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\begin{document}
\title{Mechanisms to finance multiple global public good institutions}
\author[1]{Lennart Stern}%
\affil[1]{École normale supérieure}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
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\sloppy
\section{\emph{Introduction}}
Global institutions with the mandate to promote specific global public
goods (which we shall refer to as 'GPGIs' for 'Global Public Good
Institutions') have so far mostly been financed through voluntary
contributions. One notable exception is UNITAID, a UN organisation
dedicated to AIDS, TB and malaria control that is financed through
taxes on flights by 10 countries so far.
The litterature on 'innovative financing' for global public goods
has discussed taxes and global resources (see for example \href{http://www.lse.ac.uk/GranthamInstitute/wp-content/uploads/2014/02/PP-climate-finance-fund-Romani-Stern.pdf}{http://www.lse.ac.uk/GranthamInstitute/wp-content/uploads/2014/02/PP-climate-finance-fund-Romani-Stern.pdf}
and \href{http://www.un.org/en/development/desa/policy/wess/wess_current/2012wess_overview_en.pdf}{http://www.un.org/en/development/desa/policy/wess/wess\_{}current/2012wess\_{}overview\_{}en.pdf}).
In this article we will provide analysis for both taxes (\ref{3 sec: The-Multiple-Cause})
and global resources (\ref{5 sec:Multiple-Cause-Treaties}).
The most widely discussed proposals for financing mechanisms for global
institutions on the basis of international taxes are based on aviation
taxes. For example the Group of Least Developed Countries (LDCs) made
a proposal (\href{http://www.oxfordclimatepolicy.org/publications/documents/ecbiBrief-IAPAL13Q&As.pdf}{www.oxfordclimatepolicy.org/publications/documents/ecbiBrief-IAPAL13Q\&{}As.pdf})
in 2008 for financing an adaptation fund. Similar proposals have been
made for financing the Green Climate Fund that finances both mitigation
and adaptatio (\href{http://www.lse.ac.uk/GranthamInstitute/wp-content/uploads/2014/02/PP-climate-finance-fund-Romani-Stern.pdf}{http://www.lse.ac.uk/GranthamInstitute/wp-content/uploads/2014/02/PP-climate-finance-fund-Romani-Stern.pdf}).
However, for now the world has only agreed on an offset based mechanism
for international aviation. (The above-mentioned mechanism to finance
UNITAID still has only 10 participating countries) The CORSIA agreement,
signed in October 2016, obliges airlines to offset a certain part
of their future emissions. The International Civil Aviation Organization
will determine by the end of 2018 what kind of offsets are going to
be accepted. It seems likely that various kinds of offsets will be
eligible, including those for reducing tropical deforestation and
those for promoting renewable energy in developing countries (see
\href{https://www.firstclimate.com/corsia/sarp-update/}{https://www.firstclimate.com/corsia/sarp-update/}
\href{https://www.dehst.de/SharedDocs/downloads/EN/project-mechanisms/GMBM-abschlussbericht.pdf?__blob=publicationFile&v=3}{here},
).
In this article we will propose an alternative to this offset based
mechanism which we will call the MCT mechanism (for ``Multiple Cause
Treaty) \ref{3 sec: The-Multiple-Cause}. In our proposal, flights
would instead be taxed and the money would be given to supranational
institutions. Treaty participants would first agree on a list of supranational
institutions that would be considered as `eligible' within the treaty.
Treaty participants would be able to influence the allocation of money
across the various supranational institutions, in ways that will we
detailed below. Each supranational institution would have a specific
mandate to contribute to a particular global public goods. One supranational
institution could have the mandate of reducing tropical deforestation
whilst positively affecting the welfare of the people in the corresponding
jurisdictions (see \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3087881}{emph{https://papers.ssrn.com/sol3/papers.cfm?abstract\_{}id=3087881}}).
Another such supranational institution could have the mandate of promoting
renewable energies in developing countries (see \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3086299}{emph{https://papers.ssrn.com/sol3/papers.cfm?abstract\_{}id=3086299}}).
Further such supranational institution could be the global health
institution UNITAID, which is already financed through taxes on international
flights \href{https://unitaid.eu/\#en}{emph{https://unitaid.eu/\#{}en}}and
the newly established CEPI \href{http://cepi.net/}{emph{http://cepi.net/}}that
focuses on the prevention on future epidemics. Our proposal could
be transposed to the alternative tax base CO2 emissions from international
shipping, based on a recent proposal by Heine et al. (2015) (https://papers.ssrn.com/sol3/papers.cfm?abstract\_id=2512747).
It might also be possible to define a similar mechanism based on taxes
on luxury goods.
We will define a class of treaties that would create participation
incentives based on the fact that countries derive different marginal
payoffs from increasing funding for different public goods. One source
of differences in payoffs is the heterogeneous effect of climate change
across the world as well as the heterogeneity in beliefs of different
governments about its expected impact. In any treaty contracting only
on emissions reductions, a dispersion in payoffs is likely to be a
major obstacle for achieving participation: If a government expects
to not be significantly affected or even be positively affected by
climate change, then it is hard to get it to join such a treaty.
However, in the class of treaties that we will define, this heterogeneity
in payoffs can actually be harnessed to \emph{increase} participation
incentives. For example, the coaltion for Rainforest Nations would
have an incentive to join the treaty so as to be able to direct more
money to the global institution (such as the Forest Carbon Partnership's
Carbon Fund) that incentivises countries to reduce defforestation,
given that any such mechanism will generate informational rents for
these countries. A government or a coalition of governments thinking
that it will benefit from an increase in funding for a global institution
for promoting renewable energy in developing countries would have
an incentive to join so as to direct more funding to that global institution.
On the other hand, a country or coalition of countries that thinks
that it would be even negatively affected (e.g. since its agriculture
might benefit from higher temperature or because accelerating renewable
energy will reduce its rents from fossil fuels), would have an incentive
to join in the treaty, since by doing so it would be able to redirect
funding to a different institution, for example CEPI which works to
prevent future epidemics, a cause that clearly benefits everyone.
Methodologically, our approach is ad hoc: We propose a mechanism that
appears to have some attractive properties, as shown in section \ref{3 sec: The-Multiple-Cause}.
We will proceed similarly for the other proposed source of new financing
for global public goods: global resources such as the seabed resources
from international waters (see \ref{5 sec:Multiple-Cause-Treaties}).
We will define a simple mechanis \ref{subsec: PMF} that could incentivise
countries to increase their voluntary contributions to global public
goods, by again strategically harnessing the fact that countries preferences
over different institutions providing global public goods diverge.
We start in section \ref{2 GPGIs} by giving a partial taxonomy of
global institutions providing particular global public goods. The
examples we provide are of particular interest for the mechanisms
that we will define in sectoin \ref{3 sec: The-Multiple-Cause}. In
section \ref{4 sec:Global-Public-Good} we will develop a theory for
the countries' voluntary monetary contributions to global institutions
providing global public goods. We will obtain formulas for the crowding
out and potentially for the crowding in of voluntary contributions
\ref{5 sec:Multiple-Cause-Treaties} induced by the financing mechanisms
already mentioned. The formulas will depend on certain elasticities.
In section \ref{6 sec:How-to-learn} we provide reasons for thinking
that estimating these elasticities might be valuable for learning
about crowding out and crowding in. In section \ref{7 sec:Combining-the-MCT}
we define a way of combining the two types of mechanisms analysed
previously.
\section{\label{2 GPGIs}Global Public Good Institutions (GPGIs): A partial
taxonomy and some examples of existing and proposed GPGIs}
\begin{defn}
\label{Donor-independent-Global-Public} Donor-independent Global Public
Good Institution (donor-independent GPGI).
A donor-independent GPGI is an global institution dedicated to a particular
global public good whose action only depends on its available money
(i.e. it is in particular independent of who contributed the funding).
\end{defn}
Throughout this article we will be assuming that all the global institutions
that we will be studying are donor-independent GPGIs.
\begin{defn}
\label{Direct-Purchase-Global} Direct Procurement Global Public Good
Institution (Direct Procurement GPGI)
A Direct Procurement GPGI is a donor-independent GPGI dedicated to
a particular global public good that uses its available budget to
procure goods and services contributing to that global public good
without conditioning its actions on the actions of governments.
\end{defn}
\begin{example}
The Coalition for Epidemic Preparedness Innovation (CEPI) \label{CEPI}
CEPI (http://cepi.net/mission) has the stated mission of ``stimulating,
financing and co-ordinating vaccine development against diseases with
epidemic potential in cases where market incentives fail''. The founding
members clearly decided on the approach taken by the institution.
However, going forward, it might be appropriate to model CEPI as a
donor independent GPGI. Indeed, the first vaccine targets that the
institution will pursue were chosen from the WHO\textquoteright s
R\&D Blueprint for Action to Prevent Epidemics, which is independent
of who is funding CEPI.
The institution does not (at least not explicitly) condition its actions
on the countries actions. Thus it seems appropriate to model CEPI
as a Direct Procurement GPGI. \ref{Direct-Purchase-Global}.
\end{example}
\begin{defn}
Reward Payment Based Global Public Good Institutions (Reward Payment
Based GPGI)
A Reward Payment Based GPGI is a donor-independent GPGI that uses
its entire budget to make reward payments to countries depending on
variables that they can influence (typically outcomes and policies).
\end{defn}
\begin{example}
A proposed Global Fund for Pandemic Preparedness
Pandemic prevention expert Elizabeth Cameron explains \href{https://80000hours.org/podcast/episodes/beth-cameron-pandemic-preparedness/}{https://80000hours.org/podcast/episodes/beth-cameron-pandemic-preparedness/}
the benefits that a global fund for Pandemic Preparedness could have:
``I think one of the major things that would be incredibly helpful
in this space would be to have a fund that would allow countries that
want to put more money into pandemic preparedness to be able to draw
upon it as levarage to get some additional funding in their own budget
to improve pandemic preparedness and then to be able to measure over
time that they have actually been able to fill the gaps as a way to
leverage and increase their own financing from their host ministeries.
I think that there are a tremendous number of organisations that are
interested in putting funding forward for pandemic preparedness but
people are not quite sure where to put that money, where it is going
to be measured. It is easier to see an effect for a specific drug
or vaccine or treatment: number of people treated, number of people
vaccinated. For health security there is a large number of indicators
and it is harder to see how a bit of funding makes a difference, so
having a constructive way to do that, which is organised, I think
is really important.''
This analysis suggests that the measurement of pandemic preparedness
that would be necessary to define a Reward Payment Based GPGI for
this cause would increase donors' incentives to contribute due to
the greater clarity about the impact of their spending. In \ref{reward payment based GPGI}
we will provide a further theoretical argument for why establishing
Reward Payment Based GPGIs might be very valuable.
\end{example}
\begin{example}
The Forest Carbon Partnership's Carbon Fund \label{FCCP: The-Forest-Carbon}
This Fund rewards countries with tropical forest as a function of
the policies they pursue to conserve their tropical forests: \href{https://www.forestcarbonpartnership.org/carbon-fund-0}{https://www.forestcarbonpartnership.org/carbon-fund-0}
. In the future this institution could also condition its payments
on results (amount of tropical forests conserved), as Norway already
does in bilateral mechanisms. In either case, these mechanisms fit
our definition of Reward Payment Based GPGIs.
