Do control rights determine the optimal extension of liability to investors? The case of environmental policy for mines

Abstract

Using a Pigovian tax to provide incentives for mine rehabilitation may be ineffective if limited liability, (judgement-proof) firms can declare themselves bankrupt to avoid the tax and rehabilitation costs. This paper introduces a model of environmental policy for mining that accounts for bankruptcy risk, profit risk, and a mobilization cost that applies if, following bankruptcy, rehabilitation is funded by the regulator or investor. The results show that making deep-pocketed investors who are never judgement-proof liable for a share of rehabilitation cost is optimal. The share of extended liability depends on the policy setting, control rights over the firm and the firm’s rent. If the firm has control rights over bankruptcy, optimal liability is at least the full cost of rehabilitation. If the investor has control rights then optimal liability is partial. Partial liability also applies when the policy is based on an insurance contract. Mobilization costs for a regulator to engage in rehabilitation mean it is socially optimal for an investor to “prop-up” a firm making moderate losses to complete mine rehabilitation.

Appendices

Appendices

1.1 Appendix 1: Welfare loss due to a lenient cut-off profit

The difference between the regulator’s objective function at the socially optimal cut-off and the firm’s preferred cut-off:

$$\begin{aligned} J(e,\bar{{\pi }}^{s})-J(e,\bar{{\pi }}^{f})= & {} \int _{\bar{{\pi }}^{s}}^{\pi ^{\max }} {g(\pi )\pi d\pi } -(1-G(\bar{{\pi }}^{s}))(c_f (e)\\&+\,D(1-e))-G(\bar{{\pi }}^{s})(c_r (e)+D(1-e)) \\&-\,\Bigg (\int _{\bar{{\pi }}^{f}}^{\pi ^{\max }} {g(\pi )\pi d\pi }-\,(1-G(\bar{{\pi }}^{f}))(c_f (e)\\&+\,D(1-e))-G(\bar{{\pi }}^{f})(c_r (e)+\,D(1-e))\Bigg ) \end{aligned}$$

Using the results that:

$$\begin{aligned}&\int _{\bar{{\pi }}^{s}}^{\bar{{\pi }}^{f}} {g(\pi )\pi d\pi } =\int _{\bar{{\pi }}^{s}}^{\pi ^{\max }} {g(\pi )\pi d\pi } -\int _{\bar{{\pi }}^{f}}^{\pi ^{\max }} {g(\pi )\pi d\pi } \end{aligned}$$

and that $c_r (e)=\hbox {cm}+c_f (e),$ gives:

$$\begin{aligned}&-\,(1-G(\bar{{\pi }}^{s}))(c_f (e)+D(1-e))-G(\bar{{\pi }}^{s})(c_r (e)+D(1-e))\\&\quad =-c_f (e)-G(\bar{{\pi }}^{s})\hbox {cm}-D(1-e) \\ \end{aligned}$$

and simplifying:

$$\begin{aligned}&J(e,\bar{{\pi }}^{s})-J(e,\bar{{\pi }}^{f})=\int _{\bar{{\pi }}^{s}}^{\bar{{\pi }}^{f}} {g(\pi )\pi d\pi } -(G(\bar{{\pi }}^{s})-G(\bar{{\pi }}^{f}))\,\hbox {cm}\,\ge 0 \end{aligned}$$

1.2 Appendix 2: Cut-off profit where the firm determines bankruptcy

The firm’s objective function is:

$$\begin{aligned} \mathop {Maximize}\limits _{\bar{{\pi }},\beta } \beta \int _{\bar{{\pi }}}^{\pi ^{\max }} {g(\pi )\pi d\pi } -(1-G(\bar{{\pi }})) \Bigg (c_f (e)+\tau (\hat{{e}}-e)\Bigg )-A \end{aligned}$$

(25)

A binding borrowing constraint (15) implies:

$$\begin{aligned} \beta= & {} \left\{ -G(\bar{{\pi }})\upsilon c_r (e)+\int _{\bar{{\pi }}}^{\pi ^{\max }} {g(\pi )\pi d\pi }\right. \nonumber \\&\left. -\,(1-G(\bar{{\pi }}))\Bigg (c_f (e)+\tau (\hat{{e}}-e)\Bigg )-(I-A) \right\} \Bigg /\int _{\bar{{\pi }}}^{\pi ^{\max }} {g(\pi )\pi d\pi } \end{aligned}$$

(26)

Substitute (26) into (25) for $\beta$ and cancel to give:

(27)

Take the derivative with respect to the cut-off profit $\bar{{\pi }}$ and set equal to zero gives:

$$\begin{aligned} \bar{{\pi }}^{f}=2\Bigg (c_f (e)-(\upsilon /2)c_r (e)+\tau (\hat{{e}}-e)\Bigg ) \end{aligned}$$

If this is set equal to the condition for the first best cut-off profit $\bar{{\pi }}^{s}=-c_m$ if an internal solution is feasible this implies an extended liability:

$$\begin{aligned} \upsilon =\frac{2c_f (e)+c_m }{c_r (e)}=1+\frac{c_f (e)}{c_f (e)+c_m}; \quad 1\le \upsilon \le 2. \end{aligned}$$

1.3 Appendix 3: Insurance contract

If the premium $p$ is actuarially fair then $p=G(\bar{{\pi }})\upsilon ^{In}c_r$, the firm’s objective function is:

$$\begin{aligned} \mathop {Maximize}\limits _{\bar{{\pi }},\beta } \beta \int _{\bar{{\pi }}}^{\pi ^{\max }} {g(\pi )\pi d\pi } -(1-G(\bar{{\pi }}))\Bigg (c_f (e)+\tau (\hat{{e}}-e)\Bigg )-A-p\qquad \end{aligned}$$

(28)

or equivalently:

$$\begin{aligned} \mathop {Maximize}\limits _{\bar{{\pi }},\beta } \beta \int _{\bar{{\pi }}}^{\pi ^{\max }} {g(\pi )\pi d\pi } -(1-G(\bar{{\pi }}))\Bigg (c_f (e)+\tau (\hat{{e}}-e)\Bigg )-A-G(\bar{{\pi }})\upsilon ^{In}c_r \end{aligned}$$

A binding borrowing constraint implies:

$$\begin{aligned} \beta \!= & {} \!\left\{ {-G(\bar{{\pi }})\upsilon ^{In}c_r +\int _{\bar{{\pi }}}^{\pi ^{\max }} {g(\pi )\pi d\pi } -(1-G(\bar{{\pi }}))(c_f (e)+\tau (\hat{{e}}-e)-B} \right\} \nonumber \\&\Bigg /\int _{\bar{{\pi }}}^{\pi ^{\max }} {g(\pi )\pi d\pi } \end{aligned}$$

(29)

Substitute (29) into (28) and simplify to give:

(30)

and take the derivative with respect to the cut-off profit $\bar{{\pi }}$ and set equal to zero gives:

$$\begin{aligned} \bar{{\pi }}=2\Bigg (c_f (e)-\upsilon ^{In}c_r (e)+\tau (\hat{{e}}-e)\Bigg ). \end{aligned}$$