(e) = ke^2/2\), \(\pi = \gamma e\). Further, we assume that \(Q(p) = a-p\). The choice of p and e is given by the solution to \(\max _{p,e} \mathcal {U}\), which is given by the system of equations

$$\begin{aligned} p = \frac{a+c + e}{2}, \end{aligned}$$

and (9). Let \({\tilde{e}}(f, \alpha ^M)\) and \({\tilde{p}}(f, \alpha ^M) = (a+c+{\tilde{e}}(f, \alpha ^M))/2\) be the solution to this system of equations. Specifically, we have

$$\begin{aligned} {\tilde{e}}(f, \alpha ^M) = \frac{F - \sqrt{F^2 - 4EG}}{2E}, \end{aligned}$$
(21)

where

$$\begin{aligned} E&= \frac{3}{4}\gamma (1-\delta ) \\ F&= k + \gamma (1-\delta )(a-c) - \frac{\delta }{2}\alpha ^M \\ G&= \gamma (1-\delta )f + \frac{\gamma (1-\delta )}{4}(a-c)^2 - \frac{\delta }{2}(a-c)\alpha ^M. \end{aligned}$$

Consider the level of f that is required to achieve the effort level e, which is given by

$$\begin{aligned} {\tilde{f}}(e, \alpha ^M) = \left( \frac{k - \frac{\delta }{2}\alpha ^M}{\gamma (1-\delta )} + a-c\right) e - \frac{3}{4}e^2 - \frac{(a-c)^2}{4} + \frac{\delta }{2}\alpha ^M(a-c) \end{aligned}$$
(22)

We have

$$\begin{aligned} \frac{\partial {\tilde{f}}}{\partial e} = \frac{k - \frac{\delta }{2}\alpha ^M}{\gamma (1-\delta )} + a -c - \frac{3}{2}e \end{aligned}$$

Note that \(\partial ^2 {\tilde{f}} / \partial e \partial \alpha ^M <0\), while \(\partial ^2 {{\hat{f}}} / \partial e \partial \alpha ^M = 0\). Because a high price is linked to high effort, inducing a high effort leads to a small production scale. When the monopolist is optimistic, small output implies that the cost of paying effort become small. Thus, the more optimistic the monopolist is, the less steep the incentive the regulator needs to induce higher level of preventive efforts. This might reduce the social cost of increasing the preventive effort, and if this dominates the direct reduction in the effort due to the optimism, the equilibrium preventive may effort increase.

Next, consider the regulator’s problem of choosing f to maximize welfare. The first-order condition is given by

$$\begin{aligned} \frac{\partial \mathcal {W}}{\partial e} =&\frac{\partial {\tilde{f}}}{\partial e} \frac{\partial {{\bar{W}}}}{\partial f} + \frac{\partial {{\bar{W}}}}{\partial e} + \frac{1}{2}\frac{\partial {{\bar{W}}}}{\partial p} \nonumber \\ =\,&[(1-\delta )\gamma e + \delta \alpha ^R]\left[ -\lambda \left( \frac{k-\frac{\delta }{2}\alpha ^M}{\gamma (1-\delta )} + a -c - \frac{3}{2}e\right) + \frac{1}{2}(a-c-e)\right] \nonumber \\&+ \gamma (1-\delta )[ CS(p) + \Pi (p,e)- \lambda f+ D]\nonumber \\&- \frac{a-c-e}{2}[(1-\delta )\gamma e + \delta \alpha ^R] - ke \nonumber \\ =\,&-[(1-\delta )\gamma e + \delta \alpha ^R]\lambda \left( \frac{k-\frac{\delta }{2}\alpha ^M}{\gamma (1-\delta )} + a -c - \frac{3}{2}e\right) \nonumber \\&+ \gamma (1-\delta )\left\{ \frac{(a-c-e)^2}{2} - \lambda \left[ \left( \frac{k - \frac{\delta }{2}\alpha ^M}{\gamma (1-\delta )} + a-c\right) e - \frac{3}{4}e^2\right. \right. \nonumber \\&\left. \left. - \frac{(a-c)^2}{4} + \frac{\delta }{2}\alpha ^M(a-c)\right] + D\right\} - ke \nonumber \\ =\,&0 \end{aligned}$$
(23)

We have

$$\begin{aligned} \frac{\partial ^2 \mathcal {W}}{\partial e \partial \alpha ^M} = \frac{\lambda \delta }{2}\left[ 2e + \frac{\delta \alpha ^R}{\gamma (1-\delta )} - \gamma (1-\delta )(a-c)\right] , \end{aligned}$$
(24)

which may either positive or negative, depending on the equilibrium value of effort level e and other parameters.

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Kishishita, D., Sato, S. Optimal risk regulation of monopolists with subjective risk assessment. J Regul Econ 59, 251–279 (2021). https://doi.org/10.1007/s11149-021-09429-0

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