\end{example}
\begin{example}
IRENA and a proposed Renewable Energy Support Fund (RES-Fund) \label{RES-Fund IRENA}
The International Renewable Energy Agency (IRENA) is an intergovernmental
organization to promote adoption and sustainable use of renewable
energy. Thus far, IRENA has been conducting research and disseminating
information \href{https://www.irena.org/en/aboutirena}{https://www.irena.org/en/aboutirena}.
It has been proposed that under the auspices of IRENA a global fund
be established that would reward developing countries as a function
of their policies to support renewable energy, for example by reimbursing
a given fraction of subsidies that any developing country government
pays for renewable energy. (\href{https://api.istex.fr/document/E2D4731EF6FC6E91AE71624EDF3A29683EF1F7C6/fulltext/pdf?sid=sfx/psl,istex-view}{https://api.istex.fr/document/E2D4731EF6FC6E91AE71624EDF3A29683EF1F7C6/fulltext/pdf?sid=sfx/psl,istex-view})
This would be a mechanism contracting on policies but in geneneral
such a mechanism could contract jointly on policies and outcomes (\href{https://www.cgdev.org/sites/default/files/cash-on-delivery-aid-energy.pdf}{https://www.cgdev.org/sites/default/files/cash-on-delivery-aid-energy.pdf}).
In either case, such a fund would fit our definition of Reward Payment
Based GPGI. We will refer to such a GPGI througout this article as
an 'RES-Fund'.
\end{example}
\begin{example}
A Proposed Global Fund to reward countries as a function of carbon
emissions and carbon taxes
Such a GPGI has been proposed by Gersbach (\href{https://www.econstor.eu/bitstream/10419/161824/1/cesifo1_wp6385.pdf}{https://www.econstor.eu/bitstream/10419/161824/1/cesifo1\_{}wp6385.pdf}).
This proposal fits our definition of Reward Payment Based GPGI.
Such a mechanism could fit in well with the pledge an review framework
of the Paris agreement: If countries are rewarded for taxing carbon
emissions, then they are likely to be willing to make more ambitious
pledges in the next round of pledge and review. Moreover, if the international
institution implementing such a global fund had perfect information,
then the money it would require per emissions reductions achieved
through the reward payments would be small, at least as long as the
level of emissions reductions attained is not so large. (see \ref{reward payment based GPGI}).
However, such a global institution might be bound to not taylor its
mechanism to the information that it has about the payoff functions
of countries. Since countries differ in their subjective costs of
implementing carbon taxes or reducing their emissions in other ways,
informational rents will arise, which can greatly limit how much emissions
reductions the mechanism can achieve (Martimort and Sand-Zantman 2016,
\href{https://academic.oup.com/jeea/article/14/3/669/2194880}{https://academic.oup.com/jeea/article/14/3/669/2194880}).
In the MCT mechanism that we will now define such informational rents
will be strategically harnessed: They will increase the incentives
to participate in (the first stage of) the mechanism which finances
the various GPGIs.
\end{example}
\section{\emph{\label{3 sec: The-Multiple-Cause}The Multiple Cause Treaty
(MCT): Analysis of a Proposal}}
\subsection{Definition of the proposed Multiple Cause Treaty (MCT)}
The MCT defines a \emph{list of eligible GPGIs,} $A_{1},\ldots,A_{n}$.
We denote the set of all countries by $I$. Each participant $i\in I$
to the MCT is required to levy a tax with rate $x$ on carbon emissions
from all outgoing international flights and also on incoming international
flights coming from a country that does not participate. We shall
denote the corresponding tax revenue by $M_{i}$\emph{.} For simplicity,
we assume that air travel is completely inelastic to the carbon tax
applied. We denote by $f_{\text{ij}}$ the carbon emissions from flights
from country $i$ to country $j$, by $I$ the set of countries, by
$J$ the set of participant, we have:
\begin{equation}
u_{i}=x(\sum_{j\in J}f_{\text{ij}}+\sum_{j\in I\backslash J}f_{\text{ij}}+f_{\text{ji}})
\end{equation}
Each participating country can choose among the following two options:
\begin{enumerate}
\item \emph{\label{enu:Be-a-participant}Be a participant without the right
to influence.} In this case the participant can retain a fraction
$r$ of $M_{i}$ for himself. We shall call $r$ \emph{the} \emph{retention
rate parameter} . The remaining amount, $\left(1-r\right)M_{i}$,
is added to the money at the disposal of all the participants who
have chosen to be a participant with right to influence, in a way
that scales up these amounts proportionally. Formally, let $J_{\text{wtihout}}$be
the set of participants without the right to influence and $J_{\text{with}}$
the set of participants with the right to influence. For all participants
with the right to influence, we define $G_{i}=M_{i}\ \frac{\left(1-r\right)\sum_{i\in J_{\text{without}}}M_{i}+\sum_{i\in J_{\text{with}}}M_{i}}{\sum_{i\in J_{\text{with}}}M_{i}}\ $\emph{.}
\item \emph{Be a participant with right to influence.} In this case the
participant can decide on the allocation of the amount $G_{i}$ to
the $n$ causes, i.e. he has to choose a vector $\left(A_{1}^{i},\ \ldots,\ A_{n}^{i}\right)\in\mathbb{R}_{\geq0}^{n}$
such that $\sum_{k=1}^{n}A_{k}^{i}=G_{i}$.
\end{enumerate}
The following diagram illustrates the rules for taxation inscribed
in the treaty for the case of a world consisting of the four countries
$(a,a',b,b')$ in a situation where$a$ and $a'$ participate whilst
$b$ and $b'$ do not:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/taxationrules/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
\label{enu:taxationrules}
Here the green strokes near $a$ indicate the flights that this country
is obliged to tax by the treaty and similarly for country $a'$. Each
participating country can allocate this money to one of the global
institutions that are specified in the list $A_{1},...,A_{n}$ in
the treaty.\\
If a country $b$ does not comply with all the rules of the treaty
then by the treaty it is to be considered a 'non-participant'. For
example, if a country taxes international flights in line with the
treaty rules but does not comply with either option 1 or option 2
(e.g. by keeping a greater fraction of the tax revenues collected
than the retention rate parameter $r$), then the participating countries
are obliged by the treaty to tax the emissions from all flights connecting
it with country $b$ at the rate $x$.
\subsection{Defining rules for changing participation status and withdrawing
from treaty}
One possible rule would be to allow any country to change between
the three actions (participate without right to influence, participate
without right to influence, not participate) at any time. We will
assume this rule for the purpose of this article and define a model\ref{subsec:A-model}taylored
to this version of the treaty.
\subsection{\emph{\label{sec:An-illustrative-simulation}An illustrative simulation}}
We consider the case where the list of eligible GPGIs in the Multiple
Cause Treaty has two elements: An institution for Global Pandemic
Prevention (e.g. CEPI), which we will refer to as the 'GPP-Fund' and
an institution dedicated to supporting renewable energy in developping
countries, which we will refer to as the 'RES-Fund'.
\subsubsection{\emph{\label{subsec:A-model}A model}}
We shall assume that each country $i$'s utility is given by:
\begin{equation}
u_{i}=\sum_{k=1}^{n}\theta_{k}^{i}\sum_{j\in J_{\text{with}}}A_{k}^{j}+t_{i}-c_{i}x(\sum_{j\in I\backslash J}f_{\text{ij}}+f_{\text{ji}})
\end{equation}
Here $t_{i}$ is the transfer that country $i$ receives and $\theta_{k}^{i}$
is a constant rate at which country $i$ derives utility per unit
of money allocated to the GPGI $k$. As above, we denote by $f_{\text{ij}}$
the carbon emissions from flights from country $i$ to country $j$.
Countries simultaneously choose their action from the set \{participate
without right to influence, participate without right to influence,
not participate\}. We will study the set of pure strategy equilibria
of this game. This is motivated by the fact that we are considering
the version of the treaty where countries can change between the three
actions at any time.\\
\begin{prop}
\label{Suppose-.-Then} Suppose $c_{i}=\frac{1}{2}$ and suppose that
$\theta_{k}^{i}\geq\text{0}$ for some $k$. Suppose $r=0.5$. Then
not participating is a dominanted strategy for country $i$.
proof:
By participating with right to influence county $i$ can keep half
of the tax revenue, which already compensates it fully for the cost
of the taxes. Actually, if some other countries participate, then
the additional tax burden born by participating as compared to not
participating is actually lower, so the country will be strictly better
off by participating without right to influence. Moreover, since $\theta_{k}^{i}\geq\text{0}$
the additional funding to the global institutions can only affect
country $i$'s payoff positively. $\square$
\end{prop}
\begin{prop}
Suppose $r=0.$ Then the profile of actions where each country participates
with right to influence is a Nash equilibrium.
proof:
No country gains from switching to participating without right to
influence, since $r=0$. Moreover, by switching to not participating
a country does not change the taxes weighing on its flights. Moreover,
it looses the possibility to decide on the allocation of the tax revenue
collected from its outgoing international flights. $\square$
\end{prop}
The Nash equilibrium with full participation that we have for $r=0$
is, however, very brittle: Coalitions of countries will reduce their
tax burden by withdrawing from the treaty. Therefore, if each country
in a given coalition does not care much about the allocation across
the different GPGIs, then each of them will benefit from withdrawing.
We will assume that the countries act in exogenous coalitions:
\subsubsection{Partitioning the world into players}
In the literature analyzing the climate negotiations it is common
to model countries as being organized in typically around 12 blocks,
with each block acting as a single agent (Hovi, Ward, Grundig 2015),
(Eyckmans, Finus 2009), (Carraro 2009), (Chander, Tulkens 2006). We
shall model the countries of the world as acting in the 12 following
blocks: United States, European Union, Japan, Russia, Eurasia, China,
India, Middle East, Africa, Latin America, Other high Income, Other
non-OECD Asia. The alternative partitions used in the literature on
climate negotiations do not differ substantially from this, which
suggests that it might represent a consensus view of the appropriate
partition for modelling climate negotiations.
Admitting this, the question still arises as to whether this partition
is the appropriate for analysing the MCT treaty. In fact, the partition
into players might be endogenous to the mechanism. We will analyse
this further in section \ref{sec:Endogenous-coalition-formation}.
\subsubsection{A method for estimating the payoffs in the game\label{subsec:A-method-for}}
\paragraph{Cost due to the tax burden}
We will assume that $c_{i}=0.5\forall i$. In other words: We will
assume that each country values 1 dollar of additional taxes weighing
on international flights landing on or departing from its territory
in the same way as it would value the obligation to pay 0.5 additional
dollars from its general budget. This would be correct if the burden
of the tax on any internaitonal flight is born in equal proportions
by the country of origin and the country of destination and if the
cost per dollar raised via taxes on international flights is equal
to the cost of raising an additional dollar via alternative taxes.
To estimate the network of emissions from international flights I
have used the open source data from the section \textquoteleft Global
Flight Data\textquoteright{} from http://www.worldpop.org.uk/data/data\_sources/.
From this data I computed for each pair of airports A and B the available
seat kilometers by multiplying the estimated number of seats flying
directly from A to B by the spherical distance computed from the longitudes
and latitudes. I took these numbers as an estimate for the associated
greenhouse gas emissions. In the future I will try to construct better
estimates of the greenhouse gas emissions by using data that I obtained
from the author of (Lawyer 2016) which contains information on the
aircraft used and their seat capacity.
\subsubsection{Effects of an additional dollar given to the RES-Fund}
Here we assume an additional dollar given to the RES-Fund accelerates
the transition to renewable energies. We assume that this has two
effects on countries' payoffs: Firstly, it decreases the damages from
climate change. We denote by ACD (for 'average climate damage averted')
the sum of these effects over all the players. To compute the distribution
of these effects over the players, we use the results from the RICE
model (ref).
