Issue linkage versus Ringfencing in Trade Agreements
Ngo Van Long∗, Martin Richardson†and Frank Stähler‡
Version of October 10, 2019

Abstract
Issue linkage is often regarded as a means to achieve or improve international
cooperation in the presence of a sovereignty constraint. This constraint implies thata country can leave an agreement whenever it likes, and it will do so if the perceived
gains from leaving are larger than the gains from staying in the agreement. We set
up a model of trade agreements in which it is uncertainty about future gains fromcooperation that may lead to exit. In this environment, we show that ringfencing
dominates issue linkage, even in the absence of complimentarities between separate
issues, if the degree of uncertainty is sufficiently large.
JEL Classification: F13, F15.
Keywords: issue linkage, ringfencing, international agreements, exit.

∗

Department of Economics, McGill University, Montreal, Quebec, Canada H3A 2T7. Email:ngo.long@mcgill.ca.
†
Research School of Economics, The Australian National University, Canberra ACT 2601, Australia.
Email: martin.richardson@anu.edu.au.
‡
School of Business and Economics, University of Tübingen, University of Adelaide, CESifo and NoCeT, Tübingen, Germany. Email: frank.staehler@uni-tuebingen.de.

1

1Introduction

Countries typically find it gainful to enter into international agreements on trade andrelated issues, such as cross border labor mobility, capital mobility, intellectual property
rights protection, and so on.1 An interesting fact is that many of these agreements aredesigned as a “package deal” (treating several issues in the same treaty) rather than asa collection of separate agreements (say, one treaty would cover trade while a separate
treaty would deal with cross boder labor mobility). Once a country has signed a package
deal treaty, it is practically impossible for it to withdraw from a subset of clauses of thetreaty while keeping the remaining subset. The UK’s difficulty with Brexit is a striking
illustration of what we call “the impossibility of partial exit” principle (or ImPEx for
short). Arguably, without the ImPEx principle, the UK would want to keep free goods
trade with the rest of the EU, while dropping out of the cross border labor mobility
clauses.
While almost all treaties allow members to drop out of the package deal agreement (atsome stage in the future), they typically do not allow a partial exit. Our research question
is: under what conditions do countries find it optimal (ex ante) to require members of a
package deal treaty to be (at any stage in the future) either completely “in”, or completely“out”? On one hand, allowing a country to drop out is a good provision, because allcountries recognize that any member may, at some stage in the future, face countryspecific adverse shocks which make it individually rational to leave. On the other hand,when a country leaves, it potentially imposes externalities on the remaining members,
about which it no longer cares. To mitigate against this, ImPEx serves as a counterincentive against leaving. We may view ImPEx as a “constitutional clause” that partiallyties the hands of future politicians.
This general setup can be thought of as in the standard approach to issue linkage –tying two not necessarily related issues together in a policy package to induce cooperation
over both, where neither might stand alone – but also applies to the question of the scopeof a preferential trading agreement, for instance: do two countries agree on trade policy
alone (internal and, in a customs union, external), or do they also have agreements on
labour mobility, bilateral investment, technical and professional standards, monetary pol1

For example, NAFTA – now USMCA – is a comprehensive treaty that covers not only trade in goods
and services, but also cross-border investment.

2

icy, emissions controls and so on? In this paper, we focus on the problem of choice between
linking issues in a single overarching agreement versus having separate agreements, in thecontext of free trade agreements.
There is a significant number of papers on issue linkage. The basic idea that linking
two issues might make an agreement succeed where separate agreements would not goesback at least to Tollison and Willett (1979): essentially, in a world of certainty, if country
A (B) gains a lot on issue 1 (2) but loses a bit on issue 2 (1) then, absent side-payments,
no agreement would be reached on each issue alone but a joint agreement on 1 and 2
combined would succeed. It has been applied most frequently in the context of environmental economics (see, e.g., Geng and Woodland, 2018). One of the main concerns of thisliterature is whether making participation in an environmental agreement a co-requisite
for participation in a trade agreement will induce more countries to agree and, if so, if thenature of the agreement is changed.2 Alesina et al. (2005) look at heterogeneous countries
forming a union and consider the forces that determine the equilibrium size and composition of unions in a couple of cases. Their concern is primarily with the membership of theagreement, rather than the nature of it. Finally, Maggi (2016) reviews and attempts todraw together a lot of the literature in a unifying framework. He distinguishes between therole of issue linkage for enforcement, for negotiations and for participation, and he notesthat models of the first of these typically focus on repeated games, models of the secondon bargaining games and models of the third on participation games.3 His model captures
all of these but, critically, there is no uncertainty in the model at all. He notes that there
is a general presumption in the literature that linkage is desirable and he addresses thefact that it is not so frequent in reality as the analysis suggests it should be, primarily byreferring to the existence of negotiation transaction costs.
In contrast, our approach emphasizes the role of uncertainty and the possibility of exit.
Given uncertainty about future country-specific shocks that would affect the payoffs ofmember countries, we consider whether countries wish to enter into linked (pooled) or
unlinked (ringfenced) agreements. The main contribution of our model is its focus on theadvantages and disadvantages of issue linkage in a context of uncertainty where parties2

It has been argued that while linkage might increase the number of participants in an agreement on
an issue with a free-riding problem (like emissions control), it might reduce the number involved in thelinked area (like trade policy). See, e.g., Carraro and Marchiori (2003).
3
Our focus is on a mix of the second and third: negotiation over participation.

