Environmental risks, the judgment-proof problem and financial responsibility

Abstract

This paper examines a setting in which a firm is liable to pay environmental damages caused by its activity but may not have sufficient wealth for repair of damages. In order to induce the full internalization of the environmental cost, the firm is required to demonstrate a financial guarantee from a solvent party that covers this cost. Since the firm and the guarantor are joint liable for the harm caused by the firm, it is in the interest of the guarantor to design the guarantee contract in order to induce the firm to take an adequate level of prevention. First, I show that financial responsibility regime may achieve the social optimum. Secondly, I identify a particular form of contract in the set of contracts which induce the socially optimal level of prevention. This contract is closed to an alternative risk transfer product referred to as the spread loss treaty.

Appendix

1.1 Proof of lemma 1

The social optimum e* is the solution of the following problem:

$$ \mathop {\min }\limits_{e} \int\limits_{0}^{L} {\ell f\left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell + c\left( e \right)} $$

The associated first-order condition is given by:

$$ c_{e} \left( {e*} \right) = \int\limits_{0}^{L} {F_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)d\ell } . $$

(6)

The firm’s problem can be written as:

$$ \mathop {\max }\limits_{e} \pi - c\left( e \right) - \int\limits_{0}^{\pi - c(e)} {\ell f\left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell - \left[ {1 - F\left( {\pi - c\left( e \right)} \right)} \right]\left[ {\pi - c(e)} \right]} $$

If we denoted by e ^P the interior solution to the above problem, it solves the following first-order-condition:

$$ c_{e} \left( {e^{P} } \right)F\left( {\pi - c\left( {e^{P} } \right)} \right) = \int\limits_{0}^{{\pi - c\left( {e^{P} } \right)}} {F_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell {e^{P} }}} \right. \kern-\nulldelimiterspace} {e^{P} }}} \right)d\ell } . $$

(7)

The left-hand-side term of Eq. 6 (7) represents the social (private) expected marginal cost of prevention and the right-hand-side represents the social (private) expected marginal benefit. From the comparison of (6) and (7) e ^P can be lower or higher than e*.

1.2 Proof of proposition 2

Part 1: $u \in \left[ {\underline{u,} \pi - c\left( e \right) - B} \right]$

Every level of utility u is given by the following expression:

$$ u = \pi - c\left( e \right) - \int\limits_{0}^{L} {t(\ell )f\left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell } $$

Taking into account this expression, the objective function of the guarantor becomes:

$$ \pi - c(e) - u - \int\limits_{0}^{L} {\ell f\left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell } $$

Moreover, (2) and (3) imply:$\pi - c\left( e \right) \ge \int_{0}^{L} {t(\ell )} f\left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell \ge B;$ thus $0 \le u \le \pi - c\left( e \right) - B$

Consequently, the existence of a transfers scheme verifying (1), (2) and (3) implies that the utility of the firm is bounded: $u \in \left[ {\underline{u,} \pi - c\left( e \right) - B} \right]$. Note that the principal’s objective function depends only on the expected transfer (by u). Therefore, all solutions that verify the agent’s incentive constraint and that have the same expectation are equivalent from the principal’s point of view. However, the existence of such solutions is not guaranteed. Indeed, if the problem does not admit a solution, then it is not possible to implement a given level of prevention e for a given level of utility u.

Part 2: $\left[ {\pi - c\left( e \right) - B} \right]F_{e} \left( {{{\hat{\ell }} \mathord{\left/ {\vphantom {{\hat{\ell }} e}} \right. \kern-\nulldelimiterspace} e}} \right) \ge c_{e} \left( e \right)$

Let us assume that $u \in \left[ {\underline{u,} \pi - c(e) - B} \right]$, then the next step consists to establish conditions under which the incentive constraint (4) is satisfied. Let $\Im = \left\{ {t(\ell )/B \le t(\ell ) \le \pi - c(e)\forall \ell } \right\},$ be the set of admissible transfers. Let us define:$G\left[ {t\left( \cdot \right)} \right] = \int_{0}^{L} {t\left( \ell \right)f_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell } ;$ $m = \mathop {\min }\limits_{t\left( \ell \right) \in \Im } \int_{0}^{L} {t\left( \ell \right)f_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell }$ and $M = \mathop {\max }\limits_{t\left( \ell \right) \in \Im } \int_{0}^{L} {t\left( \ell \right)f_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell }$.