The second effect of the acceleration of the transition to renewable
energy that we consider is redistributive: Fossil fuel prices will
decrease and this will positively affect net fossil fuel importers
and negatively net fossil fuel exporters. We denote by ORR the reduction
in the fossil fuel rents per additional dollar put into the RES-Fund.
To estimate the distributional effect of this, we compute for each
player the ratio of net oil exports to total global oil production
and then multiply it by ORR.
We have not found any estimates for the value of the parameter ACD.
For that reason we have considered conservative estimates in the simulations
explained in section \ref{subsec:Some-results}.
\paragraph{Effects of an additional dollar given to the GPP-Fund}
Here we assume that an additional dollar increases each country's
payoff by an amount proportional to its GDP. We denote by PP the parameter
that gives the aggregate benefit of an additional dollar given to
the GPP-Fund.
\paragraph{Ideas for improving the estimates of countries' payoffs}
It seems plausible that including informational rents will be important.
Consider the example of the RES-Fund: This fund would reward developing
countries for their policies (e.g. the aggregate subsidies paid for
renewable energy) and their results (e.g. the total amount of renewable
energy produced) according to some formula. Under perfect information
the fund would pay each developing country just enough to make it
willing to participate. However, in reality the subjective costs of
implementing renewable energy support policies are private information.
As a result, countries will reap some informational rents.
\subsubsection{Some results\label{subsec:Some-results}}
The following graphs summarize important aspects of the game that
the MCT treaty induces. On the x-axis we have the retention rate parameter
$r$ (see \ref{enu:Be-a-participant}). On the y-axis we plot 5 different
characteristics of the game. Here are the definitions of the characteristics
plotted:
Maximal Proporiton of Participants: Amongst all the Nash equilibria
we look at the one with the largest number of participants. Then we
divide this number by the number of players (which in our simulation
is 12).
Maximal Proportion of feasible Welfare gains reached: the ratio of
welfare gains achieved relative to what could be achieved if all players
participated and all the tax revenue was allocated to the best global
institution
Maximal Proportion of full aggregate contributions: the ratio between
the total amount of money given to the global institutions relative
to the total amount of tax revenue that would be collected if all
players participated
As a first example, we have taken the case with ORR=1, PP=2 and ACD=2
(see \ref{subsec:A-method-for} for the definitions).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/20stepsORR1andPP2andACD2/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
We see that for retention rate parameters $r$ smaller than 10\% there
is only one Nash equilibrium consisting of the profile of actions
where no player participates. For $r=15\%$ there is exactly one Nash
equilibrium. It has a single participant. For r=20\% and r=25\% there
is no Nash equilibrium. A drastic change occurs once we set r=30\%:
Now there exists a Nash equilibrium with full participation. Moreover,
there exists a Nash equilibrium where more than 80\% of the total
tax revenue collected are allocated to the global institutions. We
thus see the incentives to influence the allocation across global
institutions at work, since most countries choose to exert their right
to influence.
In this example the global welfare gains are equal across the two
global institutions. For that reason the red line does not show up
on the graph, since the proportion of feasible welfare gains achieved
(i.e. the ratio of welfare gains achieved relative to what could be
achieved if all the tax revenue was allocated to the best global institution)
always equals the proportion of full aggregate contributions (i.e.
the ratio of money allocated to the global institutions to the total
amount of tax revenue raised).
We observe that for $r=0.3$ and $r=3,5$ there are 5 and 11 Nash
equilibria, respectively, some of which have only 3 countries participating.
However, let us for the moment make the optimistic assumption that
the Nash equilibrium with the highest aggregate payoff is selected.
We see that this aggregate payoff is highest at $r=0.3$. For the
lower values there are insufficient participation incentives to sustain
a Nash equilibrium with full participation. However, increasing $r$
beyond $0.3$ decreases the maximal aggregate welfare that is achievable.
Firstly, there is a mechanical effect: Given a profile of strategies,
increasing $r$ leads to the players keeping more of the tax revenue
for themselves, thus decreasing the amount that is allocated to global
institutions. Secondly, the higher $r$ makes it more attractive for
players to choose the participation status without influence rather
than with influence. This accounts for the segments of the yellow
line that are more steeply downwards than the others: Here increasing
the $r$ has led to the previously welfare maximising Nash equilibrium
to no longer be a Nash equilibrium, as for some countries deviating
to the status without the right to influence has become profitable.
Interestingly, however, we do not see any very sudden decreases in
the yellow line. This can be explained by the following observation
about the mechanics of the mechanism: When a player renouces on his
right to influence, this increases the amount of money that the countries
that choose the status with the right to influence can allocate, which
increases their incentive to choose to have the status with the right
to influence. This stabilisation effect occurs not only in the comparative
statics with respect to the parameter $r$ but but to comparative
statics more generally, in particular with respect to the payoffs.
Thus, for example if one player's payoff changes (e.g. due to a change
in government), this might lead it to switch to the participation
status without right to influence, but we probably need to fear too
much that this will lead to other countries doing the same.
As a second example, we have taken the case with ORR=2, PP=2.5 and
ACD=1.5 (see \ref{subsec:A-method-for} for the definitions of these
parameters). Here the payoffs are more polarized than in our previous
example, for two reasons: Firstly, we are assuming that the redistributive
effects of the RES-Fund are higher and secondly, we are assuming that
pandemic prevention is on average valued more highly than climate
change mitigation. However, we find a rather similar pattern:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/20stepsORR2andPP2\lyxdot-5andACD1\lyxdot-5/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
Again, in order to get Nash equilibria with full participation, we
need to set $r$ at least to 0.3. However, now we already have Nash
equilibria with substantial participation starting at $r=0.2$.
One peculiar phenomenon arises at $r=0.7$: There is no Nash equilibrium!
However, this should not worry us too much in light of proposition
\ref{Suppose-.-Then}: Not participating is a strictly dominated strategy
at $r=0.7$. In any case, if we think that $c\leq\frac{1}{2}$ then
by proposition \ref{Suppose-.-Then} we probably should not even consider
setting $r$ above $0.5$, as the following arguments suggest: Setting
$r\geq\frac{1}{2}$ is unnecessarily high to get all players to participate
by proposition \ref{Suppose-.-Then} and it erodes the total amount
of money mobilized for global institutions through the effects that
we already mentioned: Firstly, there is a mechanical effect: Given
a profile of strategies, increasing $r$ leads to the players keeping
more of the tax revenue for themselves, thus decreasing the amount
that is allocated to global institutions. Secondly, the higher $r$
makes it more attractive for players to choose the participation status
without influence rather than with influence. The following consideration
might potentially weigh in favor of a higher $r$: It will cause countries
that do not have a strong preference over the different causes to
renounce on their right to influence. Thus those with a stronger preference
across causes will have a greater influence over the allocation, potentially
improving efficiency. However, our numerical results do not support
this argument: We do not see a systematic narrowing of the red and
yellow line as we increase $r$. Moreover, with the exception of the
step from 0.6 to 0.65 the red line is not increasing anywhere.
\subsubsection{Considerations for the choice of $r$}
Our previous discussion suggests that $r$ should be chosen just high
enough to enable full participation as a Nash equilibrium. As our
results illustrated, increasing $r$ beyond that will lower welfare.
A further consideration regarding the retention rate parameter $r$
concerns the legitimacy of the mechanism, which will be important
to ensure if a subset of the countries start implementing the mechanism
(with the prospect of inducing the other countries to participate
subsequently). For a given set of participating countries and a given
level at which they tax international flights, the legitimacy of the
mechanism is an increasing function in the the amount of funding provided
via this mechanism to global institutions. To the extent that the
CCGF is perceived to mainly serve the national interests of the participating
countries rather than the global public goods, the non-participating
countries are likely to oppose it, given how many countries opposed
the EU\textquoteright s attempt to price the carbon emissions of international
flights. Indeed, the degenerate case of a MCT with $r=1$ would likely
meet opposition from countries that do not participate that could
be similar to the opposition the EU experienced in 2012 until it exempted
international flights from its emissions trading scheme. Hence the
legitimacy concern reinforces our conclusion that $r$ should not
be chosen higher than the level necessary to achieve full participation.
\subsubsection{\emph{\label{sec:Endogenous-coalition-formation}Endogenous coalition
formation}}
In the simulation in section \ref{subsec:Some-results} we have been
assuming that there are 12 players given by the partition of the world's
countries used in the RICE model. The following questions arise: How
should expect the results to change as a function of the partition
of countries into players? What endogenous partition of the set of
countries into players should we expect as a result of the MCT mechanism?
Consider the profile with full participation. Consider a coalition
of countries acting as a single player that is of size $x$. The gain
of participating with the status without right to influence is roughly
proportional to $x$. Consider the gain from participating with the
status with the right to influence. The money that the coalition can
influence is roughly proportional to $x$. Moreover, per dollar the
gain that the coalition derives from moving it to its preferred cause
is roughly proportional to $x$. Hence the gain from participating
with the status with the right to influence is roughly proportional
to $x^{2}$. This suggests that typically small players will participate
without the right to influence and large players will participate
with the right to influence rather than with the right to influence.
The cost of participating is proportional to the ghg emissions from
flights between members of the coalition, which is roughly proportional
to $x^{2}$. This means that sufficiently small coalitions will be
better off participating without the right to influence than not participation
at all.
Another property to note is that any move of countries choosing to
retain money by selecting the participation status without right to
influence is somewhat self-limiting: The more countries have selected
this option, the more money the remaining countries that have selected
the participation status without right to influence will be able to
direct, and hence the greater the benefit of selecting this option.
So far we have been assuming that the set of players is exogenous.
However, let us now turn to analysing the incentives to form coalitions
that the MCT treaty would create. Consider a given set of players
and a given strategy profile. What incentives are there for two players
to merge so as to act as a single player? Consider first the case
where both players have the same preferred global institution and
suppose both participate with the status without right to influence.
In this case switching to the status with right to influence exerts
a positive externality on the other player, who will now also benefit
from the reallocation of money to their preferred global institution.
Cooperating in this way will also typically increase global welfare
(This is true as long as the global benefit of marginal dollars to
the global institution favored by the two players is not much lower
than the alternative global institution.)
Now consider the case where the two players' preferred global institution
differs and both participate and choose the status with the right
to influence. For simplicity suppose that the list of global institutions
defined as eligible in the treaty has only two elements. Then if the
situation is not very asymmetric, the reallocations that the two players
achieve by exerting their right to influence might just cancel each
other. Thus by both swtiching to the status without right to influence
they might both benefit, as they keep part of the tax revenue. Typically,
such cooperation between the two countries will decrease global welfare,
since it decreases the overall amount of money raised for global institutions
without substantially affecting the allocation across them.
Our discussion suggests that if countries who prefer the same global
institution merge to act as a single player, this will beneficial
for global welfare whereas if countries that differ in their preferred
global institution merge to act as a single player, this will be detrimental
to global welfare. There are reasons to think that the first case
is more likely to occur, since countries with similar interests are
more likely to have intergovernmental institutions that allow them
to coordinate their actions so as to act as a single player. If the
Global Forest Partnership was included in the list of eligible causes,
then this seems particularly plausible: The Coalition for Rainforest
Nations (CfRN) is an intergovernmental Organisation that includes
all the important countries with tropical forests. The CfRN appears
to have been instrumental in the establishment of the Forest Carbon
Partnership. Any mechanism to incentivise governments to conserve
tropical forests will give informational rents to the participants.
