An Economic Analysis of Earth Orbit Pollution

Abstract

Space debris, an externality generated by expended launch vehicles and damaged satellites, reduces the expected value of space activities by increasing the probability of damaging existing satellites or other space vehicles. Unlike terrestrial pollution, debris created in the production process interacts with firms’ final products, and is, moreover, self-propagating: collisions between debris or extant satellites creates additional debris. We construct a formal model to explore private incentives to launch satellites and to mitigate space debris. The model predicts that, relative to the social optimum, firms launch too many satellites and choose technologies which create more debris than is socially optimal. We discuss remediation strategies and policies, and demonstrate that Pigovian taxes can be used to internalize the debris externality.

Appendices

Appendix 1: Proof of Proposition 1

$$\begin{aligned}&\displaystyle \frac{\partial L_{Com}}{\partial t} = \frac{1}{2t} \sqrt{\frac{(\beta + (1 - k \phi ))}{(1 - k \phi )(r + F + \beta (1-k \phi ) F)}} > 0 \end{aligned}$$

(17)

$$\begin{aligned}&\displaystyle \frac{\partial L_{Com}}{\partial \beta } = \frac{1}{2\sqrt{L_{Com}}} \cdot \frac{t}{(1 - k \phi )} \cdot \frac{r + F k \phi (2 - k \phi )}{(r + F + \beta (1-k \phi ) F)^2} > 0\end{aligned}$$

(18)

$$\begin{aligned}&\displaystyle \frac{\partial L_{Com}}{\partial \phi } = \frac{1}{2\sqrt{L_{Com}}} \cdot \frac{k \beta t (r + F + 2 \beta F (1 - k \phi ) + F (1- k \phi )^2) }{[(1 - k \phi )(r + F + \beta (1-k \phi ) F)]^2} > 0\end{aligned}$$

(19)

$$\begin{aligned}&\displaystyle \frac{\partial L_{Com}}{\partial k} = \frac{\partial L_{Com}}{\partial \phi } > 0 \end{aligned}$$

(20)

$$\begin{aligned}&\displaystyle \frac{\partial L_{Com}}{\partial r} = \frac{1}{2\sqrt{L_{Com}}} \cdot \frac{- (1 - k \phi )(\beta + (1 - k \phi ))t}{[(1 - k \phi )(r + F + \beta (1-k \phi ) F)]^2} < 0\end{aligned}$$

(21)

$$\begin{aligned}&\displaystyle \frac{\partial L_{Com}}{\partial F} = \frac{1}{2\sqrt{L_{Com}}} \cdot \frac{- (1 - k \phi )(1 + \beta (1 - k \phi ))(\beta + (1-k \phi ))t}{[(1 - k \phi )(r + F + \beta (1-k \phi ) F)]^2} < 0 \end{aligned}$$

(22)

Appendix 2: Proof of Corollary 1

$\frac{\partial L_{soc}}{\partial t} = \frac{1}{2} \frac{\partial L_{Com}}{\partial t}> 0,\,\frac{\partial L_{soc}}{\partial \beta } = \frac{1}{2} \frac{\partial L_{Com}}{\partial \beta } \!>\!0,\,\frac{\partial L_{soc}}{\partial \phi } = \frac{1}{2} \frac{\partial L_{Com}}{\partial \phi } > 0,\,\frac{\partial L_{soc}}{\partial k} \!=\! \frac{1}{2} \frac{\partial L_{Com}}{\partial k} > 0,\frac{\partial L_{soc}}{\partial r} \!=\! \frac{1}{2} \frac{\partial L_{Com}}{\partial r} < 0$, and $\frac{\partial L_{soc}}{\partial F} = \frac{1}{2} \frac{\partial L_{Com}}{\partial F} < 0$.

Appendix 3: Proof of Proposition 2

Comparing $L_{Com}$ and $L_{soc}$ yields $L_{soc} = \frac{1}{2} L_{Com} < L_{Com}$.

Appendix 4: Proof of Proposition 3

Differentiating $\pi _i$ defined in (13), with respect to $\phi _i$ yields:

$$\begin{aligned} \frac{\partial \pi _i}{\partial \phi _i}&= - h'(\phi _i) - \frac{\beta k}{L} \left( \frac{t}{((1- k \bar{\phi })L)^2} - F\right) + \frac{2 k \beta t}{L^3 (1 - k \bar{\phi })^2} \end{aligned}$$

(23)

$$\begin{aligned}&= - h'(\phi _i) + \frac{ k \beta t}{(1 - k \bar{\phi })^2 L^3} + \frac{\beta k F}{L} > 0 \end{aligned}$$

(24)

This implies that firm $i's$ profit is strictly increasing in $\phi$. Therefore, it is optimal for each firm $i$ to choose $\phi _i = \phi _H$ regardless of what other firms choose. This completes the proof.

