# International cooperation and competition in orbit-use management

Aditya Jain
Becker Friedman Institute for Economics  Akhil Rao
Middlebury College Department of Economics, Warner Hall, Middlebury College,
05753<EMAIL_ADDRESS>
(This version: September 10, 2022)

###### Abstract

Orbit-use management efforts can be structured as binding national regulatory
policies or as self-enforcing international treaties. New treaties to control
space debris growth appear unlikely in the near future. Spacefaring nations
can pursue national regulatory policies, though regulatory competition and
open access to orbit make their effectiveness unclear. We develop a game-
theoretic model of national regulatory policies and self-enforcing
international treaties for orbit-use management in the face of open access,
regulatory competition, and catastrophe. While open access limits the
effectiveness of national policies, market-access control ensures the policies
can improve environmental quality. A large enough stock of legacy debris
ensures existence of a global regulatory equilibrium where all nations choose
to levy environmental regulations on all satellites. The global regulatory
equilibrium supports a self-enforcing treaty to avert catastrophe by making it
costlier to leave the treaty and free ride.

JEL codes: F53, H77, Q53, Q54, Q58

Keywords: open-access commons, satellites, space debris, international treaty,
regulatory competition

##  1 Introduction

Orbital space is a critical resource for global economic development. Indeed,
a recent assessment found that satellite services make significant
contributions to 13 of 17 UN Sustainable Development Goals (UNOOSA, 2018). Yet
orbital space is a global commons under an open access regime. International
law limits the degree to which orbit users can exclude each other from
valuable orbital slots, while the laws of physics prevent multiple objects
from occupying the same slots. Objects attempting to occupy the same slots
will collide, creating long-lived and fast-moving debris which further
increases collision risk. As debris proliferates, collisions between debris
objects may cross a tipping point beyond which debris growth becomes self-
sustaining, a condition known as “Kessler Syndrome”. While the relevant
physical dynamics operate over years and decades, crossing the tipping point
would be a global catastrophe which would severely limit human use of outer
space.

How can orbital space be managed through binding policies at the national or
international levels? Are there trade-offs between national and international
approaches? To what extent does the risk of Kessler Syndrome support or hinder
these policy approaches? And how does open access to orbital space—a
fundamental constraint imposed by the Outer Space Treaty (OST), a foundational
pillar of international space law—affect the prospects for national or
international orbital-use management? This paper addresses these questions
through a game-theoretic model of rational orbit use. While components of
these questions have been studied separately, there is relatively little work
exploring the interactions between national and international orbital-use
management approaches given the relevant physical and institutional
constraints. Understanding these interactions is critical to designing
effective orbital-use management policies and predicting policy
outcomes.111Specifically, Articles I, II, and VI of the OST specify free
access to outer space (I), non-appropriation of outer space (II), and national
supervision of space activities by non-governmental entities (VI). Articles I
and II render orbital space an open-access commons, while Article VI enables
national satellite regulatory policies.

Our paper connects to three literatures in economics and environmental
management. First, we extend the literature on orbital-use management. Prior
work in this area has established several stylized facts: the inefficiency of
decentralized orbit use (Adilov, Alexander and Cunningham, 2015; Rouillon,
2020); the importance of tipping points and decentralized rational actors’
inability to avoid crossing them (Lewis, 2020; Rao and Rondina, 2022); and the
potential for satellite tax-like policies to efficiently limit collision risk
and debris accumulation while increasing the value generated by satellites
(Rao, 2019; Rao, Burgess and Kaffine, 2020; Béal, Deschamps and Moulin, 2020).
However, this literature has not yet considered how management policies can be
structured given international open access and national competition for
satellite services, and with the exception of Singer and Musacchio (2011) it
has largely ignored the potential of debris abatement treaties.

Second, we contribute to the broader literature about environmental federalism
and decentralized commons management. Prior work in this area has established
conditions under which regulatory competition between jurisdictions can
effectively manage environmental quality (Oates, 1999; Millimet, 2013),
particularly in the face of limited regulatory control over spatially-mobile
resources and public bads (Costello, Quérou and Tomini, 2015, 2017). This
literature has largely focused on cases where resource users and public bads
are spatially separated and there are no catastrophic thresholds, and has not
studied the interactions between national policies and international treaties.
Orbital space is novel in that “partial enclosure of the commons” (i.e.
limiting access to a specific physical space to prevent resource collapse) is
physically and legally infeasible.222The notion of commons enclosure relevant
to orbital space is better described as “limited enclosure”. In the fisheries
context of Costello, Quérou and Tomini (2015), “limited enclosure” would
involve one nation preventing boats under their flag from fishing in the high
seas. Such attempts are unlikely to be as effective in preserving marine
populations as limiting access to the high seas by all nations, since boats
could fly flags of convenience to circumvent the regulations and sell their
catch at the world market price. However, a key difference between orbit use
and many other terrestrial natural resources is that satellite services
typically can’t be resold like fish or physical goods, giving nations an
additional locus of control unavailable in most other resource contexts..

Third, we contribute to the literature on self-enforcing international
environmental treaties. Prior work in this area has considered how uncertainty
over a catastrophe threshold affects treaty formation (Barrett, 2013; Barrett
and Dannenberg, 2014) and even how a space debris abatement treaty could be
structured (Singer and Musacchio, 2011). However, the literature has not yet
considered pollutant abatement treaties in the face of catastrophe, national
policies, and open access to the polluted commons by profit-maximizing firms.

In this paper we build a game-theoretic model of orbit use by commercial
satellite operators and spacefaring nations. Our model features profit-
maximizing firms competing to provide satellite services under open access,
debris accumulation and collision damages with a catastrophe threshold,
national efforts to attract satellite services and regulate them, and national
cooperation to abate debris and avert catastrophe. Our model enables us to
identify three key features about orbital-use management through national
policies and international treaties. First, despite satellite operators
migrating to nations with less-stringent policies, national policies can
improve all nations’ welfare from satellite services and orbital environmental
quality. Open access therefore will not render national regulatory policies
fruitless. Second, if the legacy debris stock is large enough, a global
regulatory equilibrium can exist where all nations impose regulations on their
own and each others’ satellite fleets without transfers of surplus between
nations. Third, by altering the marginal benefits of debris abatement,
national regulatory policies can strengthen a self-enforcing treaty to prevent
Kessler Syndrome.

Our results bring together several threads from the existing literature on
commons governance, both in general and in the space debris context. Weeden
and Chow (2012) draw on Ostrom’s principles for commons governance to suggest
that a nested system of governance may be a successful approach to orbital-use
management. Johnson-Freese and Weeden (2012) express reservations about the
potential for actually implementing such an overlapping system of orbital
governance. Wouters, De Man and Hansen (2016) argue that a new debris
abatement treaty is unlikely in the near future and instead focus on policy
actions European nations can take to manage orbit use. Similarly, Migaud
(2020) and Gilbert and Vidaurri (2021) focus on actions the US can take to
manage orbit use using only existing national policy authorities. Muñoz-
Patchen (2018) points to OST interpretations which support treating debris as
salvage to facilitate debris removal. We offer guidance on implementing these
ideas while accounting for the responses of satellite operators and other
nations, as well as indications of how they may support further improvements
in orbital-use management. Our results are also informative about other global
common-pool resources subject to global open access, environmental tipping
points, and a mix of national and international management practices. These
include the atmosphere, the North Pole, and the lunar surface.

This paper is organized as follows. In section 2 we describe the model and
present key results. We begin with interactions between firms and the physical
environment, then consider national incentives and regulations, and finally
explain the international treaty for net debris abatement. Throughout we
provide results illustrating properties of each component of the system. We
discuss the significance of our results, their connections to existing
literatures, and their relation to unmodeled national security issues in
section 3. We conclude in section 4. All proofs are shown in the Appendix.

##  2 Model and results

We focus on long-run orbit use by competitive satellite operators.333“Long-
run” in this context should not be interpreted as “steady state”, as we allow
for tipping points to be crossed and damages to accrue. Instead, it should be
interpreted as a period sufficiently far in the future that we can consider
the present value of accumulated streams of benefits and costs. Similar
approaches are taken in Weitzman (1998) and DeLong et al. (2012), where
catastrophe, long-run damages or output gaps, and policies to mitigate them
feature prominently. As such, we abstract from the details of orbital-use
dynamics and focus on two properties of orbit use established in Somma, Lewis
and Colombo (2019), Lewis (2020), and Rao and Rondina (2022). First, the long-
run collision risk is a function of the long-run launch rate, with higher
launch rates implying greater collision risk. We therefore treat collision
risk to satellites as an increasing function of the debris stock and the
debris stock as an increasing function of the number of satellites maintained
in orbit. Larger satellite stocks imply greater collision-related damages.
Second, there is a threshold level of debris beyond which debris growth
becomes self-sustaining. Such debris growth can eventually render a region of
orbital space unusable on the order of years or decades, and crossing the
threshold will not immediately stop rational economic agents from launching
satellites. We therefore model the long-run damages from crossing the
threshold as a lumpsum value incurred immediately, reflecting the present
discounted value of a stream of damages which slowly increase over
time.444Following Weitzman (1998), we assume these streams are discounted at
the lowest possible rate.

Operators provide a homogeneous output, e.g. telecommunications, and face a
constant exogenous price in each nation in which they operate.555While
telecommunications products in reality are differentiated by factors like
bandwidth and coverage, these features are unlikely to change our key messages
regarding open access, regulations, and treaties. The constant price
assumption reflects the existence of a terrestrial substitute good and a
competitive market, both limiting satellite operators’ ability to set prices.
These prices vary across nations to reflect their different market sizes and
attractiveness to satellite operators. Operators can use the same satellite to
provide services in multiple nations. The lack of excludability over orbital
slots creates an open-access problem, wherein operators launch satellites
until the marginal operator receives zero profits (Adilov, Alexander and
Cunningham, 2015; Rouillon, 2020; Rao, Burgess and Kaffine, 2020).666Paths in
low-Earth orbit are dynamic objects in 3-dimensional space, requiring 6
parameters—three positions and three velocities—to completely specify. Despite
this, they can be characterized as “slots” in a higher-dimensional space
through appropriate coordinate transformations (Arnas et al., 2021). In
addition to prices for satellite services, nations are also distinguished by
heterogeneous costs of launching and operating satellites from their
territories (e.g. due to different launch technology or labor force
availabilities).