3

can renege if, due to unfavorable shocks, things are anticipated to turn out poorly forthem if they were to remain with the original agreement. This is an important question
as we will show that it is not a priori obvious whether pooling two risky issues together
is better than ringfencing them away from each other and dealing with them separately– the linkage affects the riskiness of exit (in different ways for different distributions, as
our examples show) but also affects the expected payoff in states where exit does notoccur. In our analysis there is no general presumption that issue linkage is preferable toringfencing.

2

A simple model of exit

In this section, we consider a model in which countries can agree only on a single issue, andthe agreement is subject to country-specific, idiosyncratic shocks.4 These shocks imply thepossibility of two different realizations. We assume two countries, denoted by h (the homecountry) and f (the foreign country). Since there is only one trade issue on the table inthis first model, countries must evaluate the desirability of entering a bilateral agreementon this issue. The countries are identical ex ante, and the agreement covers two periods.
In period 1, the countries decide on whether to sign a trade agreement that will be in force
in period 2. The agreement requires that policies are to be coordinated so as to maximize
the expected value of the sum of their (period two) payoffs. Each country realizes thatits partner may renege and “exit” upon experiencing an unfavourable shock in period 2,
which would render its payoff under cooperation less attractive than otherwise.
To fix ideas, we suppose each country is subject to a country-specific shock at the
beginning of period 2 that determines the realization of country-specific benefits fromcooperation. In this and in the subsequent section, the resulting realization is binary:
it can take one of only two values, “good”or “bad”, and the economic environment inperiod 2 is as follows. At the beginning of period 2, country c ∈ {h, f } observes its
country-specific realization, rc , which would affect the country’s period 2 payoff underpolicy coordination as prescribed by the agreement.5 The country-specific realizations
4

This model is a simplified version of the exit model of Richardson and Stähler (2019). They also endogenize the degree of cooperation and exit risks, and they show that exit risks will never be optimallycompletely avoided if they exist for full cooperation.
5
We assume the shock does not affect the country’s period 2 payoff if it exits. The payoff under “exit” is4are assumed to be independent and each may be a good realization, g, or a bad one,b. The vector of realizations, (rh, rf ), can thus take one of four possible values, (gh , gf ),(gh , bf ), (bh , gf ), or (bh , bf ). The probability that a country has a good (respectively, bad)realization is q (respectively, 1 − q), where 0 < q < 1. As this is a 2-country agreement, if
a country exits the agreement, the realized payoffs of both countries are zero (regardlessof the realizations of the shocks).
We also assume that the country-specific realizations are private information. A country
may “report” its observed realization to the other country, but the latter cannot verify
such information, so this reporting is cheap talk. This has two implications: first, theagreement cannot specify transfer payments conditional on the realization of the randomvariable and, second, it is not possible that a country that has a good realization and isthus interested in keeping the agreement alive will be able to compensate the other country
for staying in the agreement. Under asymmetric information, any “Coasian solution” isinfeasible, as each country would have an incentive to pretend to have a bad realizationin order to receive compensation.
Hence, it does not matter which state each country would report or not. What mattersis that countries may either elect to “remain” members of the agreement or to exit. If
both remain, the resulting realized payoffs vector (yh , yf ) is given by
(G, G)
(G, B)(yh , yf ) = π(rh , rf ) = (B, G)(B, B)

if (rh , rf ) = (gh , gf )if (rh , rf ) = (gh , bf )if (rh , rf ) = (bh , gf )if (rh , rf ) = (bh , bh )Here, G is a positive real number and B is a negative real number, and we assume thatcooperation makes sense ex ante such that the expected gain from cooperation is positive,i.e., qG + (1 − q)B > 0. Furthermore, we assume that B + G > 0 such that cooperation
is jointly desirable even if a single country has a bad realization. This means that, withthe exception of the case (bh , bf ), the sum of payoffs is higher under “remain” than under“leave”. Therefore, ideally both countries should commit to “remain” when their reportsof the realizations are not both bad, and to “exit” only when their reports are (bh , bf ).
Let P r [(rh , rf )] denote the probability of the realization (rh , rf ). If each country couldcommit to stay in the agreement whenever their realizations are not both bad ones, thezero by normalization.

