Economic Incentives for Managing Filterable Biological Pollution Risks from Trade

Abstract

Infectious livestock disease problems are “biological pollution” problems. Prior work on biological pollution problems generally examines the efficient allocation of prevention and control efforts, but does not identify the specific externalities underpinning the design of efficiency-enhancing policy instruments. Prior analyses also focus on problems where those being damaged do not contribute to externalities. We examine a problem where the initial biological introduction harms the importer and then others are harmed by spread from this importer. Here, the externality is the spread of infection beyond the initial importer. This externality is influenced by the importer’s private risk management choices, which provide impure public goods that reduce disease spillovers to others—making disease spread a “filterable externality.” We derive efficient policy incentives to internalize filterable disease externalities given uncertainties about introduction and spread. We find efficiency requires incentivizing an importer’s trade choices along with self-protection and abatement efforts, in contrast to prior work that targets trade alone. Perhaps surprisingly, we find these incentives increase with importers’ private risk management incentives and with their ability to directly protect others. In cases where importers can spread infection to each other, we find filterable externalities may lead to multiple Nash equilibria.

Appendix

This appendix contains proofs of the major results stated in the main article text.

Proposition 1

The private optimum involves interior solutions when $\Lambda _i >0$.

We have already shown that $a_i^0 =0$ in the private optimum, and so we focus on the remaining choices. The marginal expected profits from $z_{i}$ and $x_{i}$, respectively, are:

$$\begin{aligned} \frac{\partial \pi _i }{\partial z_i }= & {} -w\left( {1-x_i } \right) h_z \left( {z_i } \right) +P^{I}\left( {x_i } \right) \frac{\partial P^{C}\left( {z_i } \right) }{\partial z_i }\left[ {\lambda -\delta \left( {1-x_i } \right) } \right] \end{aligned}$$

(19)

$$\begin{aligned} \frac{\partial \pi _i }{\partial x_i }= & {} \left( {wh\left( {z_i } \right) -v} \right) -\frac{\partial P^{I}\left( {x_i } \right) }{\partial x_i }\left[ {P^{C}\left( {z_i } \right) \left( {1-x_i } \right) \delta +\left( {1-P^{C}\left( {z_i } \right) } \right) \lambda } \right] \nonumber \\&+P^{I}\left( {x_i } \right) P^{C}\left( {z_i } \right) \delta . \end{aligned}$$

(20)

The relations in (19) and (20) involve both variables. Suppose $x_i \rightarrow 1$. Then (19) implies $\partial \pi _i /\partial z_i \rightarrow 0\,\forall z_i$, which means $z_i \rightarrow 0$ as there is no economic reason to set $z_i >0$. However, if $z_i \rightarrow 0$, then (20) implies $\partial \pi _i /\partial x_i <0$ as $x_i \rightarrow 1$. These contradictory results imply that $x_i^0 <1$. Moreover, our assumptions about $P^{I}\left( {x_i } \right)$ ensure that $x_i^0 >0$: hence $x_i^0 \in \left( {0,1} \right)$.

Next, suppose $z_i \ge \bar{z}$, where $\bar{z}<1$ is defined such that $wh\left( {\bar{z} } \right) =v$. Then (20) implies $\partial \pi _i /\partial x_i >0$, so that $x_i \rightarrow 1$. But we have already shown that $x_i \rightarrow 1$ implies $z_i \rightarrow 0$, resulting in a contradiction of our premise that $z_i \ge \bar{z}$. Further, given $x_i <1$, (19) implies that $\partial \pi _i /\partial z_i >0$ when $z_i \rightarrow 0$. These last two results imply an interior solution for $z_i$: $z_i^0 \in \left( {0,\bar{z} } \right)$. $\square$

Proposition 2

The efficient outcome involves interior solutions.