In the presence of the MCT mechanism the member countries would therefore
have an incentive to cooperate and all participate with the status
with right to influence. It seems likely that the CfRN will enable
member countries to cooperate again when it comes to the MCT treaty.
This analysis also has implications for the design of global institutions
that would be financed via the MCT. It suggests that more value should
be attached to informational rents accruing to participants, since
rents create the strategic benefit of increasing their incentives
to participate in the MCT mechanism.
\subsubsection{\emph{Considerations weighing potentially for or against the proposal
of reforming CORSIA to move to an MCT treaty}}
Our results so far suggest that amending the CORSIA treaty to move
from the envisaged offset based system would have the advantage of
increasing participation incentives. In fact, in the CORSIA treaty
participating countries cannot influence the allocation of money across
the different offset projects. Once it has been fixed what offsets
are eligible under the treaty, it will the companies who will determine
which offsets to buy, whether it be projects to reduce tropical deforestation,
to promote renewable energy in developing countries or other projects.
Thus the allocaiton of funds will probably be determined by the equalisation
of marginal costs.
It has been argued that countries ended up participating in CORSIA
because of the credible threat from the EU to unilaterally (see \href{https://www.redshawadvisors.com/corsia-and-the-inclusion-of-aviation-in-the-eu-ets/}{emph{https://www.redshawadvisors.com/corsia-and-the-inclusion-of-aviation-in-the-eu-ets/}})
reintegrate international aviation in the EUTS, which would result
in the other countries bearing the implicit incidence without benefiting
from any global public good provision. This effect might not be very
large but still sufficient to induce participation in CORSIA, given
its low level of ambition. In fact, CORSIA is estimated to generate
a demand for offsets of only 2,5 billion tons from 2021-2035 (see
\href{https://unfccc.int/files/na/application/pdf/04_current_cer_demand_cdm_and_art__6_of_the_pa_nm.pdf}{emph{https://unfccc.int/files/na/application/pdf/04\_{}current\_{}cer\_{}demand\_{}cdm\_{}and\_{}art\_{}\_{}6\_{}of\_{}the\_{}pa\_{}nm.pdf}}).
If the offset price is 10 dollars per ton, this corresponds to 1,7
billion dollars per year, which corresponds to a ticket tax rate of
only 0,3\% \href{https://www.statista.com/statistics/278372/revenue-of-commercial-airlines-worldwide/}{emph{https://www.statista.com/statistics/278372/revenue-of-commercial-airlines-worldwide/}}.
However, ramping up the ambition of CORSIA by setting higher offset
obligations might lead to some countries quitting. With the MCT treaty
that we have proposed above, it might be possible to mobilize much
more funding whilst still keeping all countries participating.
Another consideration suggesting that reforming CORSIA to move to
an MCT treaty is the fact that supranational institutions can contract
on variables that offset based mechanisms cannot in the same way.
Specifically, a supranational institution with the mandate to achieve
reductions in deforestation could contract with government jointly
on deforestation reductions and on universal cash transfers to the
people in their jurisdiction. The analysis provided in \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3087881}{emph{https://papers.ssrn.com/sol3/papers.cfm?abstract\_{}id=3087881}}suggests
that there might be efficiency gains to be reaped from contracting
jointly on the two variables.
Thirdly, contrary to the CORSIA treaty, the MCT treaty would allow
governments to use taxes that could increase efficiency even more
than carbon taxes (see \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2744402}{emph{https://papers.ssrn.com/sol3/papers.cfm?abstract\_{}id=2744402}}).
In fact, for simplicity of exposition we included in the definition
of the MCT treaty given above the requirement that each participating
country impose carbon taxes at a rate $x$ on the routes explained
there. This requirement could be relaxed by only requiring each country
to collect enough taxes such that the ratio of the tax revenue collected
to the carbon emissions is at least $x$. Some countries might then
chose `per plane taxes' or taxes on vacant seats. (see \ref{subsec:International-Aviation})
The question arises as to whether there are any drawbacks to reforming
CORSIA towards an MCT treaty. One potentially relevant consideration
pointing in this direction is given by Victor (2011), who argues that
having multiple parallel offset standards in the world could induce
each standard to experiment. Donors would evaluate the performance
of the various offset mechanisms and select the best. Via the process
of selecting eligible offsets currently underway, CORSIA seems to
already contribute to the incentives for standards to show environmental
integrity \href{http://www.v-c-s.org/icaos-corsia-and-the-case-for-the-open-sourced-carbon-market/}{emph{http://www.v-c-s.org/icaos-corsia-and-the-case-for-the-open-sourced-carbon-market/}},
giving relevance to this argument for our context.
However, it is unclear how much learning from direct observation can
take place, as far as the counterfactual impact of the offset systems
is concerned. Moreover, the modeling provided in \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2957896}{emph{https://papers.ssrn.com/sol3/papers.cfm?abstract\_{}id=2957896}}suggests
that by contracting with governments instead of with private agents
(as is mostly the case in offset mechanisms), one can achieve greater
emissions reductions for a given budget.
A second argument made in the literature in favor of offset based
mechanisms is that they might increase the political acceptability
of transfers to other countries by concealing them (Gollier, Tirole
2017). If a government pays monetary transfers, so the argument goes,
then domestic opposition voices can accuse it of not maximising the
country's payoff. However, in the case of the MCT treaty, if there
is full participation then a government of a participating country
can hardly be accused of `not putting their own country's interest
first'. Firstly, by withdrawing from the treaty, the country will
not save on any tax burden it endures. Secondly, by participating
the country can move funding to the international institution from
which it derives most benefits.
In terms of administrative costs, the offset mechanism might be cheaper
than the mechanisms by which a global institutions incentivise governments
to reduce emissions (by conserving tropical forests, subsidizing renewable
energy, respectively). Indeed, for the offset mechanisms these administrative
costs are proportional to the amount of projects covered. Moreover,
existing offset standards will be used (\href{https://www.firstclimate.com/corsia/sarp-update/}{https://www.firstclimate.com/corsia/sarp-update/}),
which have already incurred fixed costs for being developed. In the
case of mechanisms implemented by Reward Payment Based GPGIs there
is a new and potentially larger fixed cost involved: Baselines have
to be computed for the entire jurisdictions that are participating.
The CORSIA agreement might therefore be a good start for starting
cooperation with a low level of funds. However, once cooperation is
deepened and more funds are raised, it is likely to be more efficient
to move to financing Reward Payment Based GPGIs. The MCT proposal
could be an opportunity to both deepen cooperation and to realise
efficiency gains that become available at that deeper level of cooperation.
\subsubsection{Further global institutions that could be added to the list of eligible
institutions in the MCT treaty}
We have considered the case where the list of eligible GPGIs comprises
only a GPGI for pandemic prevention and a GPGI for supporting reneawable
energy in developing countries. Here are some other GPGIs that could
be added to the list of eligible GPGIs:
\paragraph{UNITAID/ The Global Fund to fight AIDS, TB and Malaria}
Since UNITAID is already being financed through taxes on international
flights, it is a natural candidate of a GPGI that at least some countries
would like to include in the list of eligible GPGIs. The Global Fund
to fight AIDS, TB and Malaria has a similar mission.
\paragraph{A Fund to reward jurisdictions for reducing deforestation by paying
universal transfers to their populations}
In \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3087881}{https://papers.ssrn.com/sol3/papers.cfm?abstract\_{}id=3087881}
we have argued that there are efficiency gains to be reaped from bundling
the two objectives, namely poverty alleviation and forest conservation
in a single mechanism.
\subsection{Transposition of the MCT mechanism to alternative tax bases}
We have focused so far on mechanisms based on the taxation of greenhouse
gas emissions from international aviation. However, the mechanism
can be transposed to other tax bases:
\subsubsection{International Aviation \label{subsec:International-Aviation}}
For simplicity of exposition we have considered a treaty that would
oblige participating countries to introduce carbon taxes at rate $x$
per ghg emissions for international flights. An alternative version
of the treaty would instead allow participating countries to choose
the kind of taxes to levy on the international flights, with the only
obligation that for all routes the ratio of the revenue collected
to the overall emissions be at least $x$. The options for the allocation
of the corresponding money would be as in the original version of
the proposal that we laid out above.
Allowing such flexibility might allow countries to reap additional
efficiency gains, as we have shown in related work (\href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2744402}{https://papers.ssrn.com/sol3/papers.cfm?abstract\_{}id=2744402}).
A further way to facilitate participation would be to allow for countries
to use a cap-and-trade system. Here, the obligation could be defined
as follows: The prices at which the permits were auctioned (either
entirely or partially) would have to be at least $x$. Again, the
options for the allocation of the money, $xE$, where $E$ are the
emissions from international flights, would be as in the original
version of the proposal that we laid out above.
\subsubsection{International Maritime Shipping}
It was long considered to be infeasible to have a mechanism for taxing
maritime shipping that could be initiated by a subset of countries,
given the ease with which ships can avoid fuel taxes by taxing in
jurisdictions that do not participate, and given a host of international
legal constraints.
However, thanks to a recent concerted effort of legal scholars and
economists (Heine et al 2015) a proposal has been elaborated that
navigates all these constraints and appears to be waterproof: https://papers.ssrn.com/sol3/papers.cfm?abstract\_id=2512747.
The MCT treaty proposal that we laid out here can be combined with
the taxation mechanism defined by by Heine et al. (2015).
As far as the incidence of carbon taxes on shipping is concerned,
poor countries are expected to be negatively affected (https://www.aeaweb.org/articles?id=10.1257/pol.20150168),
if each country retains all its collected tax revenue. For the MCT
treaty this conclusion might be reversed if at the equilibrium sufficient
money is allocated to GPGIs such as UNITAID that provide a global
public good that particularly benefits poor countries. However, to
ensure that the treaty will lead to a Pareto improvement, modifications
to the MCT as defined above might be required. One approach would
be to determine exemptions to allow the poorest countries to not tax
their shipping at all. This is the approach that has been taken in
the 2016 CORSIA agreement on aviation. The MCT treaty could in the
case of aviation simply adopt the same exemptions as in the CORSIA
agreement. Alternatively, the MCT mechanism could be modified to allow
the poorest countries to keep all the tax revenue that they collect.
\subsubsection{Luxury goods}
Taxes on luxury goods have been discussed as a potential source for
financing global public goods (http://www.un.org/en/development/desa/policy/wess/wess\_current/2012wess\_overview\_en.pdf).
In another article (forthcoming) we explore some speculative ideas
for how a taxation mechanism could be defined that would have similar
strategic properties as the one that we defined here for the case
of aviation (or maritime shipping).
\section{Global Public Good Institutions \label{4 sec:Global-Public-Good}financed
via voluntary contributions}
So far we have been implicitly assuming that the voluntary contributions
that countries make to the GPGIs are unaffected by the contributions
made via the MCT mechanism. We will later try to better understand
how these voluntary contributions might change endogenously in response
to changes in financing from other sources such as the MCT. In preparation
for that task we will study the question of how voluntary contributions
to a given GPGI will endogenously change in response to a change in
an exogenous additional funding for that GPGI. This question is of
course also relevant in its own right, for example for assessing existing
proposals to use global resources to finance particular GPGIs, as
we will discuss later \ref{subsec:Examples-of-proposals for capturing global resources}.