Appendix 5: Proof of Proposition 4

First, we note that the problem defined in (15) has a solution because the value function $V$ is continuous in $\phi$ on its closed and bounded domain. Differentiating $V$ with respect to $\phi$, while noting that $\frac{\partial V}{\partial L} = 0$, yields:

$$\begin{aligned} \frac{d V}{d \phi } = L(\phi ) h'(\phi ) + \frac{\beta k}{4 L(\phi ) (1 - k \phi )} (t - 4 F (1 - k \phi ) (L(\phi ))^2) \end{aligned}$$

(25)

To prove that $\phi _{soc} < \phi _H$, it suffices to show that $\frac{d V}{d \phi } |_{\phi _H} > 0$. Evaluating (25) at $\phi _H$ yields $\frac{d V}{d \phi } |_{\phi _H} = \frac{\beta k}{4 L(\phi _H) (1 - k \phi _H)} (t - 4 F (1 - k \phi _H) (L(\phi _H))^2)$. This expression is positive if $t - 4 F (1 - k \phi _H) (L(\phi _H))^2 > 0$.

$$\begin{aligned}&t - 4 F (1 - k \phi _H) (L(\phi _H))^2 \\&\quad =t - 4 F (1 - k \phi _H) \frac{(\beta + (1 - k \phi _H)) t}{4 (1- k \phi _H) (h(\phi _H) + F + \beta (1 - k \phi _H) F)} \\&\quad =\frac{(h(\phi _H) + (1 - \beta ) k \phi _H F) t}{h(\phi _H) + F + \beta (1 - k \phi _H) F} > 0 \end{aligned}$$

Next, we compare the launch rates. According to Corollary 1, $L_{soc} (\phi _{soc}) < L_{soc} (\phi _{H})$ because $\phi _{soc} < \phi _H$. Furthermore, according to Proposition 2, $L_{soc} (\phi _{H}) < L_{Com} (\phi _{H})$. Therefore, $L_{soc} (\phi _{soc}) < L_{Com} (\phi _{H})$. This completes the proof.

Appendix 6: Proof of Proposition 5

Suppose the regulator selects a variable tax rate $\gamma ^*(\phi ) = \{ \gamma (\phi ) = 0$ if $\phi = \phi _{soc},\,\gamma (\phi ) = s$ if $\phi \ne \phi _{soc} \}$. Because $s$ is the maximum consumer surplus value for the market (as introduced in Sect. 2.1.1), $\phi = \phi _{soc}$ is the only choice for firms that could yield non-negative profits. Thus, any firm that chooses to launch a satellite would select $\phi = \phi _{soc}$. Next, suppose the regulator sets a per launch tax rate $w^* = 3(h(\phi _{soc}) + F + \beta (1 - k \phi _{soc}) F )$. Then, the tax schedule equals $T^* = 3(h(\phi _{soc}) + F + \beta (1 - k \phi _{soc}) F ) + \gamma ^*(\phi )$. Equation (5) from Sect. 2.2 describes the number of launches in the competitive market for a given $\phi$. Then, under the tax schedule $T^*$, the number of launches in the competitive market would equal:

$$\begin{aligned} L_{Com}&= \sqrt{\frac{(\beta + (1 - k \phi _{soc}))t}{(1 - k \phi _{soc})(h(\phi _{soc}) + T^* + F + \beta (1-k \phi _{soc}) F)}} \\&= \sqrt{\frac{(\beta + (1 - k \phi _{soc}))t}{(1 - k \phi _{soc}) (4) (h(\phi _{soc}) + F + \beta (1-k \phi _{soc}) F)}}\\&= \frac{1}{2} \sqrt{\frac{(\beta + (1 - k \phi _{soc}))t}{(1 - k \phi _{soc})(h(\phi _{soc}) + F + \beta (1-k \phi _{soc}) F)}} = L_{soc} \end{aligned}$$

Thus, $L_{Com} = L_{soc}$ and $\phi _{com} = \phi _{soc}$ under the tax schedule $T^*$.