The marginal expected net social values of $z_i$, $x_i$, and $a_i$, respectively, are:

$$\begin{aligned} \frac{\partial V}{\partial z_i }= & {} \frac{\partial \pi _i }{\partial z_i }+\Theta P^{I}\left( {x_i } \right) \frac{\partial P^{C}\left( {z_i } \right) }{\partial z_i }P_i^S \left( {a_i } \right) \end{aligned}$$

(21)

$$\begin{aligned} \frac{\partial V}{\partial x_i }= & {} \frac{\partial \pi _i }{\partial x_i }-\Theta \frac{\partial P^{D}}{\partial e_i }\frac{\partial P^{I}\left( {x_i } \right) }{\partial x_i }\left[ {1-P^{C}\left( {z_i } \right) } \right] P_i^S \left( {a_i } \right) \end{aligned}$$

(22)

$$\begin{aligned} \frac{\partial V}{\partial a_i }= & {} -f^{\prime }\left( {a_i } \right) -\Theta \frac{\partial P^{D}}{\partial e_i }P^{I}\left( {x_i } \right) \left[ {1-P^{C}\left( {z_i } \right) } \right] \frac{\partial P_i^S \left( {a_i } \right) }{\partial a_i } \end{aligned}$$

(23)

Essentially the same arguments made above for expressions (19) and (20) can be applied to expressions (21) and (22) to show that $x_i^*\in \left( {0,1} \right)$ and $z_i^*\in \left( {0,\bar{z} } \right)$. We therefore focus here on expression (23). When $a_i =0$, then $\frac{\partial V}{\partial a_i }=-\Theta \frac{\partial P^{D}}{\partial e_i }P^{I}\left( {x_i } \right) \left[ {1-P^{C}\left( {z_i } \right) } \right] \frac{\partial P_i^S \left( {a_i } \right) }{\partial a_i }>0$, whereas $\partial V/\partial a_i =-{f}^{\prime }(a_{i})<0$ when $a_i =1$. Hence, $a_i^*\in \left( {0,1} \right)$. $\square$

Proposition 3

The private externality, $e_i^0$, exceeds the first-best externality, $e_i^*$,when the restricted profit function $\pi _i \left( {e_i } \right) =B-c\left( {e_i } \right) -\Lambda \left( {e_i } \right) =\mathop {max}\nolimits _{x_i ,z_i ,a_i } \{B-c\left( {x_i ,z_i ,a_i } \right) -\Lambda \left( {x_i ,z_i } \right) |e_i \left( {x_i ,z_i ,a_i } \right) \le e_i \}$ is concave in $e_i$.

Define the generalized restricted profit function

$$\begin{aligned} \pi _i \left( {e_i ,\phi _i } \right)= & {} B-c\left( {e_i ,\phi _i } \right) -\phi _i \Lambda \left( {e_i ,\phi _i } \right) \nonumber \\= & {} \mathop {\max }\limits _{x_i ,z_i ,z_i } \left\{ {B-c\left( {x_i ,z_i ,a_i } \right) -\phi _i \Lambda \left( {x_i ,z_i } \right) \hbox {|}e_i \left( {x_i ,z_i ,a_i } \right) \le e_i } \right\} , \end{aligned}$$

(24)

where $\phi _i \in \left[ {0,1} \right]$ is a parameter representing the degree of filterability so that a larger $\phi _i$ implies more filterability. (Note that $\phi _i$ only plays a role in proving Proposition 4 below; we introduce it here to avoid re-deriving the restricted profit function.) The privately optimal level of $e_i$, denoted $e_i^0 \left( {\phi _i } \right)$, solves

$$\begin{aligned} \frac{\partial \pi _i }{\partial e_i }=0. \end{aligned}$$

(25)

Given (24), we can rewrite expected social net benefits as

$$\begin{aligned} V\left( {\mathbf{e},{\varvec{\upphi }} } \right) =\sum \nolimits _{i=1}^N \pi _i \left( {e_i ,\phi _i } \right) -D\left( {P^{D}\left( {\mathbf{e},{\varvec{\upphi }} } \right) } \right) , \end{aligned}$$

(26)

where ${\varvec{\upphi }}$ is a vector with ith element $\phi _i$. The first-best externality, $e_i^*\left( {\phi _{i}} \right)$, solves

$$\begin{aligned} \frac{\partial \pi _i }{\partial e_i }=\Theta \frac{\partial P^{D}}{\partial e_i }. \end{aligned}$$

(27)

Upon comparing (27) and (25), concavity of $\pi _i \left( {e_i ,\phi _i } \right)$ in $e_i$ implies that $e_i^0 \left( {\phi _i } \right) >e_i^*\left( {\phi _i } \right)$. $\square$

Proposition 4

Other things equal, the filterability of producer i’s externality reduces (i) both the privately-produced externality and the efficient externality of producer i when $\pi _i \left( {e_i } \right)$ is concave in $e_i$ and $\Lambda \left( {e_i } \right)$ is increasing $e_i$, and (ii) the difference $e_i^0 -e_i^*$ if, additionally, $\pi _i \left( {e_i } \right)$ is reasonably approximated by a quadratic relation (so that $\partial ^{2}\pi _i \left( {e_i } \right) /\left( {\partial e_i } \right) ^{2}$ is essentially constant) and $\Lambda \left( {e_i } \right)$ is convex in $e_i$.