Nations seek to regulate satellite operators in order to maximize the net
benefits they receive from satellite services. These regulations are imposed
as conditions for market access, i.e. the right to transmit and receive
signals (more generally, sell and purchase products, e.g. satellite imagery)
to and from customers located within a nation’s jurisdiction. We model binding
national regulatory policies as “virtual satellite taxes”, reflecting that any
policy which alters firms’ behavior does so by imposing costs on the firms.
These taxes can therefore include policies such as regulatory barriers (e.g.
compliance costs due to arms control or environmental regulations) or explicit
taxes which generate revenues (e.g. orbital-use fees as described in Rao,
Burgess and Kaffine (2020)).777While policies regulating orbiting satellites
and satellite launches are equivalent in long-run models, their dynamics vary
considerably, with satellite-focused policies being more efficient than
launch-focused policies (Rao, 2019). We use the “satellite tax” terminology to
avoid short-run dynamic inefficiency issues. We assume these taxes are
“bilateral” between nations, i.e. levied by one nation on all satellites
operated under another nation’s jurisdiction, rather than differentiated by
individual operators.888Operator-specific taxes would add more complexity
without much additional insight regarding the questions considered here.
Nations receive two types of benefits from implementing regulatory policies.
First, by reducing the total stock of satellites in orbit, satellite operators
incur fewer collision-related satellite losses. Nations then receive more
satellite services as existing satellites are destroyed less frequently.
Second, to the extent that these taxes generate revenues, nations can use
those revenues to provide public goods or reduce distortionary taxes. We
assume that the taxes generate no revenues so as to keep the focus solely on
benefits received through environmental management.

Finally, nations also negotiate a treaty for net debris abatement, and choose
whether or not to be party to the treaty. We depart from the setting
considered in Singer and Musacchio (2011) by microfounding the national
benefits from debris abatement, incorporating open-access behavior by
satellite operators, and introducing the tipping point for Kessler Syndrome.
While we derive the national benefits of debris abatement from the national
net market benefits from satellite services, it is straightforward to include
additional terms reflecting non-market economic benefits from satellite
services.999Incorporating strategic interactions between nations, e.g. due to
national security or dual-use concerns, is less straightforward. Such
interactions introduce technical challenges relating to interdependent
preferences. We briefly discuss this issue in section 3 but leave a full
modeling treatment for future research.

###  2.1 Satellites, debris, and catastrophe

Identical satellites can provide services to any of the $N$ nations (markets)
in the world. There are $n_{s}\leq N$ nations from which satellites are
operated. We refer to these as spacefaring nations (even if they do not
possess their own launch capabilities). The satellite sector in spacefaring
nation $i$ controls $S_{i}$ satellites. Launching and maintaining these
satellites costs $m_{i}S_{i}^{2}$, where $m_{i}$ is a sector efficiency
parameter reflecting the cost-minimizing mix of launch services, capital, and
labor supply available to satellite operators based in nation $i$.101010Firms
may procure inputs internationally, in which case $m_{i}$ will also reflect
exchange rates and relevant international regulations facing the satellite
sector in nation $i$. The price of a unit of satellite service in nation $j$
is a constant $p_{j}$, reflecting the size of the market for satellite
services in nation $j$.111111Constant prices approximate the case for
satellite services with terrestrial substitutes, e.g. broadband. Subscripted
dots indicate arrays over the omitted index, e.g.
$p_{\cdot}=(p_{1},\dots,p_{N})$, while negative indices indicate sums over all
entries except the index shown, e.g. $S_{-i}=\sum_{i^{\prime}\neq
i}S_{i^{\prime}}$.

Each active satellite produces $d$ units of debris over its lifecycle. The
long-run stock of debris in orbit, $D$, is determined by the long-run total
satellite stock, $S=\sum_{i=1}^{n_{s}}S_{i}$, the “legacy” stock of debris
unrelated to ongoing orbit use by active satellites $D_{0}$, and net debris
abatement by all nations $Q$, which is defined relative to the benchmark level
of debris due to ongoing orbit use and the legacy debris stock, $dS+D_{0}$.
The long-run debris stock is shown in equation 1:

$D=dS+D_{0}-Q.$ (1)

Debris abatement $Q$ contains two components: reduction of debris produced by
satellites and debris removal activities. It is sometimes useful when
discussing debris abatement (particularly in section 2.3) to express the
Kessler Syndrome threshold $\bar{D}$ in terms of the debris abatement required
to avert it, $\bar{Q}$. The catastrophe-averting level of abatement,
$\bar{Q}$, can be derived from equation 1 as follows:

$\displaystyle D>\bar{D}\implies$ $\displaystyle dS+D_{0}-Q>\bar{D}$
$\displaystyle\implies$ $\displaystyle Q<dS+D_{0}-\bar{D}$
$\displaystyle\implies$ $\displaystyle\bar{Q}=dS+D_{0}-\bar{D}.$ (2)

Debris damages active satellites. The probability a satellite survives in
orbit given debris stock $D$ is shown in equation 3:

$\displaystyle Pr(\text{survive})$ $\displaystyle=1-kD,$ (3) $\displaystyle
k:Pr(\text{survive})$ $\displaystyle\in[0,1].$ (4)

$k$ is the probability of collision per unit of debris, and condition 4 is a
physical restriction imposed by this interpretation. If debris levels exceed a
threshold $\bar{D}$, self-sustaining debris growth occurs. This is a slow-
moving catastrophe: over many years, collisions between debris generate more
debris, and eventually that region of orbital space is rendered unusable.
Satellite sectors incur common long-run damages of $X$ from crossing the
threshold. There may be years or decades after the threshold is crossed during
which satellite sectors continue to accrue profits before collision risk
becomes large enough to make satellite operation unprofitable.121212Estimates
of damages to modern economies from the loss of satellite services suggest the
damages parameter from Kessler Syndrome in valuable orbits can be large
(Bradley and Wein, 2009; Schaub et al., 2015; O’Connor et al., 2019), though
many relevant structural parameters (e.g. substitution possibilities, indirect
and induced demand effects) are uncertain and may reduce the damages (Van Camp
and Peeters, 2021; Highfill and MacDonald, 2022). Given the potential for
large damages and “tail-fattening” effects of structural parameter uncertainty
(Weitzman, 2009), we ignore idiosyncratic national variation in the damages
parameter. Since our focus is on long-run outcomes we also abstract from the
dynamics of debris growth and focus on whether or not the threshold is
crossed. Following Rao and Rondina (2022), we assume satellite operators do
not internalize the costs of crossing the self-sustaining debris growth
threshold.

Each nation receiving satellite services can impose regulatory burdens on
satellite sectors. These include radio spectrum licensing procedures, explicit
Pigouvian taxes on debris creation, satellite design requirements, and any
other costly condition for market access. These regulatory burdens are
satellite taxes, $\tau_{ij}$, imposed by market $j$ on sector $i$.131313Rao
(2019) shows that the optimal instrument in a dynamic context where objects
can be deorbited and both active satellites and debris can induce collisions
is a “stock control”: a tax levied on all objects in orbit. Rao, Burgess and
Kaffine (2020) describe these controls as “orbital-use fees”, since they are
in effect fees for using orbital space. Guyot and Rouillon (2021) show that
orbital-use fees can be decomposed into separate streams of fees levied on
launch debris and satellites at different stages of their lifecycle. In our
setting, where we abstract from dynamics, make the debris stock linearly
related to the satellite stock, and only allow debris to cause collisions,
orbital-use fees are equivalent to satellite taxes. Satellite taxes reduce
service delivery from the taxed sector to the taxing market, so the quantity
of satellite services received in market $j$ is $\sum_{i}(1-\tau_{ij})S_{i}$.
Denial of satellite sector $i$’s access to market $j$ implies $\tau_{ij}=1$.
The net profits accrued by sector $i$, $Y_{i}$, are shown in equation 5.

$\displaystyle Y_{i}$
$\displaystyle=(1-kD)\sum_{j=1}^{N}p_{j}(1-\tau_{ij})S_{i}-m_{i}S_{i}^{2}.$
(5)

The inclusion of collision risk in front of the satellite stock in equation 5
reflects the long-run character of the model: of the $S_{i}$ satellites
launched (reflected in cost term $m_{i}S_{i}^{2}$), a fraction $1-kD$ will be
destroyed in collisions and therefore unable to deliver services.141414We
focus on the costs of destructive collisions in this paper, but collision-
avoidance maneuvers create another type of cost with similar effects. We
abstract from these costs since they would introduce additional complexity
without changing our results. We use this long-run property to abstract from
the dynamics of launch and destruction. The size of satellite sector $S_{i}$
is determined by an open-access condition, shown in equation 6:

$S_{i}^{\star}(S_{-i},\tau_{i\cdot},Q):Y_{i}=0.$ (6)

Equation 6 reflects the fact that satellite operators in each nation are
unable to secure exclusive property rights to orbital space, and therefore
launch satellites until there are no further profits to doing so. Solving
equation 6 yields an expression for the size of satellite sector $i$ as a
function of the satellite taxes $\tau_{i\cdot}$, the total size of all other
satellite sectors $S_{-i}$, and the global level of debris abatement
$Q$.151515Since equation 6 is quadratic in $S_{i}$ with no intercept, there
are two roots: one where $S_{i}=0$ and one where $S_{i}>0$. We ignore the root
where $S_{i}=0$. The expression is shown in equation 7:

$S_{i}^{\star}(S_{-i},\tau_{i\cdot},Q)=\frac{\sum_{j}p_{j}(1-\tau_{ij})(1-k(dS_{-i}+D_{0}-Q))}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}.$
(7)

The congestibility of orbital space suggests satellite sectors will compete
with each other for physical space in orbit, i.e. national satellite fleets
will be strategic substitutes. Differentiating equation 7 for arbitrary
sectors $i$ and $j\neq i$ reveals this to be the case:

$\partialderivative{S^{\star}_{i}}{S_{j}}=-\frac{kdp_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}<0.$
(8)

This presents a challenge to national regulatory approaches, as some of the
fleet size reductions achieved by one sector will be partially offset by fleet
size increases from others. This does not imply that reductions by sector $j$
have no effect on the total fleet size—as long as sector $i$ faces positive
launch costs they will not fully offset $j$’s reduction
($m_{i}>0\implies\partialderivative{S^{\star}_{i}}{S_{j}}>-1$).

A global open-access equilibrium, formalized in Definition 1, is a
distribution of satellites such that every satellite sector earns zero
profits—a fixed point of all $i$ best-response functions in equation 7.

######  Definition 1.

(Global open-access equilibrium) A global open-access equilibrium is a
distribution of satellites $\\{S_{i}^{\star}\\}$ such that, given a debris
abatement level $Q$, a legacy debris stock $D_{0}$, distribution of satellite
taxes $\\{\tau_{ij}\\}$, market sizes $\\{p_{j}\\}$, sector efficiencies
$\\{m_{i}\\}$, and technologies $(k,d)$, equation 6 holds for all $i$.

Lemma 1 establishes existence and a characterization of the global open-access
equilibrium with an arbitrary number of nations. Lemma 2 establishes that as
long as the equilibrium to the $n_{s}$-player game exists, there exists a
$2$-player game such that the equilibrium satellite stock for sector $i$ is
identical to $S^{*}_{i}$ in the $n_{s}$-player game, with the satellite fleet
of the remaining nations being identical to $S^{*}_{j}$. This reduction allows
us to use a $2$-player game to establish properties of the $n_{s}$-player
game. Intuitively, the reduction is possible because any firm in nation $i$
only needs to know how many satellites are launched from other firms in nation
$i$ (with whom it competes for satellite launch and operations resources as
well as orbital space) and those launched from all other sectors (with whom it
competes for orbital space).161616That is, Lemma 2 establishes that economic
competition for orbital space has a form of “diffuse reciprocity” (Keohane,
1986).