\subsection{The neutrality theorem in a model of a world with only Donor Independent
GPGIs}
The set of all countries is denoted by $I$. There is a private good
and a set $K$ of donor independent GPGIs. Each country $i\in I$
chooses its consumption $c_{i}$ of the private good and its vector
$q_{i}$ of voluntary (monetary) contributions to the (donor independent)
GPGIs (where $q_{ik}\geq0$ denotes country $i$'s contribution to
GPGI $k$). Country $i$'s utility is given by: $u_{i}(c_{i},Q)$,
where $Q$ denotes the vector of aggregate (monetary) contributions,
the aggregate contribution to the GPGI $k\in K$ being given by $Q_{k}=\sum_{i\in I}q_{ik}$.
Country $i$'s budget constraint is given by $c_{i}+\sum_{k\in K}q_{ik}\leq y_{i}$,
where $y_{i}$ is exogenous. Each country takes the other countries'
choices as given and chooses $(c_{i},\{q_{ik},k\in K\})$ so as to
maximise $u_{i}(c_{i},Q)$ under the budget constraint $c_{i}+\sum_{k\in K}q_{ik}\leq y_{i}$
.
\begin{prop}
\label{prop:(restatement-of-the} (restatement of the neutrality theorem
from Sandler and Posnett (1991) \href{http://journals.sagepub.com/doi/pdf/10.1177/109114219101900102}{http://journals.sagepub.com/doi/pdf/10.1177/109114219101900102}
in our terminology) Suppose that at the status quo the GPGIs are entirely
financed through voluntary contributions. Suppose that each country's
payoff from a GPGI is a concave function of the money available for
that GPGI. Consider a reform consisting of imposing lump sum taxes
on a set of countries each of which voluntarily contributes some money
to some GPGI, such that no country is taxed more than its status quo
aggregate contribution to GPGIs. Suppose all the money raised through
these lump sum taxes is allocated to the GPGIs, with none of the transfers
exceeding the total funding at the status quo for any GPGI. Then the
reform has no effect.
\end{prop}
proof:
Country $i$ faces the problem to maximise $u_{i}(c_{i},Q)$ subject
to the constraint that $c_{i}+\sum_{k\in K}(Q_{k}-(\sum_{j\neq i}q_{jk}+G_{k}))\leq y_{i}-L_{i}$with
the further constraint that $Q_{k}\geq G_{k}+\sum_{j\neq i}q_{jk}$.
We can rewrite this problem by eliminating $c_{i}$:
maximise $u_{i}(y_{i}-L_{i}-\sum_{k\in K}(Q_{k}-(\sum_{j\neq i}q_{jk}+G_{k})),Q)$
s.t. $Q_{k}\geq G_{k}+\sum_{j\neq i}q_{jk}$.
Consider the situation without the transfers, i.e. with $L_{i}=0\forall i$.
Suppose $(\tilde{q_{ik}})_{i\in I,k\in K}$ is a Nash equilibrium.
Now consider a global mechanism that consists of contributing $G_{k}$
to the GPGI $k$, financed by the lump sum taxes $L_{i}\leq y_{i}-\tilde{c_{i}}$.
Suppose $G_{k}\leq\sum_{i}\tilde{q_{ik}}$. Now consider any $(\hat{q_{ik}})_{i\in I,k\in K}$
with $\hat{q_{ik}}\geq0$ and $\hat{q_{ik}}>0\,iff\,\tilde{q_{ik}}>0$
such that $\sum_{i}\hat{q_{ik}}+G_{k}=\sum_{i}\tilde{q_{ik}}$ and
$\sum_{k}\hat{q_{ik}}+L_{i}=\sum_{k}\tilde{q_{ik}}\forall k\in K$.
Suppose there is a profitable deviation for player $i$ from $\hat{q_{ik}}$
to $q_{ik}^{*}$ . Then by the asssumed convexity of the payoff from
$k$ for country $i$, there would also be a profitable deviation
for player $i$ from $\tilde{q_{ik}}$ consisting of decreasing it
marginally. $\square$
\begin{prop}
(Almost a neutrality theorem)\label{(Almost-a-neutrality} Suppose
that at the status quo the GPGIs are entirely financed through voluntary
contributions. Consider a reform that raises an amount $\text{\ensuremath{F\leq\sum_{i\in I,k\in K}\tilde{q_{ik}}}}$
of money from non-contributors (i.e. from countrie that do not contribute
to any of the GPGIs) and and allocates this money to the GPGIs in
such a way that for each of the GPGI the transfer does not exceed
the funding available at the status quo. Let $x$ denote the resulting
allocation and $J$ denote the set of contributors to global public
goods at $x$. Then there exists a set of lump sum transfers $\{t_{j}\geq0,j\in J\}$
distributing the money $F$ to the contributors, i.e. with $\sum_{j\in J}t_{j}=F$
such that the resulting allocation is $x$.
\end{prop}
proof\\
The intervention 'raising money for the GPGIs from non-contributors
in a lump sum way' can be written as a sum of 'raising money for the
GPGIs from contributors in a lump sum way' + 'redistributing money
from the non-contributors to the contributors'. By proposition \ref{prop:(restatement-of-the}
the first of these interventions satisfies neutrality. $\square$\\
\\
\\
Let us analyse the MCT mechanism at taxation rate $\text{\ensuremath{x^{\#}}}$.
To do that, consider starting at $x=0$ and increasing the rate to
$dx>0$. Let $p$ be the proportion of money raised from countries
that do not contribute to GPIs. Proposition \ref{(Almost-a-neutrality}
predicts that this is equivalent to some lump sum transfer from the
countries that at the status quo do not make any voluntary contributions
to global institutions. Also, it predicts that it does not matter
to which contributor the money is given. Supposing that this transfer
does not change the proportion $z$ of money that the contributors
give to GPGIs, this would mean a dollar raised via the MCT for GPGIs
would only result in $pz$ dollars of actual increase in funding for
GPGIs. Since $z$ is much smaller than 1\%, this shows that the conclusion
is almost as pessimistic as in a situation in which the neutrality
theorem applies.
\subsection{A model for the game of voluntary contributions to a Global Public
Good Institution (GPGI)}
\subsubsection{A model for the case of Direct Procurement GPGIs}
$I$ denotes the set of all countries. Each country $i\in I$ chooses
the voluntary contribution $q_{i}$ to the GPGI in question. The GPGI
then takes an action $z(Q)$. Here we are using that by the definition
of Direct Procurement GPGIs, the action $z$ of the GPGI depends only
on the total money $Q$ at its disposal. Country $i$'s payoff is:
$U_{i}=u_{i}(z(Q))-h_{i}(q_{i})$.
\subsubsection{A model for the case of Reward Payment Based GPGIs}
In the first stage of the game each country $i\in I$ chooses the
amount $q_{i}\geq0$ that it contributes to the global institution.
In the second stage of the game each country $i$ chooses $x_{i}\in\mathbb{R}_{\geq0}$,
which we shall interpret as the domestic effort that the country makes
towards the global public good, as interpreted in the examples below.
The global institution functions by paying out reward payments $R_{i}(x,Q)$
to the countries, where $Q:=\sum_{i}q_{i}$. Country $i$'s payoff
is: $U_{i}=u_{i}(x_{i},x_{-i})-h_{i}(q_{i})+g_{i}(R_{i})$.
If we denote by $x(Q)$ the outcome in the second stage of the game,
then we can write country $i$'s payoff as follows:
$U_{i}=u_{i}(x(Q))-h_{i}(q_{i})+g_{i}(R_{i}(x(Q),Q))$
\begin{prop}
\label{reward payment based GPGI} Suppose that $g_{i}$ is continuously
differentiable and suppose that $g_{i}'(0)>0\forall i$. Suppose that
the global institution has complete information and suppose that for
any given level of aggregate funding $Q=\sum_{i}q_{i}$ the institution
maximises the aggregate benefit generated by the global public good,
$\sum_{i}u_{i}(x_{i},x_{-i})$, subject to the participation constraints
for all players. Suppose that this optimization problem always has
a unique solution, which we shall denote by $(x(Q),R(Q))$. Suppose
that in the absence of the global institution some country $i$ is
at a strictly interior solution at the Nash equilibrium, i.e. suppose
that $x_{i}(0)>0$ for some $i$. Also, suppose that $u_{j}$ is twice
continuously differentiable at $0$ and we have $\frac{\partial^{2}u_{j}}{\partial x_{i}^{2}}|_{0}<0$
and that $\sum_{j}\frac{\partial u_{j}}{\partial x_{i}}|_{0}>0$ .
Moverover, let us assume also that each player $j$ has a unique best
response (that we will denote by $\tilde{x_{j}}(x_{-j})$ ) for his
outside option, i.e. the level of effort that maximises his payoff
if he gives up on getting the reward payments. Then the marginal cost
of increasing the aggregate welfare $\sum_{j}u_{j}$ by putting money
into the Reward Payment based GPGI, starting at the Nash equilibrium
that arises in the absence of the GPGI is 0. Moreover, at any Subgame
Perfect Nash equilibrium we have $\sum_{i}q_{i}>0$.
\end{prop}
proof:
The global institution's problem is:
\begin{equation}
max\sum_{j}u_{j}
\end{equation}
subject to the following constraints:
\begin{equation}
u_{j}(x_{j},x_{-j})+g_{j}(R_{j})=u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})\forall j\label{eq:-3}
\end{equation}
\begin{equation}
\sum_{j}R_{j}=Q\label{eq:-1-1}
\end{equation}
$\tilde{x_{j}}(x_{-j})$ denotes the country $j$'s optimal choice
if it forgoes the reward payment $R_{j}$. Using \ref{eq:-3} in \ref{budget constraint}
lets us capture the constraints in a single one:
\begin{equation}
\sum_{j}g_{j}^{-1}(u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})-u_{j}(x_{j},x_{-j}))=Q\label{budget constraint}
\end{equation}
Now we can apply the maximum theorem (Theorem M.K.6 in Mas-Colell)
and our assumption that the optimum is unique to conclude that $x(Q)$
and $\tilde{x_{j}}(x_{-j})$ are continuous. Also, by the envelope
theorem we have $\forall j\neq i\frac{d}{dx_{i}}u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})=\frac{\partial}{\partial x_{-j}}u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})\frac{dx_{-j}}{dx_{i}}$.
Let $i$ be a player that is at a strictly interior solution at the
Nash equilibrium in the absence of the global institution. By continuity,
this is still the case for all $Q\geq0$ sufficiently close to 0.