We can establish that m is strictly negative and M strictly positive.¹¹ Thus the function $G\left[ {t(.)} \right]$ is bounded in the set of admissible transfers. Then the validity of the incentive constraint depends on the value taken by m as follows.

Lemma 2

the incentive constraint is satisfied for a given e and u if and only if:

$$ \min \int\limits_{0}^{L} {t\left( \ell \right)f_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell \le - c_{e} \left( e \right)} $$

Lemma 3

the scheme of transfers$\hat{t}(\ell )$which minimizes the function$G\left[ {t\left( \cdot \right)} \right] = \int_{0}^{L} {t\left( \ell \right)f_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell }$has the following form¹²:

$$ \hat{t}(\ell ) = \left\{ {\begin{array}{*{20}c} B \hfill & {\forall \ell < \hat{\ell }} \hfill \\ {\pi - c\left( e \right)} \hfill & {\forall \ell > \hat{\ell }} \hfill \\ \end{array} } \right.\quad {\text{with}}\,\hat{\ell } = F^{ - 1} \left[ {{{\frac{u}{\pi - c\left( e \right) - B}}}} \right][] $$

The second part of proposition 2 follows from lemmas 2 and 3.

1.3 Proof of proposition 3

From proposition 2, we can derive that when the guarantor’s problem (P1) admits at least one solution, it is equivalent to the following problem (P1bis):

$$ \mathop {\max }\limits_{t(\ell ),e} \pi - c\left( e \right) - u - \int\limits_{0}^{L} {\ell f\left( {{\ell \mathord{\left/ {\vphantom {\ell e}} \right. \kern-\nulldelimiterspace} e}} \right)d\ell } $$

subject to

$$ \underline{u \, } \le u \le \pi - c\left( e \right) - B $$

(8)

$$ \left[ {\pi - c\left( e \right) - B} \right]F\left( {{{\hat{\ell }} \mathord{\left/ {\vphantom {{\hat{\ell }} e}} \right. \kern-\nulldelimiterspace} e}} \right) = u $$

(9)

$$ \left[ {\pi - c\left( e \right) - B} \right]F_{e} ({{\hat{\ell }} \mathord{\left/ {\vphantom {{\hat{\ell }} e}} \right. \kern-\nulldelimiterspace} e}) \ge c_{e} \left( e \right) $$

(10)

Conditions (9) and (10) imply proposition 3.

1.4 Proof of proposition 4

From the proposition 3 we know that the socially optimal prevention level can be achieved if ${\frac{{F_{e} \left( {{{\hat{\ell }} \mathord{\left/ {\vphantom {{\hat{\ell }} {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}{{F\left( {{{\hat{\ell }} \mathord{\left/ {\vphantom {{\hat{\ell }} {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}} \ge {\frac{{c_{e} \left( {e*} \right)}}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }}}$. Moreover, we can demonstrate that the function ${\frac{{F_{e} \left( {{\ell \mathord{\left/ {\vphantom {\ell {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}{{F\left( {{\ell \mathord{\left/ {\vphantom {\ell {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}}$ is not increasing in ℓ.¹³ Consequently, if ${\frac{{F_{e} \left( {{{\hat{\ell }} \mathord{\left/ {\vphantom {{\hat{\ell }} {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}{{F\left( {{{\hat{\ell }} \mathord{\left/ {\vphantom {{\hat{\ell }} {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}} \ge {\frac{{c_{e} \left( {e*} \right)}}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }}}$, there is a level of damages $\hat{\hat{\ell }} > \hat{\ell }$ such that ${\frac{{F_{e} \left( {{{\hat{\hat{\ell }}} \mathord{\left/ {\vphantom {{\hat{\hat{\ell }}} {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}{{F\left( {{{\hat{\hat{\ell }}} \mathord{\left/ {\vphantom {{\hat{\hat{\ell }}} {e*}}} \right. \kern-\nulldelimiterspace} {e*}}} \right)}}} = {\frac{{c_{e} \left( {e*} \right)}}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }}}$.