######  Lemma 1 (Global open-access equilibrium).

Let $S$ be the $n_{s}\times 1$ vector of satellite sector sizes and $I$ be the
$n_{s}\times n_{s}$ identity matrix. Let $A$ be the $n_{s}\times 1$ vector
with typical elements

$A_{i}=\frac{\sum_{j}p_{j}(1-\tau_{ij})(1+k(Q-D_{0}))}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}$
(9)

and $B$ be the $n_{s}\times n_{s}$ matrix with $0$ on the main diagonal and
rows containing copies of

$B_{i}=-\frac{kd\sum_{j}p_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}.$
(10)

Then, if $det(I-B)\neq 0$, the global open-access equilibrium exists and is

$S=(I-B)^{-1}A.$ (11)

######  Lemma 2 (Reduction).

For every $n_{s}$-player global open-access equilibrium $S^{\star}$, there
exists a $2$-player global open-access equilibrium $S^{\star,2}$ with players
$i$ and $j$ such that

$\displaystyle S_{i}^{\star}$ $\displaystyle=S_{i}^{\star,2},$ (12)
$\displaystyle S_{-i}^{\star}$ $\displaystyle=S_{j}^{\star,2}.$ (13)

The equilibrium satellite distribution with two spacefaring nations is shown
in equation 14:

$(S_{i}^{\star},S_{j}^{\star})=\left(\sigma_{i}r_{i},~{}\sigma_{j}r_{j}\right),$
(14)

where

$\displaystyle\sigma_{i}$
$\displaystyle\equiv\frac{(1+k(Q-D_{0}))(1-kdr_{j})}{kd\sum_{j=1}^{n_{s}}p_{j}(1-\tau_{ij})+m_{i}},$
(15) $\displaystyle r_{i}$
$\displaystyle\equiv\frac{\sum_{j=1}^{n_{s}}p_{j}(1-\tau_{ij})}{kd\sum_{j=1}^{n_{s}}p_{j}(1-\tau_{ij})+m_{i}}.$
(16)

$\sigma_{i}$ reflects the benefit-cost ratio of marginal debris abatement—the
two terms in the numerator reflect the effects of abatement itself
($1+k(Q-D_{0})$) and the effects of additional collision risk due to others
($1-kdr_{j}$). $\sigma_{i}$ has percentage units. $r_{i}$ reflects the
benefit-cost ratio of operating another satellite, accounting for the costs of
forgone revenues due to collision risk
($kd\sum_{j=1}^{n_{s}}p_{j}(1-\tau_{ij})$). It has units of satellites, and
can be interpreted as the maximum economically-sustainable satellite sector
size for nation $i$. Lemma 3 establishes that the decomposition in equation 14
is valid even in the $n_{s}$-player case.

######  Lemma 3 (Decomposition).

The global open-access equilibrium satellite distribution can be decomposed
into $S=\sigma\odot r$, where $\sigma$ and $r$ are $n_{s}\times 1$ vectors,
$\odot$ is the Hadamard product, and only $\sigma$ depends on $Q$.

As long as $r_{i},r_{j}<(kd)^{-1}$, then $\sigma_{i},\sigma_{j}>0$. The global
open-access equilibrium in this case exists as long as
$r_{i}r_{j}\neq(kd)^{-2}$. Since $m_{i},m_{j}>0\implies r_{i},r_{j}<1$, both
requirements can be satisfied when $m_{i},m_{j}>0$ and $k$ is small enough.
These conditions are important for understanding the intuition of the results
that follow, so we describe them briefly. $kd$ is the probability of satellite
destruction from the lifetime debris created by a single new satellite, and
has units of probability per satellite. The condition $r_{i}<(kd)^{-1}$
therefore states that the long-run maximum economically-sustainable satellite
fleet maintainable by nation $i$ be less than the number of satellites
destroyed per incremental satellite launch. If this condition were violated
for any nation $i$ then the open-access equilibrium fleet size for nation $j$
would be zero. To see this, note that

$\displaystyle r_{i}\geq(kd)^{-1}\implies$ $\displaystyle 0\geq 1-kdr_{i}$
(17) $\displaystyle\implies$ $\displaystyle\sigma_{j}=0,S^{*}_{j}=0.$ (18)

Imposing $r_{i}<(kd)^{-1}$ for all $i$ therefore implies that the economic net
benefits of orbit use for all national satellite sectors are not so large that
they will naturally (i.e. due to purely-economic considerations) crowd all
other nations out of orbital space. If there were a nation which crowded all
others out in this fashion, the questions of this paper would reduce to
questions of regulatory policy within that nation. Such questions are
addressed in the existing economic literature, e.g. Rouillon (2020); Rao,
Burgess and Kaffine (2020), so we impose this condition here. We state this
formally in Assumption 1.

######  Assumption 1.

The economic net benefits of orbit use for all national satellite sectors are
not so large that any nation will crowd all other nations out of orbital space
due to purely economic considerations, i.e.

$r_{i}<(kd)^{-1}~{}\forall i.$

While not necessarily dispositive, evidence for Assumption 1 being satisfied
can be seen in that commercial satellite fleets are operated from multiple
nations.171717For example, Planet (a US company) and ICEYE (a Finnish company)
both operate remote sensing satellite fleets, albeit of different sizes.

We establish three key results about open-access orbit use under national
regulations and international agreements in Propositions 1, 2, and 3. First,
taxes targeted at a particular fleet are effective at reducing the targeted
satellite fleet size, but do so at the cost of increasing the size of other
satellite fleets—an “open-access tax rebound effect”. This rebound effect
follows from equation 8: satellite fleets compete with each other for rents
from using orbital space, and one fleet’s withdrawal releases rents for others
to claim. This rebound effect can be interpreted as international capital
mobility from more-taxed to less-taxed jurisdictions in the spirit of Oates
(1999) or simply as foreign sectoral expansion. Under mild conditions the
open-access tax rebound is smaller than $1$ in magnitude, i.e. a satellite tax
increase _will_ reduce the total number of satellites in orbit despite an
open-access rebound.

######  Proposition 1 (Taxes and rent dissipation).

Under Assumption 1,

$\displaystyle\forall i,j,~{}~{}\partialderivative{S_{i}^{\star}}{\tau_{ij}}$
$\displaystyle<0,$ (19) $\displaystyle\forall i\neq
j,~{}~{}\partialderivative{S_{j}^{\star}}{\tau_{ij}}$ $\displaystyle>0,$ (20)

and $\forall i\neq j$,

$\displaystyle-\partialderivative{S_{i}^{\star}}{\tau_{ij}}>\partialderivative{S_{j}^{\star}}{\tau_{ij}}.$
(21)

Corollary 1 follows immediately.

######  Corollary 1.

Given Proposition 1, satellite taxes reduce the long-run debris stock and
therefore the abatement level required to avert catastrophe. Formally,

$\partialderivative{\bar{Q}}{\tau_{ij}}=d\left(\partialderivative{S^{\star}_{i}}{\tau{{}_{ij}}}+\partialderivative{S^{\star}_{j}}{\tau{{}_{ij}}}\right)<0.$
(22)

Second, greater debris abatement increases the size of every nation’s
satellite sector, with sectors facing lower tax rates growing more than
sectors facing higher tax rates. Intuitively, debris abatement increases the
amount of usable orbital volume. Open-access sectors take advantage of this by
launching more satellites. Sectors facing lower tax rates are then able to
profitably expand relatively more (though potentially still less in absolute
terms) than sectors facing higher tax rates.

######  Proposition 2 (Debris abatement and sectoral expansion).

Under Assumption 1

$\displaystyle\forall i,~{}~{}\partialderivative{S_{i}^{\star}}{Q}$
$\displaystyle>0,$ (23) $\displaystyle\forall
i,j,~{}~{}\frac{\partial^{2}S_{i}^{\star}}{\partial Q\partial\tau_{ij}}$
$\displaystyle<0$ (24)

Third, in order for greater debris abatement from a treaty to actually reduce
the stock of debris in orbit despite open-access responses inducing additional
launches, the marginal collision risk from a new satellite must be bounded by
a constant (stated precisely in Assumption 2). Intuitively, if the marginal
collision risk is sufficiently small then the rent-dissipating behavior of
Propositions 1 and 2 won’t fully erode the gains from greater debris
abatement.

######  Assumption 2.

The marginal collision risk from a new satellite is bounded by a constant,
i.e.

$kd<\frac{1}{2}.$ (25)

######  Proposition 3 (Effective abatement treaties).

Under Assumptions 1 and 2 and if $m_{i}>0~{}~{}\forall i$, then

$\partialderivative{D^{\star}}{Q}<0,$ (26)

where

$D^{\star}=d\sum_{i}S^{\star}+D_{0}-Q.$ (27)

Assumption 2 is not very restrictive, and is likely satisfied now and in the
future. Rao, Burgess and Kaffine (2020) offer supporting evidence on this
point. Using a calibrated physico-economic model, they show that the rebound
effect from debris removal that costs satellite operators nothing (analogous
to abatement here) generally does not induce a launch response strong enough
to overturn the benefits of debris removal.