5social value of the agreement would be equal to
W = Pr [(g, g)] × (2G) + Pr [(g, b)] × (G + B) + Pr [(b, g)] × (B + G)
= 2Gq 2 + 2(B + G)q(1 − q) = 2q [qG + (1 − q)(B + G)] ,and consequently each country would realize V = 12 W = q [qG + (1 − q)(B + G)] in
expected terms.
However, each country only cares about its own payoff in period 2 in the absence ofcommitment. It will exit after learning that its country-specific realization is b. Therefore,
under the non-commitment scenario, the agreement will survive only if both countries getgood realizations, so the value of the agreement to each country is only equal to
VN = Gq 2 < V
where the subscript N refers to the non-commitment scenario. When a country exits while
the other’s country realization is g, the exit inflicts a “negative externality” on the othercountry. In this benchmark model, it is not possible to punish a country for imposing thisnegative externality.

3

The scope of issue linkage

Now suppose there are two issues (e.g., the free flows of goods and the free flows of labor)
each of which can yield, on average, higher benefits to each country if they can coordinate
their policies. The two countries can now have two separate agreements, one for each issue(ringfencing), or they can pool the two issues together in a single agreement (pooling or
issue linkage). Ringfencing is the same as having two independent single-issue agreements
as described in the previous section but things are different for issue linkage/pooling: inthis case, we assume the ImPEx principle holds and we suppose that period 1 provisions
for policy coordinations in period 2 are so much intertwined that any country that wishes
to stop cooperating on one issue in period 2 must exit the pooled agreement altogether,thus obtaining a payoff of zero for both issues.
Under what conditions, if any, is pooling preferable to ringfencing? Obviously, if therealizations were strongly negatively correlated, issue linkage would provide insurance. If
the realizations were strongly positively correlated, issue linkage becomes less attractive as
the exit risk will increase with including a second issue. We do not consider any correlation,6

as we want to focus squarely on the role of uncertainty; thus, as noted, we assume thatthe realizations are independently distributed. As before, countries are ex ante identicalin terms of gains and losses. For any country c, let Gi > 0 and Bi < 0 denote respectively
the good outcome and the bad outcome relating to issue i, where i = 1, 2. We denoteby qic the probability that a country-specific realization affecting country c on issue i isgood, and we assume that q1c = q2c = q, for c = h, f . However, we allow the possibility
that G1 6= G2 and B1 6= B2 ; that is, that the outcomes of good and bad realizations differ
across issues. We keep the assumptions qGi + (1 − q)Bi > 0 and Gi + Bi > 0 for i = 1, 2,
which imply that the expected gains from cooperation are positive ex ante, and if one
country has a good realization and the other country has a bad realization for issue i, it
is still socially desirable that both countries would cooperate on issue i.
As demonstrated in the previous section, if the countries enter into separate agreementson the two issues, a country c will quit the agreement on issue i as soon as it learns that its
country-specific realization of that issue is bad, i.e., rci = b. Thus, the expected value of theagreement on issue i, under non-commitment, is equal to EViN = Gi q 2 , and consequently
the expected value of having two separate agreements is then given byE(V ring ) = (G1 + G2 )q 2 .

(1)

If instead the issues are pooled together in a single agreement, a country c will exit if
and only if it has realizations that indicate that its payoff under “remain” is negative forthe two issues together. Clearly, if the realizations in country c are (b, b), i.e., bad on bothissues, then it will exit. What happens if country c’s realizations are good on issue 1 andbad on issue 2? The answer is that country c will exit the pooled agreement if G1 + B2 < 0
and remain if G1 + B2 > 0 (i.e. if the good outcome in issue 1 more than compensates forthe bad outcome in issue 2). We now have to distinguish four different cases, and we doso by defining four different configurations. Configuration 1 will be defined as the mostfavorable parameter configuration, i.e., both G1 + B2 > 0 and B1 + G2 > 0 hold suchthat a bad outcome for one issue can always be made up by a good outcome for the otherone. Configuration 2 will be defined by G1 + B2 > 0 and B1 + G2 < 0 such that only
a bad realization of issue 2 can also be made up by a good realization of issue 1, but
not vice versa, and similarly G1 + B2 < 0 and B1 + G2 > 0 will define configuration 3.
Configuration 4 is the least favorable and defined by G1 + B2 < 0 and B1 + G2 < 0 suchthat each country will want to exit unless both of its realizations are good.
Consider first the parameter configuration most favorable for remaining in the agree7

ment, configuration 1, for which the probability that a country wants to exit is given by(1 − q)2 . The probability that a country ‘remains’ is then equal to 1 − (1 − q)2 = q(2 − q).
Conditional on its partner choosing to ‘remain’, the expected gain to a country fromentering into a pooled agreement under configuration 1 is given by
q 2 (G1 + G2 ) + q(1 − q)(G1 + B2 ) + (1 − q)q(B1 + G2 )= [qG1 + (1 − q)B1 ] + [qG2 + (1 − q)B2 ] .
Multiplying this expression by the probability that its partner does not exit, q(2 − q), we
get the expected value of the pooled agreement under configuration 1:
E1 (V pool ) = q(2 − q) {[qG1 + (1 − q)B1 ] + [qG2 + (1 − q)B2 ]} .