Denoting the Lagrange multiplier associated with the budget constraint
\ref{budget constraint}by $\lambda$, we obtain the first order condition:
$\sum_{j}\frac{\partial u_{j}}{\partial x_{i}}=$
$\lambda((\sum_{j\neq i}\frac{1}{g_{j}'(g_{j}^{-1}(u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})-u_{j}(x_{j},x_{-j})))}\frac{\partial u_{j}}{\partial x_{i}}(\tilde{x_{j}}(x_{-j}),x_{-j})-\sum_{j}\frac{1}{g_{j}'(g_{j}^{-1}(u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})-u_{j}(x_{j},x_{-j})))}\frac{\partial u_{j}}{\partial x_{i}}(x_{j},x_{-j}))$
$\lambda=$
$\frac{\sum_{j}\frac{\partial u_{j}}{\partial x_{i}}}{(\sum_{j\neq i}\frac{1}{g_{j}'(g_{j}^{-1}(u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})-u_{j}(x_{j},x_{-j})))}(\frac{\partial u_{j}}{\partial x_{i}}(\tilde{x_{j}}(x_{-j}),x_{-j})-\frac{\partial u_{j}}{\partial x_{i}}(x_{j},x_{-j})))-\frac{1}{g_{i}'(g_{i}^{-1}(u_{i}(\tilde{x_{i}}(x_{-i}),x_{-i})-u_{i}(x_{i},x_{-i})))}\frac{\partial u_{i}}{\partial x_{i}}(x_{i},x_{-i})}$
Now by our assumption that $u$ is twice continuously differentiable,
we can write this as:
$\lambda=$
$\frac{\sum_{j}\frac{\partial u_{j}}{\partial x_{i}}}{(\sum_{j\neq i}\frac{1}{g_{j}'(g_{j}^{-1}(u_{j}(\tilde{x_{j}}(x_{-j}),x_{-j})-u_{j}(x_{j},x_{-j})))}(x_{j}-\tilde{x_{j}}(x_{-j}))(-(\frac{\partial^{2}u_{j}}{\partial x_{i}^{2}}(\tilde{x_{i}}(x_{-j}),x_{-j})+o(x_{j})))-\frac{1}{g_{i}'(g_{i}^{-1}(u_{i}(\tilde{x_{i}}(x_{-i}),x_{-i})-u_{i}(x_{i},x_{-i})))}\frac{\partial u_{i}}{\partial x_{i}}(x_{i},x_{-i})}$
Now let us consider the limit as $Q\rightarrow0$. By the continuit
of $x$ we obtain $\lim_{Q\rightarrow0}\sum_{j}\frac{\partial u_{j}}{\partial x_{i}}=\sum_{j}\frac{\partial u_{j}}{\partial x_{i}}|_{0}$,
which by assumption is strictly positive. We will now show that the
limit of the denominator is 0.
By the continuity of $\tilde{x_{i}}(x_{-j})$ and $g_{j}'$at 0 we
have
$\lim_{Q\rightarrow0}g_{j}'(g_{j}^{-1}(u_{j}(\tilde{x_{i}}(x_{-j}),x_{-j})-u_{j}(x_{i},x_{-j})))=g_{j}'(g_{j}^{-1}(0))=g_{j}'(0)$
By assumption this is strictly positive.
By the first order condition for $x_{i}$ in the absence of the fund,
we have:
$\lim_{Q\rightarrow0}\frac{\partial u_{i}}{\partial x_{i}}(x_{i},x_{-i})=0$
Moreover, by the continuity of $\tilde{x_{i}}(x_{-j})$ we have $\lim_{Q\rightarrow0}x_{i}-\tilde{x_{i}}(x_{-j})=0$
We can thus deduce that $\lim_{Q\rightarrow0}\lambda=+\infty\,or\,-\infty$.
But $\lambda$ is the value of relaxing the budget constraint, so
we can conclude that $\lim_{Q\rightarrow0}\lambda=+\infty$. Moreover,
$\frac{1}{\lambda}$ is the marginal cost of 'purchasing an additional
unit of global welfare via the global institution's mechanism'. But
we can deduce that $\lim_{Q\rightarrow0}\frac{1}{\lambda}=0$. Since
there is a finite number of countries, there must at the situation
where $Q=0$ be at least one country for whom the marginal cost of
purchasing an additional unit of welfare via the global institution's
mechanism is 0. Hence $Q=0$ cannot happen at any subgame perfect
Nash equilibrium.$\square$
From some perspective theorem is puzzling: It shows that at any SPNE
in the first stage at least some countries pay into the fund so as
to effectively pay other countries (and itself) in the second period
to increase their efforts for the global public good. It seems natural
to wonder: Why is not each country better off by paying itself for
increasing efforts rather than paying also others via the contribution
to the fund? Moreover, paying reward payments to itself is pointless,
so does it not follow from this line of reasoning that each country
is best off setting its contribution to the fund to 0?
To see why this line of reasoning is erroneous, consider the profile
$q=0$ in the first stage of the game. In the ensuing NE in the second
stage of the game each country that is at an interior solution is
indifferent between marginally increasing or decreasing its effort.
We are assuming that there is at least one such country. Let us pick
such a country and denote it by B. Consider a country A deciding between
increasing its own effort by an infinitesimal unit or paying another
country B for reducing her effort by the same infinitesimal unit.
Per unit of additional effort that country A makes, it incurs some
finite amount of additional cost. This is a first order cost. If on
the other hand country A pays country B for increasing her effort,
the marginal amount of required compensation is of second order. Hence
we see that country A is better off paying country B for increasing
its effort. The Reward Payment Based GPGI is a vehicle for allowing
country A to pay country B for increasing its effort.
\subsubsection{Microfoundations for the payoff function $U_{i}=u_{i}(x_{i}(Q),x_{-i}(Q))-h_{i}(q_{i})+g_{i}(R_{i})$}
The benefit $u_{i}(x_{i},x_{-i})$ that the player $i$ internalises
can incorporate pure altruism. In fact, this seems necessary to explain
Norway's and Sweden's high contributions to global public goods. Given
that these countries only have populations of 5 million and 10 million,
respectively, the partial internalisation of the benefits of global
public goods can hardly account for these countries' contributions.
The function $h_{i}(q_{i})$ might write as: $h_{i}(q_{i})=c_{i}(q_{i})-s_{i}(q_{i})$,
where $c_{i}(q_{i})$ could represent the net expected political cost.
This could on the one hand include both the opportunity cost of public
funds and the risk that domestic political forces will discredit the
government for some constituencies by accusing it to not act in the
national interest. $s_{i}$ could include the benefit of earning the
support of citizens in favor of altruistic action as well as the diplomatic
recognition and reputation internationally. It is then plausible that
we have $c_{i}'>0,c_{i}''>0,s_{i}'>0,s_{i}''<0$, which would imply
that $h_{i}''(q_{i})>0$.
In the litterature modelling the voluntary contributions of states
(ref) it is assumed that in their decision how much to voluntarily
contribute to GPGIs countries only consider the benefit of marginal
funding to the GPGI and the opportunity costs of funds. In \ref{subsec:Implications-for-the opportunity costs}
we will see that this would imply that the $h_{i}$ is almost perfectly
flat on the relevant range. This is why the following proposition
will be of interest:
\begin{prop}
\label{flat costs} Suppose that $h_{i}(q_{i})=q_{i}\forall i$. Then
almost surely only one player contributes money to the GPGI in the
first stage of the game.
proof:
Consider the marginal benefits $\frac{\partial u_{i}}{\partial Q}$
of additional money provided the GPGI in question. Let $S^{*}(i)$
be the set of funding levels at which country $i$ is indifferent
between contributing a bit more or a bit less. Almost surely, each
of the $S^{*}(i)$ is finite. Hence for any $i,j\in I$ almost surely
$S^{*}(i)\cap S^{*}(j)=\emptyset$. But if $i$ and $j$ are to both
contribute at the Nash equilibrium, this can only happen at a $Q\in S^{*}(i)\cap S^{*}(j)$.
$\square$
\end{prop}
\subsubsection{Some Data on financing for global institutions}
Here is data for the Coalition for Epidemic Preparedness Innovation
\href{http://documents.worldbank.org/curated/en/311611506132045673/pdf/Coalition-for-Epidemic-preparedness-document-09112017.pdf}{http://documents.worldbank.org/curated/en/311611506132045673/pdf/Coalition-for-Epidemic-preparedness-document-09112017.pdf}:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/cepi-pledges\string-/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
By adding up all the pledges for each donor, we obtain the following:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/CEPI-donors\string-/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
http://www.globalfundadvocatesnetwork.org/wp-content/uploads/2017/04/PIIS0140673616324023.pdf\\\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/gavi-donors-2000-13\string-/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
\\\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/global-fund-donors-200-13\string-/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
data: https://www.theglobalfund.org/en/government/
Forest Carbon Partnership:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Forest-Carbon-Partnership\string-/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
\subsubsection{Implications for the countries' opportunity costs of contributing\label{subsec:Implications-for-the opportunity costs}}
Suppose that $h_{i}(q_{i})$ consists only of the opportunity cost
of the money $q_{i}$ contributed (as is the case in the models in
which the Neutrality Theorem holds (see Sandler (ref)).
$\frac{h_{i}'(q_{i})-h_{i}'(0)}{h_{i}'(0)}=\frac{F_{i}'(GDP_{i}-q_{i})-F_{i}'(GDP_{i})}{F_{i}'(GDP_{i})}$
For an illustrative computation, consider the case of the isoealestic
utility function for $F_{i}$ with elasiticity 1, i.e. consider the
case where $F_{i}(GPD_{i})=log(GDP_{i})$
$\frac{h_{i}'(q_{i})-h_{i}'(0)}{h_{i}'(0)}=\frac{\frac{1}{GDP_{i}-q_{i}}-\frac{1}{GDP_{i}}}{\frac{1}{GDP_{i}}}=\frac{q_{i}}{GDP_{i}-q_{i}}$
Below we give the numerical values for $\frac{q_{i}}{GDP_{i}-q_{i}}$
computed for the example of global institutions that we provided above:\\
\\
https://en.wikipedia.org/wiki/List\_of\_countries\_by\_GDP\_(nominal)
\begin{tabular}
{|c|c|c|c|c|c|}
\hline
country & 2017 GPD & 2017 contribution to CEPI in millions & $\frac{contributiontoCEPI}{GDP-contributiontoCEPI}$ & & \tabularnewline
\hline
\hline
Norway & 392052 & 13 & 3,316E-05 & & \tabularnewline
\hline
Germany & 3651871 & 12 & 3,286E-06 & & \tabularnewline
\hline
Japan & 4883389 & 25 & 5,119E-06 & & \tabularnewline
\hline
Canada & 1640385 & 3 & 1,829E-06 & & \tabularnewline
\hline
Australia & 1390150 & 2 & 1,439E-06 & & \tabularnewline
\hline
Belgium & 491672 & 1 & 2,034E-06 & & \tabularnewline
\hline
\end{tabular}
We see that for all countries and all global institutions for which
we have given the data here, the ratio $\frac{q_{i}}{GDP_{i}-q_{i}}$
is very close to 0. This will still be true for all the other global
institutions and for other specifications of how a government's payoff
depends on its available money. In other words: If we assume that
countries' costs of contributions is determined by their opportunity
costs of the money, then we have a constant marginal cost function
would be an extremely good approximation.
We have seen \ref{flat costs} that this would imply that we should
expect there to be only one contributing country for each GPGI. However,
for the examples we have seen (CEPI, GAVI, Global Fund for AIDS, TB
and Malaria, Forest Carbon Partnership) there are always several contributing
countries. Thus we can conclude that the function $h_{i}(q_{i})$
must contain more components than just the opportunity costs of funds.
\subsubsection{Crowding out of voluntary contributions}
We shall analyse a donor-indpendent GPGI. We shall use the utility
specification $U_{i}=u_{i}(M)-h_{i}(q_{i})$, where $M$ is the total
amount of money at the disposal of the GPGI. We will consider the
case where $M$ is the sum of an external amount $F$ and the aggregate
voluntary contributions $Q$. This specification contains as a special
case the model for the Direct Procurement GPGIs that we defined above,
$U_{i}=u_{i}(z)-h_{i}(q_{i})$, since the action $z$ that the GPGI
will take will be determined by its available money, $M$. It also
contains as a special case the specification that we used above for
analyzing Reward Payment based GPGIs, namely $U_{i}=u_{i}(x_{i}(M),x_{-i}(M))-h_{i}(q_{i})+g_{i}(R_{i})$,
since $R_{i}$ is determined by $M$, at least if we assume that there
is a unique subgame perfect Nash equilibrium.
\begin{prop}
\label{Crowding-out for exogenous donation to a given GPGI} Consider
a GPGI that is financed by an exogenous donation of $F$ and by voluntary
contributions by governments that are determined as a Nash equilibrium
given $F$. Denoting the elasticities of marginal benefits and marginal
costs by $\epsilon_{u_{i}'}$ and $\epsilon_{h_{i}'}$, respectively,
and supposing that $\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}\leq1$,
we obtain: $\frac{dQ}{dF}=\frac{\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}{1-\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}$.