###  2.2 National regulations

Each nation $j$ receives benefits from satellite services provided by all
sectors per their market access, subject to open-access behavior by satellite
operators. The benefits received by nation $j$ are shown in equation 28:

$W_{j}=(1-kD)p_{j}\sum_{i=1}^{n_{s}}(1-\tau_{ij})S_{i}^{\star},$ (28)

where $S_{i}^{\star}$ satisfies equation 6. Each nation $j$ sets its virtual
tax schedule, $\tau_{\cdot j}$ to maximize the benefits it receives from
satellite services:

$\displaystyle\max_{\tau_{\cdot j}}~{}$
$\displaystyle~{}W_{j}=(1-kD)p_{j}\sum_{i=1}^{n_{s}}(1-\tau_{ij})S_{i}^{\star}$
(29) s.t.
$\displaystyle~{}S_{i}^{\star}=\frac{\sum_{j}p_{j}(1-\tau_{ij})(1-k(dS_{-i}+D_{0}-Q))}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}~{}~{}\forall
i,$ $\displaystyle~{}D=d\sum_{i=1}^{n_{s}}S_{i}^{\star}+D_{0}-Q.$

We do not include the catastrophe damages in program 29 to reflect that
nations acting on their own do not internalize the full long-run benefits of
orbital protection. We also hold the abatement level fixed to identify the
incentives to tax in the absence of agreements regarding abatement. A typical
first-order condition characterizing the optimal tax rate imposed by nation
$j$ on sector $i$, $\tau_{ij}^{\star}(\tau_{\cdot\cdot})$, has the form in
equation 30:

$\tau^{\star}_{ij}(\tau_{\cdot\cdot}):\underbrace{-kdp_{j}\sum_{i=1}^{n_{s}}(1-\tau_{ij})S_{i}^{\star}\left[\partialderivative{S_{i}^{\star}}{\tau_{ij}}+\partialderivative{S_{j}^{*}}{\tau_{ij}}\right]}_{\text{\shortstack{Value
of services gained due to\\\ cleaner orbital environment\\\ (environmental
cleanup
channel)}}}+\underbrace{(1-kD)p_{j}(1-\tau_{jj})\partialderivative{S_{j}^{\star}}{\tau_{ij}}}_{\text{\shortstack{Value
of services gained due to\\\ sector $j$ responding to\\\ higher taxes on
$i$\\\ (fleet expansion
channel)}}}=\underbrace{(1-kD)p_{j}\left[S_{i}^{\star}-(1-\tau_{ij})\partialderivative{S_{i}^{\star}}{\tau_{ij}}\right]}_{\text{\shortstack{Value
of services lost due to\\\ higher taxes on $i$\\\ (fleet reduction
channel)}}}.$ (30)

The two terms on the left-hand side of equation 30 describe sources of
national gains through higher taxes on satellite sector $i$. There are two
channels here: an environmental cleanup channel and a fleet expansion channel.
First, when nation $j$ taxes sector $i$ (i.e. any foreign sector), sector $i$
reduces its fleet size while sector $j$ (i.e. the domestic sector) increases
its fleet size. By Proposition 1, the increase in $S_{j}^{\star}$ is smaller
than the decrease in $S_{i}^{\star}$. Increasing $\tau_{ij}$ therefore reduces
orbital collision risk and increases the value of satellite services $j$
receives. Second, the expansion of sector $j$’s fleet due to a higher tax on
sector $i$’s fleet provides some (gross) benefits to nations receiving
satellite services. The term on the right-hand side of equation 30 describes
the national losses through higher taxes on satellite sector $i$, i.e. the
value lost due to reductions in $S_{i}^{\star}$.181818If some of these
regulations take the form of explicit revenue-generating taxes, the left-hand
size of equation 30 will include an additional term for tax revenues gained
from a higher rate on $i$ and a larger tax base for $j$, while the right-hand
size will include an additional term for tax revenues lost from a smaller tax
base in fleet $i$. We abstract from these issues to focus on the environmental
considerations. A debris abatement treaty which assured sufficient abatement
to avoid catastrophe would enter here to make the environmental cleanup
channel larger (since $\partialderivative{\bar{Q}}{\tau_{ij}}<0$ from
Corollary 1). A treaty which assured only a fixed level of debris abatement
would have no effect on the environmental cleanup channel. Any level of debris
abatement would reduce the magnitude of both the fleet expansion and fleet
reduction channels, while larger legacy debris stocks would have the opposite
effect.

In a global regulatory equilibrium, all nations set their tax rates to
maximize their benefits received from satellites under open access to orbit.
Definition 2 formalizes this.

######  Definition 2.

(Global regulatory equilibrium) A global regulatory equilibrium is a schedule
of satellite taxes $\\{\tau_{ij}^{\star}\\}$ which, given a distribution of
open-access satellite sector sizes $\\{S^{\star}_{i}\\}$, net national debris
abatement $Q$, a legacy debris stock $D_{0}$, market sizes $\\{p_{j}\\}$,
sector efficiencies $\\{m_{i}\\}$, and technologies $(k,d)$, solve program 29
for all $j$.

Assumption 3 shows a condition for satellite taxes on nation $i$’s fleet to
improve nation $j$’s welfare.

######  Assumption 3.

Define the semi-elasticity of the total satellite fleet with respect to a
change in $\tau_{ij}$,

$\mathcal{S}^{\star}_{ij}\equiv\frac{\partialderivative{S^{\star}_{i}}{\tau_{ij}}+\partialderivative{S^{\star}_{j}}{\tau_{ij}}}{S^{\star}_{i}+S^{\star}_{j}}.$
(31)

The legacy debris stock, $D_{0}$, is large enough that Condition 32 holds for
nation $i$:

$-kd\mathcal{S}^{\star}_{ij}>(1-kD)\frac{S^{\star}_{i}}{S^{\star}_{i}+S^{\star}_{j}}-(1-kD)\mathcal{S}^{\star}_{ij}.$
(32)

Under Proposition 1 the semi-elasticity in Assumption 3 is negative, making
the left-hand side of condition 32 positive. The right-hand side of condition
32 is also positive. However, if the legacy debris stock is large enough, the
right-hand side of condition 32 will be small. At sufficiently large levels of
$D_{0}$ Assumption 3 can be satisfied for all nations. Identifying nations for
which Assumption 3 holds currently is an important question for future
empirical research.

Regardless, Proposition 4 shows that Assumption 3 holding for all nations is a
sufficient but not necessary condition for existence a global regulatory
equilibrium in which all nations maintain positive tax rates on all satellite
fleets from which they receive services. National incentives to have larger
domestic satellite fleets tilt the equilibrium in favor of taxing others’
fleets more than one’s own, but in general do not overturn the result.

######  Proposition 4 (Equilibrium tax distribution).

If Assumptions 1, 2, and 3 hold for all $i,j$, then there exists a global
regulatory equilibrium where all nations levy positive taxes on all satellite
sectors, i.e.

$\forall i,j,~{}~{}\tau^{\star}_{ij}>0.$ (33)

Intuitively, the larger the legacy debris stock the greater the loss of
satellite services due to debris-related collisions. This can lead nations to
regulate satellites in order to increase the value they derive from the
orbital environment. If the legacy debris stock is large enough, this
incentive applies to all nations and Proposition 4 follows.

###  2.3 International treaties

The marginal benefit of debris abatement for nation $i$ is the change in the
amount of satellite services $i$ receives in response to an increase in debris
abatement.191919We only impose open access to orbit and not optimal satellite
taxation at this stage. This serves two purposes. First, by imposing open
access we can account for how the treaty will alter the relative sizes of
national satellite sectors through capital mobility and economic expansion. In
effect we are assuming that the treaty discussed here will not undermine the
Outer Space Treaty. Second, by not imposing optimal taxation, we can study how
the treaty will interact with an arbitrary distribution of binding national
satellite regulatory policies. This is more flexible than imposing optimal
taxation conditions, and mirrors our approach in the previous section. That
is,

$b_{i}(Q)\equiv\partialderivative{W_{i}}{Q}.$ (34)

$b_{i}(Q)$ represents each nation’s marginal benefit from abating non-
catastrophic debris accumulation. Since each nation may receive different
benefits from satellite services, their marginal benefit of debris abatement
may also vary.202020Non-spacefaring nations can contribute to debris
abatement, e.g. by purchasing active debris removal services. However, since
debris abatement confers non-rival and non-excludable benefits, nation $i$
benefits from the total amount of debris abatement $Q$ and not just their
individual contribution $q_{i}$, where

$Q=\sum_{i=1}^{N}q_{i}.$ (35)

Classifying nations into two groups (nation $i$ and all others) and using the
reduction from Lemma 2 to simplify the satellite fleet responses to the
$2$-nation case, $b_{i}(Q)$ has the form shown in equation 36:

$b_{i}(Q)=\left(\frac{\sum_{j}p_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}\right)^{2}\left(\frac{k(1-kdr_{j})}{1-(kd)^{2}r_{i}r_{j}}\right)\left((1-2\sigma_{i}(Q))kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}\right).$
(36)

Notice the presence of $\sigma_{i}$, with dependence on $Q$ made explicit.
Since $\sigma_{i}$ is linear in $Q$, the marginal benefits of debris abatement
can be rewritten as a linear function, shown in equation 37:

$\displaystyle b_{i}(Q)$ $\displaystyle=\alpha_{i}-\beta_{i}Q,$ (37)

where

$\displaystyle\alpha_{i}$
$\displaystyle=\left(\frac{\sum_{j}p_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}\right)^{2}\left(\frac{k(1-kdr_{j})^{2}}{1-(kd)^{2}r_{i}r_{j}}\right),$
(38) $\displaystyle\beta_{i}$
$\displaystyle=2k\left(\frac{\sum_{j}p_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}\right)^{2}\left(\frac{k(1-kdr_{j})^{2}}{1-(kd)^{2}r_{i}r_{j}}\right)\left(\frac{kd\sum_{j}p_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}\right).$
(39)

The literature on international treaties often treats the marginal benefits of
pollution abatement as affine and increasing in total abatement, e.g. Singer
and Musacchio (2011); Barrett (2013); Barrett and Dannenberg (2014). Here, a
linear and decreasing form emerges from the value a nation receives from
satellite services in the prior stages of the game. Under Assumption 1, both
coefficients of equation 37 are positive, making the marginal benefits of
abatement decreasing in the total amount abated.212121This property is similar
to the form used in Gelves and McGinty (2016), though the benefits function
there is quadratic and concave. Intuitively, a decreasing marginal benefits
function can emerge from prioritizing the most valuable types of abatement
(e.g. removing the objects posing the greatest fragmentation risk or
passivating spent launch vehicles) and implementing them first.

We assume there exists a common set of debris abatement technologies available
to all nations, such that abating $q_{i}$ units in the cost-minimizing pattern
has marginal cost $\frac{c}{2}q_{i}^{2}$. The payoff to nation $i$ from
abating $q_{i}$ units and experiencing $Q$ units of total abatement, ignoring
catastrophic damages, are shown in equation 40:

$b_{i}(Q)-\frac{c}{2}q_{i}^{2}.$ (40)

We assume that the economic incentive to use orbital space under open access,
despite any satellite taxes, is sufficient to cause Kessler Syndrome in the
absence of debris abatement, i.e. $dS^{\star}>\bar{D}$. Since the payoffs of
abatement are quadratic, when nations avert catastrophe they will do so by
providing total abatement exactly equal to $\bar{Q}$. Recall that if
catastrophe is not averted (i.e. debris exceeds $\bar{D}$, or abatement is
less than $\bar{Q}$), then all nations experience damages of $X$. This gives
the abatement payoff to nation $i$ shown in equation 41:

$\displaystyle\pi_{i}=\begin{cases}b_{i}(\bar{Q})-\frac{c}{2}q_{i}^{2}&\text{
if }Q\geq\bar{Q}\\\ b_{i}(Q)-X-\frac{c}{2}q_{i}^{2}&\text{ if
}Q<\bar{Q}.\end{cases}$ (41)

There are two interpretations of debris abatement: reduction of debris
produced by satellites launched, and removal of debris in orbit. Reducing
debris production through a treaty is relatively straightforward: each nation
agrees to reduce debris production by an agreed-upon amount. Removing debris
is more complicated. We assume that the treaty allows debris placed by any
nation’s operators to be removed by any other nation’s operators. In practice,
rights to debris objects are held by the entities which launched them,
creating non-trivial transactions costs to debris removal (Weeden, 2011). We
maintain this assumption to illustrate the scope of possible outcomes if such
a treaty is signed, or if institutional reforms (e.g. debris salvage rights as
described in Muñoz-Patchen (2018)) facilitate the formation of competitive
markets for debris rights.