(2)

Next, consider configuration 2 for which a country will want to exit the pooled agreement if it receives a bad shock on issue 1. Under this configuration, the (conditional)
expected gain to a country for entering into a pooled agreement is equal toq 2 (G1 + G2 ) + q(1 − q)(G1 + B2 ) = qG1 + q [qG2 + (1 − q)B2 ] .
Multiplying this by the probability q that its partner does not exit, we get the expectedvalue of the pooled agreement under configuration 2, and we observe that it is smaller
than the expected value of having two separate agreements, E(V ring ):E2 (V pool ) = q 2 {G1 + [qG2 + (1 − q)B2 ]} < E(V ring ) = (G1 + G2 )q 2 .
Configuration 3 is identical to configuration 2, except that the subscripts are swapped so,clearly, E3 (V pool ) < E(V ring ). Finally, consider configuration 4 for which G1 + B2 < 0
and B1 + G2 < 0. Each country will want to exit unless both of its realizations are good.
Again, it is straightforward to show that E4 (V pool ) < E(V ring ).
It follows then that a necessary condition for pooling to be better than ringfencing isthat the two countries are in configuration 1, i.e. the good outcome in one issue would morethan compensate for the bad outcome in the other issue. However, this is not sufficient.
Comparing (2) with (A.1) shows that ringfencing welfare-dominates pooling if and only
if q(G1 + G2 ) + (2 − q)(B1 + B2 ) > 0 holds in addition to having both G1 + B2 > 0 andB1 + G2 > 0. Thus, we conclude:8

Proposition 1. Pooling is superior to ringfencing if and only if the following three conditions are simultaneously satisfied:(i) G1 + B2 > 0,(ii) B1 + G2 > 0, and(iii) q(G1 + G2 ) + (2 − q)(B1 + B2 ) > 0.
While conditions (i) and (ii) here echo the reasoning of Tollison and Willett (1979), in
an environment of uncertainty, as modelled here, these are not sufficient for issue linkage
to be preferable to ringfencing. Note, too, that an expected welfare gain for each issue,that is, qGi + (1 − q)Bi > 0, is not sufficient to satisfy condition (iii) in Proposition 1 andthus to guarantee superiority of issue linkage. The reason why pooling is more attractive
under the conditions stated in Proposition 1 is as follows. Under ringfencing, a country willexit an issue-specific agreement if it has a bad realization that adversely affects its payoff
under cooperation on that issue. Such an exit imposes an externality on the other country,
denying it the benefit of cooperation, if the other country’s realization is favorable. Under
pooling, a country does not exit provided that the bad realization in one issue is morethan compensated for by a good realization in the other issue (conditions (i) and (ii)above). However, preventing exit by means of pooling would make sense only if the loss
incurred by the country that would have chosen to exit under ringfencing is not too large
compared with the expected value of the externality inflicted on the other country by suchan exit. This is condition (iii) above. The importance of Proposition 1 is that it showsthat pooling is superior only under rather stringent conditions. Thus it provides a direct
counterexample to any presumption that issue linkage would be beneficial whenever eachof the issues has positive expected value.
So far, we have considered the case of identical probabilities and differences in payoffs
from the different issues. We now want to consider the role of issue linkage if successprobabilities of good realizations differ across issues, and for this reason we now assume
that outcomes do not, i.e., G1 = G2 = G and B1 = B2 = B holds. Let qi denote theprobability that G will materialize for issue i. In order to scrutinize the role of probabilities
for issue linkage, we now assume that the aggregate expected payoff from both issues, thatis, (q1 + q2 )G + [1 − (q1 + q2 )]B, stays constant, implying that q1 + q2 = γ, where γ < 2
is a constant. It is now clear that only configurations 1 and 4 can arise in this setup as
9B + G is either positive or negative. Again, it is straightforward to show that ringfencing
will welfare-dominate issue linkage for configuration 4.
If B + G > 0, we are in configuration 1. The expected value of issue linkage andringfencing now depends on the different probabilities q1 and q2 . We have relegated thedetails to Appendix A.1, and we find
Proposition 2. Issue linkage is more (less) likely to welfare-dominate ringfencing themore equal (different) the success probabilities are. The difference between expected issue
linkage welfare and expected ringfencing welfare is maximal if q1 = q2 = γ/2.
Proof. See Appendix A.1
Proposition 2 shows that issue linkage has the best chance to dominate ringfencing
if the success probabilities of the issues are sufficiently similar. It is not a good idea tocombine an issue with a large success probability with one with a low success probability,
given that the expected ex ante welfare across issues stays constant. In that case, thechances are too great that one issue will have a bad realization that cannot be avoided
by issue linkage. Ringfencing ensures that agreements are implemented only upon goodrealizations and, with different success probabilities, it is better not to compromise on ahigh-probability success for one issue with a high probability of a bad realization for theother issue.