\end{prop}
proof
The government of country $i$ has payoff $U_{i}=u_{i}(Q+F)-h_{i}(q_{i})$.
Suppose $i\in Contributors$. Then $q_{i}$ is determined by the FOC,
which is: $\frac{dU_{i}}{dq_{i}}=u_{i}'(Q+F)-h_{i}'(q_{i})=0$. Differentiating
this FOC with respect to F yields:$(\frac{dQ}{dF}+1)u_{i}''(Q+F)-\frac{dq_{i}}{dF}h_{i}''(q_{i})=0$
Now using again the FOC we can rewrite this as: $\frac{dq_{i}}{dF}=(\frac{dQ}{dF}+1)\frac{u_{i}''(Q+F)}{u_{i}'(Q+F)}\frac{h_{i}'(q_{i})}{h_{i}''(q_{i})}=(\frac{dQ}{dF}+1)\frac{u_{i}''(Q+F)(Q+F)}{u_{i}'(Q+F)}\frac{h_{i}'(q_{i})}{h_{i}''(q_{i})q_{i}}\frac{q_{i}}{Q+F}=(\frac{dQ}{dF}+1)\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}$
Summing over all contributors yields: $\frac{dQ}{dF}=(\frac{dQ}{dF}+1)\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}$.
From this we deduce: $\frac{dQ}{dF}=\frac{\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}{1-\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}$
$\square$
Let us consider the formula $\frac{dQ}{dF}=\frac{\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}{1-\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}$.
First consider the case where all the marginal benefits are concave,
i.e. where $\epsilon_{u_{i}'}\leq0\forall i\in Contributors$ and
where all the marginal cost curves are convex, i.e. where $\epsilon_{h_{i}'}\geq\text{0}\forall i\in Contributors$.
In this case the smaller the elasticities of the marginal costs are,
the more crowding out there is. Moreover if all the elasticities of
the marginal costs are very close to 0 then $\frac{dQ}{dF}$ is close
to $-1$, meaning that there is close to full crowding out.
Now consider the case where $\epsilon_{u_{i}'}=0\forall i\in Contributors$.
then we have $\frac{dQ}{dF}=0$. In other words: If the curves for
the marginal benefit as a function of the available money to the global
institution are flat, then the naive view that evaluates the impact
of additional funding contributed by a philantropist or raised through
international taxes is correct.
Finally, consider the case where $\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}\in(0,1)$.
(This can happen if some of the contributors' marginal benefit curves
are locally convex around the equilibrium.) In this case we see that
there is actually ``crowding in'', as $\frac{dQ}{dF}>0$. If $\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}>1$
there will even be a discontinuous jump in the equilibrium.
To see why this happens, we note that from the second order conditions
we can only establish: $u_{i}''(Q+F)\leq h_{i}''(q_{i})$, which using
the FOC can be rewritten as follows:
$\frac{u_{i}''(Q+F)}{u_{i}'(Q+F)}\leq\frac{h_{i}''(q_{i})}{h_{i}'(q_{i})}$
$\frac{\frac{u_{i}''(Q+F)}{u_{i}'(Q+F)}}{\frac{h_{i}''(q_{i})}{h_{i}'(q_{i})}}\leq1$
$\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}\leq1$
This is not sufficient to ensure that $\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}\leq1$.\\
\\
So far we have been analyzing the impact of a change in the exogenous
part $F$ of funding for a GPGI on the aggregate endogenous funding
$Q$ that comes from voluntary contributions. The exogenous part $F$
in some sense came from 'outside' our model.
Now let us try to better undestand the impact that the introduction
of an additional financing mechanism for GPGIs would have on voluntary
contributions by governments. First consider the case of a treaty
that would allocate the revenues raised from seabed resources from
the high sea to some GPGI. Switching from a regime (that currently
appears to be envisaged) where the revenue from the seabed resources
from the high sea are paid out to countries to the regime where this
money is instead paid out to some GPGI is equivalent to implementing
lump sum taxes on countries and giving the money raised to the GPGI.
However, as we indicated in the discussion following \ref{(Almost-a-neutrality},
as far as the effect on voluntary contributions is concerned this
is approximately equivalent to considering the money as coming from
'outside', i.e. as being raised from non-contributors, given how small
contributions for GPGIs are relative to GDP for any given country
\ref{subsec:Implications-for-the opportunity costs}. Thus the proposition
\ref{Crowding-out for exogenous donation to a given GPGI} seems applicable.
Now consider the introduction of a treaty that raises funds for GPGIs
via taxes, such as the MCT mechanism. One approach to modelling this
would be as a subgame perfect Nash equilibrium of a two stage game,
where in the first stage countries play the game induced by the MCT
mechanism and in the second stage countries play the voluntary contribution
game. In this case, we should dampen our optimism about the potential
of the MCT, to the extent that there is crowding out. This is because
in the first stage of the game countries will anticipate the crowding
out and thus the benefit that they ascribe to money being directed
via the MCT mechanism to a GPGI. To put it more precisely: Suppose
the benefit for country $i$ per money directed to GPGI $k$ is $\theta_{i}^{k}$.
Then the relevant payoff in the first stage of the game is actually
$(1+\frac{dQ}{dF})\theta_{i}^{k}$. Thus, in this model crowding out
would not only lessen the impact of the money raised via the MCT mechanism
for the GPGIs, it would also make it less likely that the MCT mechanism
would have full (or even any) participation.
In section \ref{7 sec:Combining-the-MCT} we will explore extensions
to MCT that could mitigate this crowding out problem or possibly even
reverse it to achieve 'crowding-in'.
\section{\label{5 sec:Multiple-Cause-Treaties}Multiple Cause Treaties financed
by capturing global revenues}
\subsection{\label{subsec:Examples-of-proposals for capturing global resources}{\normalsize{}Examples}
of proposals for capturing global resources}
\subsubsection{Special Drawing Rights}
In \href{http://www.un.org/en/development/desa/policy/wess/wess_current/2012wess_overview_en.pdf}{http://www.un.org/en/development/desa/policy/wess/wess\_{}current/2012wess\_{}overview\_{}en.pdf}
it is suggested that: ``new international {[}could be created{]}
through the issuance of special drawing rights (SDRs) by the International
Monetary Fund, to be allocated with a bias favoring developing countries
or leveraged as development financing''.
\subsubsection{Revenues from the ownership of global resources: \textquotedblleft royalties\textquotedblright{}
obtained from natural resource extraction beyond territorial limits}
In \href{http://www.un.org/en/development/desa/policy/wess/wess_current/2012wess_overview_en.pdf}{http://www.un.org/en/development/desa/policy/wess/wess\_{}current/2012wess\_{}overview\_{}en.pdf}
it is suggested that: \textquotedblleft royalties\textquotedblright{}
obtained from natural resource extraction beyond territorial limits,
such as from the oceans beyond territorial limits, Antarctica or outer
space, could be tapped for international development cooperation.
As global resources lie outside national jurisdictions, any licensing
and payment of royalties would have to involve an international authority
that was recognized as the responsible agent for managing the specific
commons. While fishing in international waters and other activities
in the \textquotedblleft commons\textquotedblright{} might be licensed
in this way, only revenues from mining the seabed has thus far been
seen as a source of international public revenue. ``
\subsection{How to use the revenue from global resources?}
In \href{http://www.un.org/en/development/desa/policy/wess/wess_current/2012wess_overview_en.pdf}{http://www.un.org/en/development/desa/policy/wess/wess\_{}current/2012wess\_{}overview\_{}en.pdf}
the authors propose to use the money captured from global resources
be used for ODA. We will now define and explore an alternative proposal:
\subsubsection{An example of a mechanism leveraging voluntary contributions: The
Proportional Matching Fund mechanism (PMF)\label{subsec: PMF} }
\begin{defn}
The Proportional Matching Fund (PMF) would be a newly created global
institution that would have a set $J$ of 'eligible Global Public
Good Institutions'. (For example, $J$ could be \{CEPI,GlobalFund,The
Forest Carbon Partnership's Carbon Fund, the proposed RES-Fund under
the auspices of the International Renewable Energy Agency\}or $J$
could encompass all GPGIs.) The PMF would pay out its available money
$F$ to the GPGIs in $J$ in proportion to the voluntary contributions
made to the various institutions . In other words, the PMF would each
year look at the budget $Q_{j}$ of each GPGI $j\in J$ and pay to
$j$ the amount $\frac{Q_{j}}{\sum_{k\in J}Q_{k}}F$.
\end{defn}
\begin{prop}
We consider the marginal effect of the first dollar of a newly introduced
Proportional Matching Fund with an associated set $J$ of eligible
GPGIs. Let us assume that at the status quo each GPGI $j\in J$ is
financed entirely by the voluntary contributions by governments. Moreover,
suppose that the profile of contributions is always a Nash equilibrium
in the game. Let us denote by $Contributors_{a}(F)$ the set of countries
making strictly positive voluntary contributions to the GPGI $a$.
Let us denote by $q_{i,a}(F)$ the voluntary contribution of country
$i$ to the GPGI $a\in J$, by $Q_{a}(F):=\sum_{i\in I}q_{i,a}(F)=\sum_{i\in Contributors_{a}(F)}q_{i,a}(F)$
the aggregate amount of contributions made to GPGI $a$ and by $Q(F):=\sum_{j\in J}Q_{j}(F)$
the aggregate amount of contributions made to all GPGIs in total.
Let us denote by $\epsilon_{u_{i}'}^{a}:=\frac{Q_{a}}{\frac{\partial u_{i}}{\partial Q_{a}}}\frac{\partial^{2}u_{i}}{\partial Q_{a}^{2}}$
the elasticity at $F=0$ of the marginal benefit of additional funding
for the GPGI $a$ accruing to country $i$ and by \textbf{$\epsilon_{h_{i,a}'}:=\frac{q_{i,a}}{\frac{\partial h_{i,a}}{\partial q_{i,a}}}\frac{\partial^{2}h_{i,a}}{\partial q_{i,a}^{2}}$
} the elasticiy at $F=0$ of the subjective marginal cost for the government
of country $i$ of making voluntary contributions to the GPGI $a$.