When nations supply abatement to maximize their individual payoffs there are
two symmetric pure-strategy Nash equilibria: one where nations ignore the risk
of catastrophe and supply no abatement ($q_{i}=0$) and one where they equally
share the burden of avoiding the catastrophe threshold
($q_{i}=\frac{\bar{Q}}{N}$). This is formalized in Proposition 5.

######  Proposition 5 (Nash abatement levels).

When

$X\geq\beta_{i}\frac{\bar{Q}}{N}+\frac{c(\bar{Q}/N)^{2}}{2},$ (42)

for all $i$, then there are at least two pure-strategy Nash equilibrium debris
abatement levels,

$\displaystyle q_{i}$ $\displaystyle=0~{}~{}~{}\forall i,$ (43) $\displaystyle
q_{i}$ $\displaystyle=\frac{\bar{Q}}{N}~{}~{}~{}\forall i.$ (44)

Only $q_{i}=\frac{\bar{Q}}{N}$ for all $i$ averts catastrophe.

Condition 42 is a lower bound on the catastrophe damages such that no nation
would prefer to be pivotal in causing catastrophe. It states that the
catastrophe damages are greater than the average marginal payoff from being
the one nation supplying insufficient abatement.222222National security
concerns could matter here. For example, if nation $i$ preferred to see
satellites operated from nation $j$ destroyed by debris despite the risk to
their own satellites, then the individual benefits term $b_{i}$ could be
increasing in the debris stock. Particularly if $b_{i}$ is relatively flat in
the amount of satellite services received (e.g. because $p_{i}$ is small),
then $i$ may prefer to incur catastrophe precisely because $j$ prefers not to.
Exploring such interdependent preferences are beyond our scope here but is an
important area for future research.

The Nash equilibria identified in Proposition 5 reflect the fact that debris
abatement is a public good. Each piece removed confers non-rival and non-
excludable benefits to all orbit users, though the cost is borne privately.
The catastrophe-averting equilibrium can be sustained if and only if each
nation faces a sufficiently-low incentive to free-ride. Condition 42 offers
some intuition for how the incentive to cooperate emerges without a treaty. As
in Barrett (2013), the damages from the catastrophic outcome act as Nature’s
incentive to cooperate—if the damages are sufficiently severe, nations will
seek to avert Kessler Syndrome even without a formal treaty to coordinate
them. But as discussed in Barrett (2016), coordination which involves
reshaping incentives (rather than relying on traditional enforcement
mechanisms) nearly always works better than voluntarism. Definition 3
describes the type of “self-enforcing” treaty we consider, and Proposition 6
describes the treaty structure.

######  Definition 3.

(Self-enforcing treaty) A self-enforcing treaty to prevent Kessler Syndrome is
a treaty which

  1. 1.

ensures total abatement supply is $Q=\bar{Q}$, and

  2. 2.

prevents nations who prefer to avert Kessler Syndrome from leaving the treaty,
by using the abatement responses of the remaining signatories.

######  Proposition 6 (Self-enforcing treaty).

A debris abatement treaty with $N$ signatories can avert Kessler Syndrome
despite defections if, in response to defections by any nation $i$, the treaty
instructs the remaining signatories to supply

$Q_{\setminus
i}=\bar{Q}+\frac{1}{c}\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right)$ (45)

units of debris abatement. If

$\frac{\bar{Q}}{N}<-\frac{1}{c}\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right)$
(46)

also holds for all $i$, then the treaty will also be self-enforcing.

As in Proposition 2 of Barrett (2013), the analysis in Proposition 6 assumes
full participation. Yet Singer and Musacchio (2011) finds that a debris
abatement treaty is likely to support only a lower level of participation,
albeit without accounting for open access to orbit. Can a debris abatement
treaty sustain full participation once open access to orbit is accounted for?
There are two ways to resolve this question affirmatively. First, if condition
42 holds, then nation $i$ would prefer averting catastrophe rather than
incurring it, and remaining within the treaty offers a way to do so. Second,
provided condition 46 holds and the treaty response is a punishment for all
potential defectors rather than a burden-reduction, no nation will prefer to
defect. Full participation then becomes a self-reinforcing (as well as self-
enforcing) equilibrium. The defector will recognize that if they leave the
treaty the remaining signatories will exploit the defector’s aversion to
Kessler Syndrome and force them to incur greater abatement costs than if they
remained in the treaty. Condition 42 shows that all nations will prefer to
avert Kessler Syndrome if either the damages are high enough or the total
abatement costs are low enough.

As abatement technologies improve and $c\to 0$, the remaining signatories’
response to defection in equation 45 becomes a punishment rather than a
burden-reduction for all potential defectors. Similarly, inspection of
condition 42 reveals that innovations which reduce the cost of debris
abatement make averting Kessler Syndrome (even without a treaty) more
attractive. Technology improvements which reduce the cost of abatement make
both remaining within the treaty and treaty-free coordination more attractive.
It is less clear how changes in national taxes ($\tau_{\cdot\cdot}$) or $i$’s
satellite sector efficiency ($m_{i}$) will affect the incentive to cooperate
through the parameter $\beta_{i}$. Lemma 4 establishes these comparative
statics, showing that $\beta_{i}$ is decreasing in $i$’s satellite sector
efficiency and all taxes.

######  Lemma 4 (Comparative statics).

$\beta_{i}$ is decreasing in $m_{i}$ and $\tau_{\cdot\cdot}$.

Using Lemma 4, we can now establish our key result about treaties and taxes:
that national satellite tax policies make an international debris abatement
treaty easier to sustain. This result operates through two channels. First,
higher national satellite tax rates make it more likely that all nations
prefer to avert catastrophe even without a treaty. Second, provided the costs
of debris abatement are low enough (or the damages from Kessler Syndrome are
high enough), higher national satellite tax rates make it more likely that
defection from the treaty is a punishment rather than burden-reduction.

######  Proposition 7 (Taxes support treaties).

Higher foreign taxes on nation $i$’s satellite sector give nation $i$ a
stronger incentive to avert Kessler Syndrome. If the costs of debris abatement
are low enough (or the damages from Kessler Syndrome are high enough), then
higher foreign taxes on $i$ also give $i$ a stronger incentive to remain
within a self-enforcing debris abatement treaty. Formally,

$\displaystyle\partialderivative{\tau_{ij}}\left(\beta_{i}\frac{\bar{Q}}{N}+\frac{c(\bar{Q}/N)^{2}}{2}\right)$
$\displaystyle<0,$ (47)

and if

$\sqrt{\beta_{i}^{2}+2cX}-c\beta_{i}>0,$ (48)

then

$\displaystyle\partialderivative{\tau_{ij}}\left(-\frac{\bar{Q}}{N}-\frac{1}{c}\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right)\right)$
$\displaystyle>0.$ (49)

A similar conclusion follows for improvements in sector efficiencies.

##  3 Discussion

Many scholars analyzing orbital-use management have established pessimistic or
concerning results. These include the high likelihood that physical tipping
points have been or are soon to be crossed (Krisko, Opiela and Kessler, 2001;
Liou and Johnson, 2008; Kessler et al., 2010; Drmola and Hubik, 2018), the
economic incentives provided by current management institutions to overuse
orbital space and induce Kessler Syndrome (Adilov, Alexander and Cunningham,
2015; Rouillon, 2020; Rao, Burgess and Kaffine, 2020; Rao and Rondina, 2022),
and the lack of incentives (both due to legal and policy barriers as well as
the natural economic structure of the problem) to abate or remove orbital
debris without free-riding (Weeden, 2011; Adilov, Alexander and Cunningham,
2015; Rao, 2019; Klima et al., 2018).

This paper offers more optimistic results. Proposition 4 shows that a
equilibrium can exist where nations regulate their own and each others’
satellites without transfers of surplus. Proposition 6 provides guidance on
how to draft a debris abatement treaty which can avert Kessler Syndrome
without requiring supranational enforcement beyond incentive-compatible
responses specified in the treaty itself. Proposition 7 shows how bilateral
national satellite taxes can increase the likelihood of treaty success. To the
extent that debris abatement treaties can be linked to other environmental
issues (e.g. climate treaties) during negotiations, there is even more scope
for treaty-based debris abatement (Adilov, Alexander and Cunningham, 2022).

We decompose the national incentives to use “bilateral satellite taxes” (i.e.
satellite regulatory policies imposed as conditions for market access) into
three channels. The first two channels, environmental cleanup and fleet
expansion, create an incentive for nations to levy satellite taxes on other
nations’ satellites. The third channel, fleet reduction, limits that incentive
from inducing an international “tax war”. These channels suggest that even in
the absence of catastrophe risk, nations will prefer to levy some rather than
no satellite taxes on each others’ fleets so as to control collision risk and
increase the long-run value of satellite services received. This echoes an
important result in Brander and Taylor (1998), where trade in a renewable
resource subject to open access can reduce welfare by hastening stock
depletion and tariffs or trade restrictions can improve welfare by mitigating
the negative externality. In our model satellite services are a tradeable good
produced using orbital space, a renewable resource.

The magnitude of global welfare gains from a globally-harmonized optimal
orbital-use fee identified in Rao, Burgess and Kaffine (2020) suggests the
existence of a global regulatory equilibrium with positive tax rates _and_
transfers between nations. Proposition 4 goes one step further, showing that a
global regulatory equilibrium with positive satellite taxes can exist
_without_ transfers and identifying a relevant sufficient condition. The
sufficient condition requires debris abatement to be decreasing in tax rates,
which can be assured by a debris abatement treaty subject to open-access
responses. Together, Propositions 4 and 7 show that a system of non-harmonized
national satellite taxes and an international treaty to abate just enough
debris to avert Kessler Syndrome can reinforce each other, increasing the
global benefits from orbit use without undermining the Outer Space Treaty.

We focus on the positive question of what equilibrium tax rates will look like
when nations act strategically to maximize the economic benefits they receive
from orbit use rather than normative questions of optimal fee system design.
The literature on environmental federalism offers some insights regarding the
efficiency properties of the system we identify. Orbital space is a fully
spatially-connected resource, with different national markets for satellite
services inducing preference heterogeneity. Open access and orbital debris
create an interjurisdictional resource externality. Decentralized national
policies (and the equilibrium identified in Proposition 4) are therefore
likely inefficient, albeit efficiency-enhancing, unless the marginal dynamic
costs of debris are sufficiently high to make it privately optimal to totally
eradicate debris and collision risk (Levinson, 2003; Costello, Quérou and
Tomini, 2015, 2017). This condition seems both unlikely to be satisfied—the
external cost decomposition in Rao and Rondina (2022) shows debris reductions
will reduce the marginal dynamic cost of debris (see the third term of
equation (30) there)—and technologically infeasible. The potential for
catastrophe and an international treaty to avert it create an
interjurisdictional fiscal externality (Wilson, 1999). Proposition 7 can
therefore be interpreted as a “second-best” result (Lipsey and Lancaster,
1956), wherein the fiscal and resource externalities mitigate each other.