4

A more general model of issue linkage and ringfencing

In the preceding sections, a country-specific realization pertaining to an issue can take on
only one of two values: good or bad. We now consider the case where the realizations are
random variables with a continuous density function. As before, at the beginning of period
2, each country c ∈ {h, f } observes its country-specific realizations (rc1 , rc2 ) pertaining toissue 1 and issue 2, and decides whether to withdraw from an agreement. The value takenby the realization rci is a perfect predictor of country c’s payoff in issue i if c honors theagreement. If it withdraws, the payoff is zero (regardless of the realizations).
Again, we assume that the two countries are ex ante identical, and the country-specificrealizations on issue i (i = 1, 2) are identically and independently distributed. Let xc
10(respectively, yc ) denote country c’s payoff on issue 1 (respectively, 2) if its realization
for this issue takes on the value xc (respectively, yc ). For tractability, we assume that thedensity functions g(x) and g(y) are uniform, defined over the interval [γ − δ, γ + δ] whereγ > 0 and γ − δ < 0.
Again, we want to compare the merit of a ringfencing regime with that of a poolingregime. Panel (a) of Figure 1 illustrates the distribution of payoffs for a single issue. Under
ringfencing, if both xh and xf are positive, then both countries will cooperate on issue
1 in period 2. If one of the xc is negative, one country will withdraw, and the payoff on
issue 1 is zero for each country. It follows that country h’s expected payoff from joining
the agreement on issue 1 is given by
Z γ+δ

Z(γ + δ)3
γ + δ γ+δ x
Pr [xf > 0] ×
xh g(xh )dxh =
dx =
2δ 0
2δ
8δ 2
0
where Pr [xf > 0] denotes the probability of no exit by partner country f . The same
calculation applies to country f . Similarly, a country’s expected payoff on issue 2 is (γ +δ)3 /8δ 2 . Summing over issues, the expected payoff of a country under the ringfencing
regime is given by(γ + δ)3.
(3)
4δ 2
We now turn to the expected payoff under the pooling regime. In this case, a countryE(V ring ) =c would leave if and only if xc + yc < 0. Define zc = xc + yc . It can be shown that thedensity function of zc , being a convolution of two identical uniform distributions, has theshape of an isosceles triangle centered on 2γ. This is illustrated in panel (b) of Figure 1.
The density function, denoted by φ(z), has the slope 1/(4δ 2 ) in the interval [2(γ − δ), 2γ]
and −1/(4δ 2 ) in the interval [2γ, 2(γ + δ)] such that(z−2(γ−δ)for 2(γ − δ) ≤ z ≤ 2γ,
4δ 2
φ(z) =
2(γ+δ)−z
for 2γ ≤ z ≤ 2(γ + δ).
4δ 2
Each country will gain only if both countries stay in. Under pooling, the expected gain
to country h is given by

E(V

pool

Z 2(γ+δ)) = Pr [zf > 0]zh φ(zh )dzh
0

=

 3
1
2
2
δ
−
γ
+
2δγ
(δ − γ 3 + 3(δ 2 γ + γ 2 δ))
4
6δ
11

(4)

Figure 1: Identical uniform probability distributions, ringfenced and linked

where Pr [zf > 0] denotes the probability of no exit under issue linkage. Comparing (3)and (4), we can show:
Proposition 3. Let k ≡ δ/γ > 1. Under the uniform distribution, pooling dominates
ring fencing if and only if δ is sufficiently small such that the following condition holds:
1 < k < k where k ≈ 3.065.
Proof. See Appendix A.2
According to Proposition 3, ringfencing dominates pooling if δ > kγ. Recall that byassumption γ − δ < 0. Thus δ − γ is the absolute value of the greatest possible loss to a
country on any issue. Condition δ > kγ is equivalent toδ − γ > (k − 1)γThus if the greatest possible loss is greater that 2.065 times the mean γ, then ringfencing isbetter than pooling. This result is consistent with that stated in Proposition 1 that shows12that pooling requires “not too bad” realizations. In this more general model, ringfencing
similarly dominates pooling if the degree of uncertainty is sufficiently large relative to themean.
How does our model extend to differences in the distributions? Suppose the support ofthe density function for x is [γ − ε, γ + ε] and that for y is [γ − δ, γ + δ] with δ > ε > γ.
Under ringfencing, the expected values of the two separate agreements are (γ + ε)3
E W1ring =
8ε2
and