Suppose that $\epsilon_{h_{i,a}'}>0$ Then if $\text{1}\text{+\text{\ensuremath{\sum_{i\in Contributors_{a}(0)}\frac{q_{i,a}}{Q_{a}}\frac{-\epsilon_{u_{i}'}^{a}}{\epsilon_{h_{i,a}'}}}}}>0$
we have:
$\frac{dQ_{a}}{dF}|_{F=0}=\frac{\text{\ensuremath{\sum_{i\in Contributors_{a}(0)}\frac{q_{i,a}}{Q}\left(\epsilon_{u_{i}'}^{a}+\frac{Q-Q_{a}}{Q}(\text{\ensuremath{1}}-\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}})\right)\frac{1}{\epsilon_{h_{i,a}'}}}}}{\text{1}\text{+\text{\ensuremath{\sum_{i\in Contributors_{a}(0)}\frac{q_{i,a}}{Q_{a}}\frac{-\epsilon_{u_{i}'}^{a}}{\epsilon_{h_{i,a}'}}}}}}$
proof:
\end{prop}
The formula for $\frac{dQ_{a}}{dF}|_{F=0}$ was obtained through long
computations using Mathematica. $\square$
Since by definition $\sum_{i\in Contributors_{a}}\frac{q_{ai}}{Q_{a}}=1$
, we can deduce from the fact that $\frac{-\epsilon_{u_{i}'}^{a}}{\epsilon_{h_{i,a}'}}\geq-1\forall i\in Contributors_{a}$
that the denominator, $\text{1}\text{+\text{\ensuremath{\sum_{i\in Contributors_{a}(0)}\frac{q_{i,a}}{Q_{a}}\frac{-\epsilon_{u_{i}'}^{a}}{\epsilon_{h_{i,a}'}}}}}$,
is positive. Let us from now on focus on the generic case in which
this is strictly positive.
Now consider the expression $\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}}$
for $i\in Contributor_{a}$. Since $\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}=1,$
the numerator, $\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}$,
is a weighted average of the marginal utility that country $i$ gets
from additional financing for the other GPGIs (i.e. excluding $a$),
weighted in proportion to the financing that these receive. Since
$i\in Contributor_{a}$, it is plausible that we have $\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}}\leq1$.
Thus we see that the sign of $\frac{dQ_{a}}{dF}|_{F=0}$ is determined
by two forces. Firstly, there is the positive term $\frac{Q-Q_{a}}{Q}(\text{\ensuremath{1}}-\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}})$,
which arises from the fact that the marginal increase of $F$ starting
from $0$ increases country $i$'s incentive to voluntarily contribute,
since by doing so it can move money to its preferred GPGI, $a$, shifting
it away from the other GPGIs. The size of this effect is proportional
to the proportion of money going to the other GPGIs, $\frac{Q-Q_{a}}{Q}$.
Secondly, there is the term $\epsilon_{u_{i}'}^{a}$. If $\epsilon_{u_{i}'}^{a}>0$,
i.e. if there are increasing marginal benefits accruing to country
$i$ from money going to the GPGI $a$, then we obtain the conclusion
that $\epsilon_{u_{i}'}^{a}+\frac{Q-Q_{a}}{Q}(\text{\ensuremath{1}}-\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}})>0$.
Thus in this case there is always 'crowding in' of voluntary contributions
to the GPGI $a$ from country $i$. We also see that the size of this
effect increases in $\frac{1}{\epsilon_{h_{i,a}'}}$. Moreover, there
is no theoretical bound on how strong the crowding-in can be, as we
will prove in corollary below.
Now consider the case where $\epsilon_{u_{i}'}^{a}<0$, i.e. when
there are decreasing marginal benefits accruing to country $i$ from
money going to the GPGI $a$. If $\epsilon_{u_{i}'}^{a}$ is sufficiently
negative then there will be crowding out of contributions from country
$i$ for the GPGI $a$. Specifically the condition for there to be
crowding out is $\text{\ensuremath{\frac{-\epsilon_{u_{i}'}^{a}}{\frac{Q-Q_{a}}{Q}}>1}}-\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}}$$\forall i\in Contributors_{a}$.
For the limit we get:
\begin{cor}
$\lim_{\epsilon_{u_{ai}}^{a}\rightarrow-\infty}\text{\text{\ensuremath{\frac{dQ_{a}}{dF}}}}=-1$
\end{cor}
This makes sense: If for some country the marginal benefit drops off
all of a sudden exactly at the current level of financing for the
GPGI $a$, then the aggregate contribution to the GPGI $a$ cannot
increase until country $i$'s contribution has been fully crowded
out.
\begin{cor}
$\sup_{set\,of\,all\,models}min\{\frac{dQ_{a}}{dF}|_{F=0}:a\in J\}=\infty$
proof: Suppose we have any model with corresponding Nash equilibrium
such that $\epsilon_{u_{i}'}^{a}+\frac{Q-Q_{a}}{Q}(\text{\ensuremath{1}}-\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}})>0\forall a\in J\forall i\in Contributor_{a}$
at this Nash equilibrium. Let us keep the model to first order unchanged
around the Nash equilibrium but multiply all the elasticities by the
same factor $\lambda<1$. Then the second order conditions for the
best responses are still satisfied. Moreover, $\frac{dQ_{a}}{dF}|_{F=0}$
increases, since $\frac{Q-Q_{a}}{Q}(\text{\ensuremath{1}}-\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}})>0$
and we can make this increase arbitrarily large if we choose $\lambda$
sufficiently close to $0$. $\square$
\end{cor}
\section{\label{6 sec:How-to-learn}How to learn from past data about crowding
in and crowding out?}
One approach would be to study the few cases where countries have
introduced taxes to finance global institutions, for example the case
of France in 2006. However, the situation for the countries who introduced
the solidarity levy on air tickets might have been quite different:
Since few countries participated in the mechanism, introducing the
tax and transferring the money to UNITAID was very similar to making
a voluntary contribution from the general budget. The governments
could thus get diplomatic credit for an act of altruism in a similar
way to making a voluntary contribution. Moreover, domestic actors
could accuse the government of giving away money without pursuing
national interest in just the same way as is the case for voluntary
contributions. As we have already pointed out, this would not be the
case for a Nash equilibrium with full participation under the MCT
mechanism. In this case the government would still need to make voluntary
contributions to GPGIs in order to get diplomatic recognition for
its altruism. Moverover, domestic actors could only accuse it for
being too generous on the basis of the amount of the voluntary contribution.
The same arguments apply to the case of contributions to GPGIs coming
from international seabed resources.
If the line of reasoning given in the previous is correct, then we
cannot rely on the past example of taxes financing GPGIs to learn
about crowding in and crowding out. Instead, a more promising route
might be to try to estimate the elasticities in $\epsilon_{u_{i}'}^{a}$
and $\epsilon_{h_{i}'}^{a}$.
\subsection{Empirical estimation of the elasticities $\epsilon_{u_{i,a}'}$ and
$\epsilon_{h_{i,a}'}$}
Yet to be done.
\section{\label{7 sec:Combining-the-MCT}Combining the MCT mechanism with
the PMF mechanisms}
In our original definition of the MCT mechanism the list of eligible
institutions only contains GPGIs. We have seen in \ref{Crowding-out for exogenous donation to a given GPGI}
that our model predicts that for each GPGI there will be partial crowding
out given by:
$\frac{dQ}{dF}=\frac{\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}{1-\sum_{i\in Contributors}\frac{\epsilon_{u_{i}'}}{\epsilon_{h_{i}'}}\frac{q_{i}}{Q+F}}$\\
However, if the $\epsilon_{u_{i}'}^{a}$ are not too negative, then
the PMF mechanism would actually allow to achieve crowding in of voluntary
contribution, as long as the voluntary contributors prefer the GPGIs
that they contribute to sufficiently strongly over the other GPGIs,
as is revealed by the formula that we obtained for the PMF:
$\frac{dQ_{a}}{dF}|_{F=0}=\frac{\text{\ensuremath{\sum_{i\in Contributors_{a}(0)}\frac{q_{i,a}}{Q}\left(\epsilon_{u_{i}'}^{a}+\frac{Q-Q_{a}}{Q}(\text{\ensuremath{1}}-\frac{\sum_{b\neq a}\frac{Q_{b}}{Q-Q_{a}}\frac{\partial u_{i}}{\partial Q_{b}}}{\frac{\partial u_{i}}{\partial Q_{a}}})\right)\frac{1}{\epsilon_{h_{i,a}'}}}}}{\text{1}\text{+\text{\ensuremath{\sum_{i\in Contributors_{a}(0)}\frac{q_{i,a}}{Q_{a}}\frac{-\epsilon_{u_{i}'}^{a}}{\epsilon_{h_{i,a}'}}}}}}$\\
The MCT mechanism could be enriched by adding PMFs to the list of
eligible institutions. Consider the following list of GPGIs:
L=\{Coalition for Epidemic Preparedness Innovation CEPI, GlobalFund,
The Forest Carbon Partnership's Carbon Fund, the proposed RES-Fund
under the auspices of the International Renewable Energy Agency\}
We could now enrich this list to obtain the following:
$L^{*}$=\{Coalition for Epidemic Preparedness Innovation CEPI, GlobalFund,
The Forest Carbon Partnership's Carbon Fund, the proposed RES-Fund
under the auspices of the International Renewable Energy Agency, PMF(\{RES-Fund,
The Forest Carbon Partnership's Carbon Fund\}), PMF(\{CEPI, Global
Fund\})\}
Here PMF(\{RES-Fund, The Forest Carbon Partnership's Carbon Fund\})
denotes the Proportional Matching Fund (PMF) whose set of eligible
GPGIs comprises the RES-Fund for supporting renewable energy in developing
countries and the Forest Carbon Partnership's Carbon Fund.
Let us study the incentives that arise if the set of eligible GPGIs
for the MCT is taken to be $L^{*}$. Consider a government that cares
a lot about mitigating climate change, so that both money going to
the RES-Fund and the Forest Carbon Partnership's Carbon Fund are greatly
valued by the government. If this government allocates the money it
raises in the MCT mechnanism to the PMF(\{RES-Fund, The Forest Carbon
Partnership's Carbon Fund\}), then it would not only make sure that
this money reaches one of the two GPGIs, but moreover, it would create
incentives for some other countries to make voluntary contributions
to one of the two.
For example, a country or coalition of countries that stand to loose
from an acceleration of renewable energy development would have an
incentive to contribute to the Forest Carbon Partnership's Carbon
Fund, since by the mechanics of the PMF this would decrease the funds
going to the RES-Fund.
Now consider a government that thinks that it will benefit from climate
change but that cares about promoting global health and preventing
pandemics. This government might be best of contributing to PMF(\{CEPI,
Global Fund\}).
These examples also suggest that it might be good to modify the rules
of the PMF to include not only voluntary contributions in the formula
for calculating how the available money is distributed to GPGIs but
to instead also include the money allocated to the GPGIs via the MCT
itself. This would actually increase the incentives to participate
in the MCT.
\begin{defn}
The Augemented MCT mechanism is defined for a given a set $S$ of
eligible GPGIs as follows: The list of eligible institutions is defined
to be $S\cup\{PMF(G):G$ a subset of S\}.
Conjecture: The augmented MCT mechanism will perform better than the
simple MCT mechanism.
\end{defn}
\section{\emph{\label{8 sec:Conclusion}Conclusion}}
We have explored some ways in which funding for global institutions
dedicated to particular global public goods could be increased. The
mechanisms do not rely on any punishments to induce countries to participate.
Instead, they use the fact that countries' payoffs from different
global public good institutions diverge. By participating countries
can influence the allocation of funds across the different global
institutions dedicated to the various global public goods in their
favor.
\appendix
\part{Matlab programs}
Here we will shortly provide the Matlab code with which the reader
can compute the summary of the game that we have used in \ref{sec:An-illustrative-simulation}.
\part{Reports of all simulations run}
As another example, we have taken the case with ORR=1, PP=1.5 and
ACD=1.5 (see \ref{subsec:A-method-for} for the definitions):\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/20stepsORR1andPP1\lyxdot-5andACD1\lyxdot-5/default-figure}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\end{document}