While our analysis focuses on long-run possibilities and outcomes, our
analysis offers suggestions on how to begin the process of implementing the
interlocking system of orbital-use management we describe here. The key is the
legacy debris stock. Lemoine (2020) finds that large legacy carbon stocks make
negative emissions policies more challenging to implement, a finding
reaffirmed in our analysis of a self-enforcing treaty—notice the ways the
required abatement level $\bar{Q}$ (which is increasing in the legacy stock
$D_{0}$) appears in Propositions 5 and 6. However, Proposition 4 shows that
larger legacy debris stocks make a global regulatory equilibrium _more_ likely
to emerge (i.e. larger $D_{0}$ makes Assumption 3 more likely to hold for all
$i$). Further, Proposition 7 shows that bilateral taxes in the global
regulatory equilibrium will support a self-enforcing debris abatement treaty.
Thus, our analysis suggests that when the legacy debris stock is large,
beginning with national regulatory policies may be a path toward implementing
an international debris abatement treaty.

Technological advances which improve access to orbit and make it cheaper to
abate debris also make it easier to sustain cooperation in preventing Kessler
Syndrome. Policies which support these technologies seem like a useful path
forward. Understanding how different types of satellite operators respond to
different regulatory policies is also important in determining the shape of a
potential global regulatory equilibrium. Empirical and experimental work in
this direction seems like another useful direction to pursue.

There remain many limitations to our analysis. Importantly, we do not consider
national security and dual-use concerns. These are key drivers of orbit use.
To the extent that nations prefer to weaken each others’ satellite sectors—or
satellite sectors serving each others’ markets—our results overstate the
potential for international cooperation and the benefits of international
competition in orbit use. But to the extent that nations with high dependence
on satellites for national security internalize the benefits of keeping the
orbital environment clean, our results understate the potential for
cooperation. Similar caveats apply in the political domain when applying our
results to atmospheric carbon management. Further, anti-satellite missile
testing may produce enough difficult-to-remove fragments that it is no longer
possible to abate enough debris to prevent Kessler Syndrome. These limitations
suggest several areas for future research. One avenue is to identify potential
non-kinetic attacks on national satellite fleets which exploit open-access
responses or national satellite tax policies so that these attack surfaces can
be mitigated when designing the policy regime. Another is to explore the scope
for decentralized or treaty-based deterrence or prevention of missile testing,
while a third is to study mechanisms to overcome strategic non-economic
orbital-use incentives.

We also ignore how the costs of debris abatement are financed. It is possible
that such abatement can be financed by satellite taxes from the global
regulatory equilibrium. This is an interesting avenue for future research. Yet
we argue that focusing on budget-balanced tax-and-spend approaches to orbital-
use management risks misunderstanding the nature of the problem. Open access
to orbit is an extremely costly status quo, reducing the global value
generated by satellites over the coming decades by an order of magnitude (Rao,
Burgess and Kaffine, 2020). This is not surprising—similar results have been
observed in other open access resources, such as oil fields and fisheries
(Libecap and Wiggins, 1984; Kelleher et al., 2009). As debris accumulates, the
damages incurred from open access will grow and eventually become costlier to
address (McKnight, 2010). Allowing open-access orbit use to continue without
economic policy responses is thus akin to allowing a depressed economy to be
scarred by hysteresis. Fiscal policy to avert hysteresis can be self-financing
through the additional growth it produces relative to the status quo (DeLong
et al., 2012). Like fiscal policy in a depressed economy, satellite taxes and
debris abatement pay for themselves.

Finally, our analysis also ignores uncertainty over the location of debris
threshold for Kessler Syndrome. Barrett (2013) finds that such uncertainty can
greatly limit the potential for self-enforcing international treaties to avert
catastrophe. However, engineering analyses are continually refining estimates
of the nature of the self-sustaining debris growth threshold, e.g. Krisko,
Opiela and Kessler (2001); Liou and Johnson (2008); Drmola and Hubik (2018);
Lewis (2020). In general, scientific understanding of the fundamental
mechanisms and interactions driving orbital debris growth seems better than
scientific understanding of the mechanisms and interactions driving planetary
warming. Further, regardless of the location of the threshold, national
regulatory policies can be used to support cooperation in debris abatement,
and the incentives we identify to implement national regulatory policies are
independent of the existence of a catastrophe threshold. Consequently,
threshold uncertainty may not be as serious a threat to international
cooperation in orbital-use management as in atmospheric carbon management.

##  4 Conclusion

Managing orbital space is critical to ensuring humanity continues to benefit
from satellite services. While international pollution abatement agreements
may be a preferred approach to managing a global commons, orbital-use
management agreements may be more difficult to construct in the near future
than national regulatory policies. In this paper we contribute to the
literature and policy discussion around orbital-use management by examining
the relationships between open access to orbit, national regulatory policies,
and international treaties to prevent self-sustaining debris growth (“Kessler
Syndrome”). We highlight three messages regarding international competition
and cooperation in orbital-use management.

First, like a mobile capital stock subject to costly regulation in one
jurisdiction, open access to orbit means that regulatory policies in one
nation will shrink their satellite sector and induce expansion of other
nations’ satellite sectors. Despite this response pattern, bilateral
regulatory policies will generally produce sustained improvements to the
orbital environment and the international economic value derived from
satellites. Second, a large legacy debris stock can induce all nations
receiving satellite services to impose environmental regulations on all
satellite fleets providing services as a condition for market access. Such a
global regulatory equilibrium is compatible with incentives to maximize
national benefits from satellite services. Third, a system of bilateral
regulatory policies as conditions for market access can be used to support the
formation and maintenance of a self-enforcing international treaty to abate
debris production and avert Kessler Syndrome. Such a system can be incentive-
compatible for all nations involved, avoiding the need for a supranational
enforcer.

There are many remaining open questions of orbital-use policy design,
particularly relating to national security and dual-use technology concerns.
Still, it is worth knowing that there is a light at the end of the tunnel. If
economic benefits are a large enough share of the value nations derive from
orbit use, international competition for orbital space and international
cooperation to manage the orbital environment can be mutually reinforcing.

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##  Appendix A Appendix

See 1

###### Proof.

Define

$A_{i}=\frac{\sum_{j}p_{j}(1-\tau_{ij})(1+k(Q-D_{0}))}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}$

and

$B_{i}=-\frac{kd\sum_{j}p_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}},$

where $A_{i}$ and $B_{i}$ reflect the long-run benefit-cost ratio and the
long-run ratio of environmental to total costs for sector $i$. Equation 7 can
then be written as

$S_{i}^{\star}(S_{-i},\tau_{i\cdot},Q)=A_{i}+B_{i}S_{-i}.$ (50)

Collecting best-response functions in this form yields the linear system

$\displaystyle S^{\star}_{1}$ $\displaystyle=A_{1}+B_{1}\sum_{j\neq
1}S^{\star}_{j}$ (51) $\displaystyle S^{\star}_{2}$
$\displaystyle=A_{2}+B_{2}\sum_{j\neq 2}S^{\star}_{j}$ (52)
$\displaystyle\vdots$ $\displaystyle S^{\star}_{n_{s}}$
$\displaystyle=A_{n_{s}}+B_{n_{s}}\sum_{j\neq n_{s}}S^{\star}_{j}.$ (53)

Letting $A$ be the $n_{s}\times 1$ vector of the $A_{i}$,

$A=[A_{1},\dots,A_{n_{s}}],$ (54)

$B$ be the $n_{s}\times n_{s}$ matrix with $0$ on the main diagonal and row
$i$ containing copies of $B_{i}$,

$\displaystyle B=\begin{bmatrix}0&B_{1}&\dots&B_{1}\\\ B_{2}&0&&\vdots\\\
\vdots&&\ddots&B_{n_{s}-1}\\\ B_{n_{s}}&\dots&B_{n_{s}}&0\end{bmatrix}$ (55)

and $S$ be the $n_{s}\times 1$ vector of equilibrium satellite levels
$S^{\star}_{i}$, the linear system above can be rewritten as

$S=A+BS.$ (56)

If $det(I-B)\neq 0$, then the global open-access equilibrium is

$S=(I-B)^{-1}A.$

∎

See 2

###### Proof.

Let $I$ be the $n_{s}\times n_{s}$ identity matrix, $B$ and $A$ as defined in
Lemma 1 for an $n_{s}$-player global open-access equilibrium, and
$I^{2},B^{2},A^{2}$ their analogs for a $2$-player global open-access
equilibrium such that $\sum_{i}A_{i}=\sum_{i}A^{2}_{i}$ and
$\sum_{i}B_{i}=\sum_{i}B^{2}_{i}$. Define $G=(I-B)^{-1}$ and
$H=(I^{2}-B^{2})^{-1}$.

$G$ can be partitioned into 4 submatrices,

$\displaystyle G=\begin{bmatrix}G_{11}&G_{12}\\\ G_{21}&G_{22}\end{bmatrix},$

where $G_{11}$ has dimension $1\times 1$, $G_{12}=G_{21}^{T}$ by the has
dimension $1\times(n_{s}-1)$, and $G_{22}$ has dimension
$(n_{s}-1)\times(n_{s}-1)$. Similarly, $A$ can be partitioned into 2 vectors,

$\displaystyle A=\begin{bmatrix}A_{1}\\\ A_{\setminus 1}\end{bmatrix},$

of dimension $1\times 1$ and $(n_{s}-1)\times 1$. Like $G$ and $A$, $H$ and
$A^{2}$ can be written as

$\displaystyle H=\begin{bmatrix}H_{11}&H_{12}\\\ H_{21}&H_{22}\end{bmatrix}$

and

$\displaystyle A^{2}=\begin{bmatrix}A^{2}_{1}\\\ A^{2}_{\setminus
1}\end{bmatrix},$

where all elements of both objects are of dimension $1\times 1$. From the
definitions of $A^{2}$ and $B^{2}$, it can be seen that

$\displaystyle G_{11}A_{1}+G_{12}A_{-1}$
$\displaystyle=H_{11}A^{2}_{1}+H_{12}A^{2}_{-1}$ (57) $\displaystyle\implies
S_{i}^{\star}$ $\displaystyle=S_{i}^{\star,2}.$ (58)

Similarly, from the definitions of $A^{2}$ and $B^{2}$, it can be seen that

$\displaystyle\sum_{-i}\left(G_{21}A_{1}+G_{22}A_{\setminus 1}\right)$
$\displaystyle=H_{21}A^{2}_{1}+H_{22}A^{2}_{\setminus 1}$ (59)
$\displaystyle\implies S_{-i}^{\star}$ $\displaystyle=S_{j}^{\star,2}.$ (60)

where the sum in $\sum_{-i}$ is over the $n_{s}-1$ elements of the
$(n_{s}-1)\times 1$ vector $G_{21}A_{1}+G_{22}A_{-1}$.