 (γ + δ)3.
E W2ring =8δ 2

and their sum is equal to ε2 (γ + δ)3 + δ 2 (γ + ε)3
E V ring =8δ 2 ε2(5)Under pooling, let z = x + y denote the realization of the sum of the two issues. The
distribution of z is a convolution of the distributions of x and y, and the density functionof z can be shown to have the shape of a trapezoid and is given by z−2γ+ε+δ
for 2γ − (δ + ε) ≤ z ≤ 2γ − (δ − ε)4δε1
for 2γ − (δ − ε) ≤ z ≤ 2γ + (δ − ε)fz (z) =2δ 2γ+δ+ε
for 2γ + (δ − ε) ≤ z ≤ 2γ + δ + ε
2δεFigure 2 shows the probability distribution, and the resulting cumulative distribution is
given byFz (z) =







(z−2γ+δ+ε)28δε
z+2γ+δ
2δ
8δε−(2γ+δ+ε−z)28δεfor 2γ − (δ + ε) ≤ z ≤ 2γ − (δ − ε)for 2γ − (δ − ε) ≤ z ≤ 2γ + (δ − ε)for 2γ + (δ − ε) ≤ z ≤ 2γ + δ + ε

To compute the expected value of the pooled agreement, we must consider two distinct
cases. In case 1, the flat segment of the density function fz (z) lies entirely to the right ofzero, i.e., 2γ − (δ − ε) > 0 and in case 2, 2γ − (δ − ε) ≤ 0. Figure 2 depicts the secondcase. For case 1, the probability that the partner country remains “in” is equal to1 − Fz (0) =8εδ − (2γ − δ − ε)2,8εδ

and we can show that13

Figure 2: Probability distribution of z

8εδ − (2γ − δ − ε)28εδ(2γ − δ + ε)2 (5ε + δ − 2γ) 2γ(δ − ε) (6γ + 3δ − ε)ε
×
+
+
.
24δε
δ
6δ

E(V pool ) =(6)

We want to find out whether the disparity between the supports of x and y matters.
For this purpose, let us define q = (δ + ε)/2 and δ = q + θ and ε = q − θ, with q ≥ θ ≥ 0. Ifθ = 0, we are back to the case of identical supports. A larger θ means a greater difference
between the two supports. It is also convenient to define κ = q/γ ≥ 0 and t = θ/γ ≥ 0.
Since γ − ε < 0 or γ − q + θ < 0, division by q implies that κ > t + 1. For case 1,
2γ > δ − ε = 2θ, so case 1 implies γ > θ which implies t < 1. Without loss of generality,
set γ = 1, so that case 1 implies κ > t + 1 and t ∈ [0, 1]. We have relegated the details of
the computations to Appendix A.3, which shows that case 1 is confined to κ ∈ [1, 3.56683](and t ∈ [0, 1]). Figure 3 shows the contour plot of the gains from issue linkage for thisrange, with κ on the horizontal axis and t on the vertical axis, taking into account that
κ > t + 1.
Issue linkage is welfare-preferred only in the brightest area below the κ > t+1 constraint,
and Figure 3 shows that the attraction of issue linkage is weakly reduced by an increase
14

Figure 3: Case 1

in t, measuring the difference in the two uniform distributions.
Case 2, assuming 2γ ≤ δ − ε, implies an even larger difference in distributions, and it
cannot include the symmetric case of δ = ε. This case now implies t ≥ 1 and κ ≥ t + 1:
2γ ≤ δ − ε holds only if t ≥ 1. Also, we require that negative payoffs occur with positive
probability, and thus δ > ε > γ, implying κ − t > 1 ⇔ κ > t + 1 (and thus also implying
κ > 2). The probability that a country stays in the pooling agreement is now given by
1 − Fz (0) = 1 −δ − 2γδ + 2γ=,2δ2δ

and thus the expected payoff from pooling is now equal to(2γ + δ − ε)2 (6γ + 3δ − ε) ε
+
4δ
6δ

1
2
2
2
=
(δ
+
2γ)
12γ
+
12γδ
+
3δ
+
ε
.
24δ 2

δ + 2γE(V pool ) =
2δ

!
(7)

The details of comparing (7) with (5) are also relegated to Appendix A.3. Figure 4 shows
the contour plot of the gains from issue linkage, taking the constraint κ > t + 1 into
account.