(i) Assume $\frac{\partial ^{2}\pi _i }{\left( {\partial e_i } \right) ^{2}}<0$ and $\frac{\partial \Lambda _i }{\partial e_i }>0$. Substitute $e_i^0 \left( {\phi _i } \right)$ into (25) to create an identity, and then differentiate this identity with respect to $\upphi _i$ to obtain

$$\begin{aligned} \frac{de_i^0 }{d\phi _i }=-\frac{\partial ^{2}\pi _i^0 /\left( {\partial e_i \partial \phi _i } \right) }{\partial ^{2}\pi _i^0 /\left( {\partial e_i } \right) ^{2}}=\frac{\partial \Lambda _i^0 /\partial e_i }{\partial ^{2}\pi _i^0 /\left( {\partial e_i } \right) ^{2}}<0, \end{aligned}$$

(28)

where the second equality in (28) is due to the envelope theorem: $\partial \pi \left( {e_i ,\phi _i } \right) /\partial \phi _i =-\Lambda \left( {e_i ,\phi _i } \right)$. Expression (28) indicates that greater filterability reduces the privately optimal externality.

Now consider the efficient level of the externality. Recall that the RHS of (27) is independent of $e_i$, as $P^{D}$ is linear in $e_i$. Substitute $e_i^*\left( {\phi _i } \right)$ into (27) to create an identity, and then differentiate this identity with respect to $\phi _i$ to obtain

$$\begin{aligned} \frac{de_i^*}{d\phi _i }=-\frac{\partial ^{2}\pi _i^*/\left( {\partial e_i \partial \phi _i } \right) }{\partial ^{2}\pi _i^*/\left( {\partial e_i } \right) ^{2}}=\frac{\partial \Lambda _i^*/\partial e_i }{\partial ^{2}\pi _i^*/\left( {\partial e_i } \right) ^{2}}<0. \end{aligned}$$

(29)

Expression (29) indicates greater filterability reduces the efficient value of the externality.

(ii) We extend the assumptions for part (i) by taking $\frac{\partial ^{2}\pi _i }{\left( {\partial e_i } \right) ^{2}}$ to be essentially constant and $\frac{\partial ^{2}\Lambda _i }{\left( {\partial e_i } \right) ^{2}}>0$. We want to show that the difference $e_i^0 \left( {\phi _i } \right) -e_i^*\left( {\phi _i } \right)$ is decreasing in $\phi _i$. We therefore derive

$$\begin{aligned} \frac{\partial \left( {e_i^0 \left( {\phi _i } \right) -e_i^*\left( {\phi _i } \right) } \right) }{\partial \phi _i }=\frac{\partial \Lambda _i^0 /\partial e_i }{\partial ^{2}\pi _i /\left( {\partial e_i } \right) ^{2}}-\frac{\partial \Lambda _i^*/\partial e_i }{\partial ^{2}\pi _i /\left( {\partial e_i } \right) ^{2}}=\frac{\partial \Lambda _i^0 /\partial e_i -\partial \Lambda _i^*/\partial e_i }{\partial ^{2}\pi _i /\left( {\partial e_i } \right) ^{2}}, \end{aligned}$$

(30)

where the second equality stems from our assumption that $\partial ^{2}\pi _i /\partial e_i^2$ is constant (independent of $e_i )$. Given our result in part (i) that $e_i^0 \left( {\phi _i } \right) >e_i^*\left( {\phi _i } \right)$, combined with convexity of $\Lambda _i$, implies the numerator of the RHS of (30) is positive. As the denominator of the RHS of (30) is negative due to concavity of $\pi _i$, this means (30) is negative. $\square$