∎

See 3

###### Proof.

From Lemma 1 we have that $S=(I-B)^{-1}A$ is the global open-access
equilibrium satellite distribution. Define $\phi$ to be the $n_{s}\times 1$
vector where each element is $1+kQ$, and $r$ to be the $n_{s}\times 1$ vector
with $i^{\text{th}}$ element

$r_{i}=\frac{\sum_{j}p_{j}(1-\tau_{ij})}{kd\sum_{j}p_{j}(1-\tau_{ij})+m_{i}}.$
(61)

Next, note that

$\displaystyle A=\phi\odot r.$ (62)

Define

$\sigma=(I-B)^{-1}\phi,$ (63)

which is a $n_{s}\times 1$ vector. Then we have

$\displaystyle S$ $\displaystyle=(I-B)^{-1}A$ (64) $\displaystyle=\sigma\odot
r,$ (65)

where only $\sigma$ depends on $Q$. ∎

See 1

###### Proof.

Differentiating equation 14 yields

$\partialderivative{S_{i}^{\star}}{\tau_{ij}}=\partialderivative{\sigma_{i}}{r_{i}}\partialderivative{r_{i}}{\tau_{ij}}r_{i}+\sigma_{i}\partialderivative{r_{i}}{\tau_{ij}}.$
(66)

Since we have assumed $r_{i}<(kd)^{-1}~{}~{}\forall i$, we have
$\sigma_{i},r_{i}>0~{}~{}\forall i$. Differentiating $r_{i}$ and $\sigma_{i}$
under these assumptions yields

$\displaystyle\partialderivative{\sigma_{i}}{r_{i}}$
$\displaystyle=\frac{(kd)^{2}(1+k(Q-D_{0}))(1-kdr_{j})r_{j}}{(1-(kd)^{2}r_{i}r_{j})^{2}}>0,$
(67) $\displaystyle\partialderivative{r_{i}}{\tau_{ij}}$
$\displaystyle=-\frac{m_{i}p_{j}}{(\sum_{j}p_{j}(1-\tau_{ij})+m_{i})^{2}}<0.$
(68)

This establishes that $\forall
i,j,~{}~{}\partialderivative{S_{i}^{\star}}{\tau_{ij}}<0$.

Again differentiating equation 14 we obtain

$\partialderivative{S_{j}^{\star}}{\tau_{ij}}=\partialderivative{\sigma_{j}}{r_{i}}\partialderivative{r_{i}}{\tau_{ij}}r_{j}.$
(69)

Differentiating $\sigma_{i}$,

$\displaystyle\partialderivative{\sigma_{j}}{r_{i}}$
$\displaystyle=-\frac{kd(1+k(Q-D_{0}))(1-kdr_{j})}{(1-(kd)^{2}r_{i}r_{j})^{2}}<0,$
(70)

establishing that $\forall i\neq
j,~{}~{}\partialderivative{S_{j}^{\star}}{\tau_{ij}}>0$.

Finally, comparing $-\partialderivative{S_{i}^{\star}}{\tau_{ij}}$ and
$\partialderivative{S_{j}^{\star}}{\tau_{ij}}$, we obtain

$\displaystyle-\partialderivative{S_{i}^{\star}}{\tau_{ij}}$
$\displaystyle>\partialderivative{S_{j}^{\star}}{\tau_{ij}}$ (71)
$\displaystyle\implies-\partialderivative{\sigma_{i}}{r_{i}}\partialderivative{r_{i}}{\tau_{ij}}r_{i}-\sigma_{i}\partialderivative{r_{i}}{\tau_{ij}}$
$\displaystyle>\partialderivative{\sigma_{j}}{r_{i}}\partialderivative{r_{i}}{\tau_{ij}}r_{j}$
(72)
$\displaystyle\implies-\partialderivative{r_{i}}{\tau_{ij}}\left(\partialderivative{\sigma_{i}}{r_{i}}r_{i}+\partialderivative{\sigma_{j}}{r_{i}}r_{j}+\sigma_{i}\right)$
$\displaystyle>0,$ (73)

which holds as long as
$\left(\partialderivative{\sigma_{i}}{r_{i}}r_{i}+\partialderivative{\sigma_{j}}{r_{i}}r_{j}+\sigma_{i}\right)>0$.
Simplifying reveals that

$\displaystyle\left(\partialderivative{\sigma_{i}}{r_{i}}r_{i}+\partialderivative{\sigma_{j}}{r_{i}}r_{j}+\sigma_{i}\right)>0\iff(1+k(Q-D_{0}))[1-kdr_{j}(1-kdr_{i}(1-kdr_{i}))]>0,$
(74)

which is ensured by $r_{i}<(kd)^{-1}~{}~{}\forall i$. ∎

See 2

###### Proof.

Differentiating equation 6 yields

$\partialderivative{S_{i}^{\star}}{Q}=\frac{k(1-kdr_{j})}{1-(kd)^{2}r_{i}r_{j}}>0,$
(75)

establishing that $\forall i,~{}~{}\partialderivative{S_{i}^{\star}}{Q}>0$.
Differentiating again yields

$\frac{\partial^{2}S_{i}^{\star}}{\partial
Q\partial\tau_{ij}}=-\frac{d^{2}k^{3}m_{i}m_{j}p_{j}\sum_{i}p_{i}(1-\tau_{ji})}{(kdm_{j}\sum_{j}p_{j}(1-\tau_{ij})+m_{i}(kd\sum_{i}p_{i}(1-\tau_{ji})+m_{j}))^{2}}<0,$
(76)

establishing that $\forall i,j,~{}~{}\frac{\partial^{2}S_{i}^{\star}}{\partial
Q\partial\tau_{ij}}<0$. ∎

See 3

###### Proof.

Differentiating $D^{*}$ with respect to $Q$, we obtain

$\displaystyle\partialderivative{D^{*}}{Q}$
$\displaystyle=d\left[\partialderivative{S^{*}_{i}}{Q}+\partialderivative{S^{*}_{j}}{Q}\right]-1$
(77)
$\displaystyle=d\left[\frac{k(1-kdr_{j})+k(1-kdr_{i})}{1-(kd)^{2}r_{i}r_{j}}\right]-1$
(78)
$\displaystyle=-kd\frac{(1-2kd)+(kd)^{2}(r_{i}+r_{j}-r_{i}r_{j})}{1-(kd)^{2}r_{i}r_{j}}.$
(79)

First, $m_{i}>0~{}~{}\forall i\implies r_{i}<1~{}~{}\forall i\implies
r_{i}+r_{j}-r_{i}r_{j}>0$. Next, $r_{i}<(kd)^{-1}~{}~{}\forall i\implies
1-(kd)^{2}r_{i}r_{j}>0$. Finally, $kd<\frac{1}{2}\implies 1-2kd>0$. These
three conditions are sufficient to ensure $\partialderivative{D^{*}}{Q}<0$. ∎

See 4

###### Proof.

First, note that the system of bilateral tax best responses defined by
equation 30 is a bijective mapping from a compact set to itself, i.e.

$\tau^{\star}_{\cdot\cdot}(\tau_{\cdot\cdot}):\mathbb{R}^{n_{s}\times
n_{s}}_{[0,1]}\to\mathbb{R}^{n_{s}\times n_{s}}_{[0,1]}.$ (80)

By the Brouwer Fixed Point Theorem, an equilibrium tax distribution exists. It
remains to be shown that an equilibrium exists with positive tax rates.
Select an arbitrary nation $j$. For conciseness, define

$V_{j}=p_{j}\sum_{i=1}^{n_{s}}(1-\tau_{ij})S_{i}^{\star},$ (81)

so that $j$’s benefits function can be written as $W_{j}=(1-kD)V_{j}$. Next,
note that

$\partialderivative{W_{j}}{\tau_{ij}}=-k\partialderivative{D}{\tau_{ij}}V_{j}+(1-kD)\partialderivative{V_{j}}{\tau_{ij}},$
(82)

where

$\displaystyle\partialderivative{D}{\tau_{ij}}\bigg{|}_{\tau_{ij}=\tau_{jj}=0}$
$\displaystyle=d\left(\partialderivative{S^{\star}_{i}}{\tau_{ij}}+\partialderivative{S^{\star}_{j}}{\tau_{ij}}\right)\bigg{|}_{\tau_{ij}=\tau_{jj}=0}<0,$
(83)
$\displaystyle\partialderivative{V_{j}}{\tau_{ij}}\bigg{|}_{\tau_{ij}=\tau_{jj}=0}$
$\displaystyle=p_{j}\left(-S^{\star}_{i}+\left(\partialderivative{S^{\star}_{i}}{\tau_{ij}}+\partialderivative{S^{\star}_{j}}{\tau_{ij}}\right)\right)\bigg{|}_{\tau_{ij}=\tau_{jj}=0}<0$
(84)

both inequalities following from Proposition 1. Rearranging and applying
Assumption 3, we obtain

$\partialderivative{W_{j}}{\tau_{ij}}\bigg{|}_{\tau_{ij}=\tau_{jj}=0}>0.$ (85)

Similarly, we compute

$\partialderivative{W_{j}}{\tau_{jj}}=-k\partialderivative{D}{\tau_{jj}}V_{j}+(1-kD)\partialderivative{V_{j}}{\tau_{jj}},$
(86)

obtaining

$\displaystyle\partialderivative{D}{\tau_{jj}}\bigg{|}_{\tau_{ij}=\tau_{jj}=0}$
$\displaystyle=d\left(\partialderivative{S^{\star}_{i}}{\tau_{jj}}+\partialderivative{S^{\star}_{j}}{\tau_{jj}}\right)\bigg{|}_{\tau_{ij}=\tau_{jj}=0}<0,$
(87)
$\displaystyle\partialderivative{V_{j}}{\tau_{ij}}\bigg{|}_{\tau_{ij}=\tau_{jj}=0}$
$\displaystyle=p_{j}\left(-S^{\star}_{i}+\left(\partialderivative{S^{\star}_{i}}{\tau_{ij}}+\partialderivative{S^{\star}_{j}}{\tau_{ij}}\right)\right)\bigg{|}_{\tau_{ij}=\tau_{jj}=0}<0$
(88)

Analysis reveals both inequalities hold whenever
$\frac{m_{j}}{kd\sum_{j}p_{j}(1-\tau_{ij})}>0$, which is always the case.
Again applying Assumption 3, we obtain

$\partialderivative{W_{j}}{\tau_{jj}}\bigg{|}_{\tau_{ij}=\tau_{jj}=0}>0.$ (89)

Conditions 85 and 89 show that any nation $j$ will prefer to levy a positive
tax rate on any nation $i$, including itself. Since existence of an
equilibrium is guaranteed by the Brouwer Fixed Point Theorem and we have shown
that all nations will prefer positive tax rates to zero tax rates (regardless
of what tax rates others levy), we have established existence of a global
regulatory equilibrium with positive tax rates. ∎

See 5

###### Proof.

There are two cases: either it is not privately optimal to avert catastrophe,
or it is privately optimal to avert catastrophe.