15

Figure 4: Case 2

We see for the ranges here that issue linkage will never welfare-dominate ringfencing.
Since κ and t are unbounded, the question is whether this could happen for very large
κ and t. Searching for a maximum numerically indicates that this maximum welfare
difference between issue linkage and ringfencing is given by κ = 2 and t = 1, but forthese values, issue linkage will be welfare-inferior.6 Thus, we conclude that ringfencing
will always welfare-dominate issue linkage for case 2.

5

Concluding remarks

This paper has made a contribution to a better understanding of why we do not see universal issue linkage. We have employed a simple model of potential exit from an agreement
to show that issue linkage does not always welfare-dominate ringfencing. Our model does
not rely on transaction and/or coordination costs as previous analyses have done. Instead
6

The Mathematica code is available upon request.

16

we show that issue linkage can harm parties because it will not allow them to drop outof a part of the agreement when both parties face a very bad realization. We have shownthat even when cooperation on each of the issues has positive value, the pooling of issues
may be inferior to ringfencing if the degree of uncertainty is large.
Our objective in this analysis was to offer a framework in which the trade-off between
ringfencing and issue linkage can be scrutinized in a simple way. For this reason, we havedeliberately left out any insurance aspect of issue linkage in our analysis, by assumingthat all random variables are independently distributed. If issue realizations are negatively(positively) correlated with each other, issue linkage becomes more (less) attractive. We
leave this aspect of issue linkage to future research.

References[1] Alesina, A., I. Angeloni and F. Etro, 2005. Economic Unions. American Economic
Review 95 No.3, 602-615.
[2] Carraro, C. and C. Marchiori, 2003. Endogenous strategic issue linkage in international
negotiations. mimeo.
[3] Geng, X. and A. Woodland, 2018. Issue linkage in International Environmental Agreements formation. mimeo UNSW.
[4] Maggi, G., 2016. Issue Linkage. In K. Bagwell and R. Staiger, eds, Handbook of Commercial Policy Chapter 9. Elsevier.
[5] Richardson, M. and Stähler, F., 2019, International Agreements, Economic Sovereignty
and Exit. mimeo.
[6] Tollison, R. and T. Willett, 1979. An economic theory of mutually advantageous issue
linkages in international negotiations. International Organization 33 No.4, 425-449.

Appendix
A.1

Proof of Proposition 2

With G1 = G2 = G and B1 = B2 = B, the expected value of ringfencing two agreements
is equal toE(Ṽ ring ) = G q12 + q22 ,
17

(A.1)

and the expected value of an issue linkage agreement is equal toE(Ṽ pool ) = [1 − (1 − q1 )(1 − q2 )] [2q1 q2 G + (G + B) (q1 (1 − q2 ) + q2 (1 − q1 ))] .

(A.2)

Since 1 − (1 − q1 )(1 − q2 ) = q1 + q2 − q1 q2 and q1 (1 − q2 ) + q2 (1 − q1 ) = q1 + q2 − 2q1 q2 ,we find that E(Ṽ pool ) > E(Ṽ ring ) requires
G+B
q12 + q22 − 2q1 q2 (q1 + q2 − q1 q2 )>.
G
(q1 + q2 − q1 q2 )(q1 + q2 − 2q1 q2 )(A.3)

The LHS of (A.3) is between zero and unity since B < 0 and gives the ratio of a mixed
outcome to a good outcome only. Since γ = q1 + q2 and q12 + q22 = γ 2 − 2q1 q2 , we canrewrite condition (A.3) as
G+B(γ 2 − 2δ) − 2δ(γ − δ)
> f (δ) =,G(γ − δ)(γ − 2δ)
where δ = q1 q2 . Note that f (0) = 1 while (G + B)/G < 1, implying that ringfencing will
always dominate if q1 q2 = 0. Differentiation yields
f 0 (δ) = −(2 − γ)(γ 2 − 2δ 2 )<0[(γ − δ)(γ − 2δ)]2because γ = q1 + q2 < 2 and γ 2 − 2δ 2 = q12 + q22 + 2q1 q2 − 2q12 q22 > 0 because q1 q2 > q12 q22 ⇔1 > q1 q2 . Thus, the RHS of condition (A.3) becomes less demanding with an increase inδ. The maximum δ = q1 q2 subject to γ = q1 + q2 is given by q1 = q2 = γ/2. Consequently,
q1 = q2 = γ/2 minimizes the RHS of condition (A.3).