If is not privately optimal to avert catastrophe: Nations choosing abatement
levels non-cooperatively will maximize equation 40 taking all other nations’
abatement levels as given. This gives the following FOC:

$\displaystyle q_{i}:\frac{\partial\pi_{i}}{\partial q_{i}}$
$\displaystyle=-\beta_{i}-cq_{i}=0$ (90) $\displaystyle\implies q_{i}$
$\displaystyle=-\frac{\beta_{i}}{c}.$ (91)

Since $\beta_{i}>0$ by Assumption 1, $-\frac{\beta_{i}}{c}<0$. But since
$q_{i}$ is bounded below at $0$, the privately optimal abatement level is
$q_{i}=0$. Since this holds for all $i$, we obtain $Q=0$ and catastrophe is
not averted.

If it is privately optimal to avert catastrophe: Since the payoff function is
quadratic except for a discontinuity at $Q=\bar{Q}$, the maximum is attained
either when $q_{i}=0$ or when $q_{i}:Q=\bar{Q}$.

Suppose all $N-1$ nations $j\neq i$ play $q_{j}=\frac{\bar{Q}}{N}$. Then $i$
is pivotal in averting catastrophe. If $i$ chooses not to play
$q_{i}=\frac{\bar{Q}}{N}$, then following the argument in the previous case
they will free-ride and play $q_{i}=0$, giving the following payoff function:

$\displaystyle\pi_{i}=\left\\{\begin{array}[]{ll}b_{i}(\bar{Q})-\frac{c(\bar{Q}/N)^{2}}{2}&\mbox{if
}Q=\bar{Q}\\\ b_{i}\left(\frac{\bar{Q}}{N}(N-1)\right)-X&\mbox{if
}Q<\bar{Q}\end{array}\right.$ (94)

Nation $i$ will not free-ride and instead play $q_{i}=\frac{\bar{Q}}{N}$ if
and only if

$\displaystyle b_{i}(\bar{Q})-\frac{c(\bar{Q}/N)^{2}}{2}$ $\displaystyle\geq
b_{i}\left(\frac{\bar{Q}}{N}(N-1)\right)-X$ (95) $\displaystyle\implies X$
$\displaystyle\geq
b_{i}\left(\frac{\bar{Q}}{N}(N-1)\right)-b_{i}(\bar{Q})+\frac{c(\bar{Q}/N)^{2}}{2}$
(96) $\displaystyle\implies X$
$\displaystyle\geq\beta_{i}\frac{\bar{Q}}{N}+\frac{c(\bar{Q}/N)^{2}}{2}.$ (42)

Equation 42 must hold for all $i$ to sustain the catastrophe-averting Nash
equilibrium without a treaty. ∎

See 6

###### Proof.

Starting from full participation in the treaty, suppose nation $i$ considers
withdrawal. If they do so, the treaty instructs the remaining $N-1$ members to
play $Q_{\setminus i}$ (which may be different from $\frac{\bar{Q}}{N}(N-1)$).
There are two cases from here:

  1. 1.

either $i$ plays $q_{i}=\bar{Q}-Q_{\setminus i}$ and catastrophe is averted
despite their defection, or

  2. 2.

$i$ plays $q_{i}=0$ and allows catastrophe to occur.

Suppose $i$ considers withdrawal and supplying $q_{i}=0$ units of abatement
instead of the treaty-specified $\frac{\bar{Q}}{N}$, and the remaining members
play $Q_{\setminus i}$. Then $i$ will prefer to avert catastrophe by supplying
$q_{i}=\bar{Q}-Q_{\setminus i}$ units of abatement instead of $q_{i}=0$ if and
only if

$\underbrace{b_{i}(\bar{Q})-\frac{c}{2}\left(\bar{Q}-Q_{\setminus
i}\right)^{2}}_{\text{\shortstack{Payoff from averting catastrophe\\\ while
outside the treaty}}}\geq\underbrace{b_{i}\left(Q_{\setminus
i}\right)-X}_{\text{\shortstack{Payoff from Nash defection\\\ and incurring
catastrophe}}}$ (97)

Since $Q_{\setminus i}$ is a choice for the remaining treaty signatories,
condition 97 shows how the treaty can be written to naturally induce any
defector to avert catastrophe despite leaving the treaty. Rearranging
condition 97, solving the quadratic inequality, and selecting the smallest
root, we obtain

$Q_{\setminus
i}=\bar{Q}+\frac{1}{c}\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right).$ (45)

Since $\beta_{i},c,X>0$, we have
$\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right)<0$, and $Q_{\setminus
i}<\bar{Q}$.

Comparing the treaty response abatement quantity $Q_{\setminus i}$ to the
treaty abatement quantity $\frac{\bar{Q}}{N}$ shows that if

$\frac{\bar{Q}}{N}<-\frac{1}{c}\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right),$
(46)

then the treaty response $Q_{\setminus i}$ coordinates the remaining
signatories to “punish” defection by nation $i$ by inducing them to supply
abatement greater than $\frac{\bar{Q}}{N}$. Else, the treaty response
coordinates the remaining signatories to avert catastrophe by reducing $i$’s
burden of cooperation. Thus, if condition 46 holds, defections by nation $i$
are costlier than remaining in the treaty and they will prefer to remain in
the treaty. ∎

See 4

###### Proof.

For conciseness, define

$\displaystyle A_{i}$ $\displaystyle\equiv\sum_{j}p_{j}(1-\tau_{ij}),$ (98)
$\displaystyle B_{i}$ $\displaystyle\equiv kdA_{i}+m_{i},$ (99) $\displaystyle
T^{i}_{ij}$
$\displaystyle\equiv\frac{2k^{3}dm_{i}m_{j}^{2}A_{i}^{2}\left(3m_{i}A_{j}+kdA_{i}\left(3m_{j}+kdA_{j}\right)\right)}{B_{i}^{3}B_{j}\left(kdm_{j}A_{i}+m_{i}B_{j}\right)^{2}}$
(100) $\displaystyle T^{j}_{ij}$
$\displaystyle\equiv\frac{2k^{4}d^{2}m_{j}^{2}A_{i}^{3}\left(kdm_{j}A_{i}+2m_{i}B_{j}\right)}{\left(A_{i}A_{j}\left(kdm_{j}A_{i}+m_{i}B_{j}\right)\right)^{2}}$
(101)

Next, computing derivatives, we obtain:

$\displaystyle\partialderivative{\beta_{i}}{\tau_{ii}}$
$\displaystyle=-p_{i}T^{i}_{ij}<0$ (102)
$\displaystyle\partialderivative{\beta_{i}}{\tau_{ij}}$
$\displaystyle=-p_{j}T^{i}_{ij}<0$ (103)
$\displaystyle\partialderivative{\beta_{i}}{\tau_{ji}}$
$\displaystyle=-p_{i}T^{j}_{ij}<0$ (104)
$\displaystyle\partialderivative{\beta_{i}}{\tau_{jj}}$
$\displaystyle=-p_{j}T^{j}_{ij}<0$ (105)
$\displaystyle\partialderivative{\beta_{i}}{m_{i}}$
$\displaystyle=-\frac{2k^{3}dm_{j}^{2}A_{i}^{3}\left(3m_{i}B_{j}+kdA_{i}\left(3m_{j}+kdA_{j}\right)\right)}{B_{i}^{3}B_{j}\left(kdm_{j}A_{i}+m_{i}B_{j}\right)^{2}}<0.$
(106)

∎

See 7

###### Proof.

We first establish condition 47, which contains the right-hand side of
condition 42:

$\displaystyle\partialderivative{\tau_{ij}}\left(\beta_{i}\frac{\bar{Q}}{N}+\frac{c(\bar{Q}/N)^{2}}{2}\right)$
$\displaystyle=\partialderivative{\beta_{i}}{\tau_{ij}}\frac{\bar{Q}}{N}+\frac{\beta_{i}}{N}\partialderivative{\bar{Q}}{\tau_{ij}}+\frac{c\bar{Q}}{N^{2}}\partialderivative{\bar{Q}}{\tau_{ij}}$
(107)
$\displaystyle=\partialderivative{\beta_{i}}{\tau_{ij}}\frac{\bar{Q}}{N}+\frac{\beta_{i}}{N}d\partialderivative{S^{\star}}{\tau_{ij}}+\frac{c\bar{Q}}{N^{2}}\partialderivative{S^{\star}}{\tau_{ij}}<0.$
(108)

The inequality follows from Proposition 1, which establishes that
$\partialderivative{S^{\star}}{\tau_{ij}}$, and Lemma 4, which establishes
that $\partialderivative{\beta_{i}}{\tau_{ij}}<0$. Since increases in
$\tau{ij}$ reduce the right-hand side of condition 42 (making the condition
more likely to hold), they increase $i$’s incentive to avert Kessler Syndrome.

Next, we establish condition 49, which contains both the sides of condition
46:

$\displaystyle\partialderivative{\tau_{ij}}\left(-\frac{\bar{Q}}{N}-\frac{1}{c}\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right)\right)$
$\displaystyle=\frac{1}{N}\partialderivative{\bar{Q}}{\tau_{ij}}-\partialderivative{\beta_{i}}{\tau_{ij}}\left(\frac{1}{c}-\frac{\beta_{i}}{\sqrt{\beta_{i}^{2}+2cX}}\right).$
(109)

While the sign of $\frac{1}{c}-\frac{\beta_{i}}{\sqrt{\beta_{i}^{2}+2cX}}$ is
ambiguous, some manipulation reveals that
$\sqrt{\beta_{i}^{2}+2cX}-c\beta_{i}>0$ is a sufficient condition to ensure
$\frac{1}{c}-\frac{\beta_{i}}{\sqrt{\beta_{i}^{2}+2cX}}>0$. Thus, again
applying Proposition 1 and Lemma 4, we obtain the desired result:

$\displaystyle\partialderivative{\tau_{ij}}\left(-\frac{\bar{Q}}{N}-\frac{1}{c}\left(\beta_{i}-\sqrt{\beta_{i}^{2}+2cX}\right)\right)$
$\displaystyle=\frac{1}{N}\partialderivative{\bar{Q}}{\tau_{ij}}-\partialderivative{\beta_{i}}{\tau_{ij}}\left(\frac{1}{c}-\frac{\beta_{i}}{\sqrt{\beta_{i}^{2}+2cX}}\right)>0.$
(110)

∎