A.2

Proof of Proposition 3

Pooling yields a higher expected payoff iff(δ 2 − γ 2 + 2δγ) (δ 3 − γ 3 + 3(δ 2 γ + γ 2 δ))(γ + δ)3>6δ 2
4⇔ 4 δ 2 − γ 2 + 2δγ (δ 3 − γ 3 + 3(δ 2 γ + γ 2 δ)) − 6δ 2 (γ + δ)3 > 0⇔ 4γ 5 − 20γ 4 δ + 2γ 3 δ 2 + 14γ 2 δ 3 + 2γδ 4 − 2δ 5 > 0.
Since δ > γ we can write δ = kγ, where k > 1. Then the above condition becomes
4γ 5 − 20kγ 5 + 2k 2 γ 5 + 14k 3 γ 5 + 2k 4 γ 5 − 2k 5 γ 5 > 0i.e., pooling is better than ringfencing iff∆(k) ≡ 4 − 20k + 2k 2 + 14k 3 + 2k 4 − 2k 5 > 0.
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Note that ∆(1) = 0, and if k is very large, then ∆(k) < 0. Also, ∆0 (k) = −20 + 4k +42k 2 + 8k 3 − 10k 4 with ∆0 (1) > 0, and ∆00 (k) = 4 + 84k + 24k 2 − 40k 3 . Calculating theroots of the polynomial ∆(k) = 0 in the relevant range, we find that ∆(k1 = 1) = 0
and ∆(k2 = 3.0654) = 0 are the relevant solutions. Furthermore, the only extremum inthe relevant range is given by ∆0 (k3 = 2.44681) = 0, and since ∆00 (k3 ) = −232.731; k3
defines a local maximum and proves that ∆(k) > 0 for 1 < k < 3.0654 and ∆(k) ≤ 0 fork ≥ 3.0654.

A.3

Extension to different uniform distributions

Case 1Comparing (6) with (5) shows that pooling welfare-dominates ringfencing iff8εδ − (2γ − δ − ε)2 (2γ − δ + ε)2 (5ε + δ − 2γ) + 48γ(δ − ε)ε + 4(6γ + 3δ − ε)ε2> 24 ε2 (γ + δ)3 + δ 2 (γ + ε)3 .
Using the definitions of κ and t, pooling dominates ringfencing iff t and κ satisfy8(κ + t)(κ − t) − (2 − 2κ)2×

(2 − 2t)2 (5(κ − t) + (κ + t) − 2) + 48(2t)(κ − t) + 4(6 + 3(κ + t) − (κ − t))(κ − t)2−24 (κ − t)2 (1 + κ + t)3 + (κ + t)2 (1 + κ − t)3 ≥ 0

for some t > 0 and κ ≥ t + 1 :, which is equivalent to−16κ5 +16κ4 +32κ3 t2 +112κ3 −96κ2 t2 +16κ2 −48κt4 −432κt2 −160κ+240t4 +208t2 +32 ≥ 0.
Take this condition for pooling to welfare-dominate, and create a new term in which(i) all terms that are negative and contain t are set equal to zero, and (ii) all terms
that are positive and contain t are evaluated at t = 1. This new term is thus larger
than the net difference in welfare from pooling versus ringfencing and given by Λ(κ) =(κ − 1)κ(κ + 2)(5 − (κ − 2)κ) + 480. It is straightforward to show that Λ(κ1 ) = 0 forκ1 = 3.56683 and Λ(κ) > 0 for κ ∈ [1, κ1 ]. Thus we can constrain our analysis to theranges κ ∈ [1, κ1 ] and t ∈ [0, 1].
Case 2Comparing (7) with (5) shows that pooling dominates ringfencing iff
ε2 (γ + δ)3 + δ 2 (γ + ε)3(δ + 2γ) (12γ 2 + 12γδ + 3δ 2 + ε2 )>24δ 2
8δ 2 ε2i.e., iff
ε2 (δ + 2γ) 12γ 2 + 12γδ + 3δ 2 + ε2 − 3ε2 (γ + δ)3 − 3δ 2 (γ + ε)3 > 0.
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Again, using δ = q + θ and ε = q − θ, this condition can be rewritten as(q − θ)2 (q + θ + 2γ) 12γ 2 + 12γ(q + θ) + 3(q + θ)2 + (q − θ)2− 3(q − θ)2 (γ + q + θ)3 − 3(q + θ)2 (γ + q − θ)3 > 0,
and after factoring out as−2q 5 + 2q 4 γ + 8q 3 θ2 − 8q 3 θγ + 18q 3 γ 2 − 4q 2 θ3 + 12q 2 θ2 γ
−36q 2 θγ 2 + 18q 2 γ 3 − 6qθ4 − 8qθ3 γ − 18qθ2 γ 2 − 48qθγ 3+4θ5 + 2θ4 γ + 36θ3 γ 2 + 18θ2 γ 3 > 0.
Again, we use q = κγ and t = θγ. Then, the condition for the welfare dominance ofpooling can be written as∆(κ, t) = −2κ5 + 2κ4 + 8κ3 t2 − 8κ3 t + 18κ3 − 4κ2 t3 + 12κ2 t2− 36κ2 t + 18κ2 − 6κt4 − 8κt3 − 18κt2 − 48κt + 4t5 + 2t4 + 36t3 + 18t2 > 0.

20