Lateral Isms
Lateral Isms
Lateral Isms
ALEXANDER THOMPSON
Thompson.1191@osu.edu
DANIEL VERDIER
Verdier.2@osu.edu
Department of Political Science
Ohio State University
Abstract
Dierent international regimes are built from legal instruments that vary in terms of whether they
are multilateral, bilateral or a combination thereof. We investigate the reasons for such variation.
The choice between multilateralism and bilateralism is a function of the trade-o between each
instruments relative aw. Multilateralism is wasteful in incentives, as the same agreement is
oered to all states regardless of their compliance costs. Bilateralism mitigates this problem by
allowing for more tailored agreements but in the process multiplies transaction costs by requiring
many of them. We use a formal model to generate propositions on the design of lateralism and
the related issue of membership size, and oer illustrations in the context of four regimes: foreign
direct investment, human rights, climate change, and international trade.
International regimes are built on instruments that vary in terms of whether they are multi-
lateral, bilateral or a combination thereof. For example, the current trade regime has a strong
multilateral component as its core, centered on the World Trade Organization (WTO), while the
foreign direct investment (FDI) regime is primarily constituted of bilateral agreements. Notably,
many regimes combine two or more instruments: the nuclear nonproliferation regime combines a
multilateral treaty with bilateral security guarantees proered mostly by the United States.
The question of "lateralism" relates to whether countries should be treated equally under inter-
national law or dierently. Multilateralism, according to Ruggie, implements "generalized principles
of conduct" whereas bilateralism "dierentiates relations case-by-case based principally on a pri-
ori particularistic grounds or situational exigencies" (1992: 571). How this issue is addressed in
practice is usually seen as a result of prevailing norms or of bargaining power, with some states
advocating more uniform rights and obligations and others calling for dierential treatment.
In an attempt to understand why, we point to two key factors: transaction costs and a new
concept, the member surplus. The member surplus captures the idea that the multilateral strategy
can be wasteful in incentives, since incentives are calculated to elicit the participation of the state
that is burdened with the highest cost of compliance, thereby creating rents for the other mem-
bers. The bilateral strategy, in contrast, allows the customization of rights and obligations to each
individual member state. But because the bilateral strategy is more wasteful in transaction costs
than the multilateral strategy, states face a trade-o.
This allows us to make three central claims: (1) multilateralism is most attractive with high
transaction costs and a low member surplus, (2) bilateralism is most attractive when transaction
costs are low and the member surplus is high, and (3) we encounter combinations of multilateralism
and bilateralism when both transaction costs and the member surplus are high. Such combinations
come in two forms, regimes that contain a mix of multilateral and bilateral agreements and regimes
that rely on multilateral agreements that are customized to the needs of dierent members. A
fourth claim is that regimes with both high transaction costs and a high member surplus exhibit
the highest rate of exclusion because of the cost of attracting members. We develop a formal model
to capture the logic behind these institutional outcomes.
The next section situates our topic in the theoretical literature. We then introduce the notions
of transaction costs and member surplus and present the model and a set of predictions that ow
1
from it. In an empirical section, we demonstrate the plausibility of our claims in the context of
four prominent regimes: the foreign direct investment regime, the human rights regime, the climate
change regime, and the trade regime.
1 Lateralisms in the Literature
We adopt the standard denition of international regimes, which grounds them in a set of norms or
goals that guide state behavior in a given issue area (Krasner 1983; Keohane 1984). Most regimes
have at their origins a market imperfection, making a mutually ecient outcome unreachable
through standard competitive behavior. Our approach also complements work on regimes that
views them as created by a hegemonic state (Krasner 1976; Keohane 1984), a small group of great
powers (Snidal 1985), the market-oriented Anglo-Saxon subset (Cowhey and Klimenko 2000), or
the North at large (Sell 2007). We assume that most regimes are organized by a small group of
founders who try to enlist the cooperation of a larger group of regular members. The founders take
the lead on account of their larger resources, greater interest in the regime, or principled concern
over the issue.
Most treatments of international regimes simply assume that they rest on multilateral agree-
ments. The impetus behind this approach is at least partly normative. International law scholars
have a general legal obligation to approach their international relations multilaterally (Dupuy 2000).
Indeed, multilateral agreements seem to carry special symbolic power and legitimacy, partly because
they promote uniformity and equal treatment of states (Blum 2008). Ruggie (1992) and Finnemore
(2003) point to a norm of multilateralism that has predominated since World War II, as reected in
the proliferation of multilateral institutions and a tendency of states to work through them. Others
see much less multilateralism in the system and explain the choice of lateralism in power-based
terms. For example, Realists point to the end of the Cold War and the unrivaled power of the
United States to explain its shift toward bilateralism and unilateralism in recent years.
1
Gruber
(2000) and Guzman (1998) describe how powerful states use bilateral bargaining to gain leverage
over weaker counterparts.
We look at variation in multilateralism versus bilateralism and address the question from
a strategic and eciency perspective. This is consistent with the approach of Yarbrough and
Yarbrough (1992), who consider the dierent forms of lateralism during dierent historical periods
2
in the context of trade cooperation. Like Kahler (2004), we also take seriously the possibility that
these outcomes are not mutually exclusive but can coexist at a given point in time and even within
a single regime. We build on the insight of Conybeare (1980) and Oye (1992) that decentralized
bargaining oers a viable substitute for multilateral agreements in some situations.
Some authors do try to account for the choice of lateralism at a given time. Distinguish-
ing between coordination and PD-like problems, Ruggie (1992) suggests that the former generate
transaction costs that can easily be solved through multilateralism, whereas the latter require a
normative change in conjunction with multilateralism. Studying the double taxation regime, Rixen
(2010) endorses Ruggies proposition that a multilateral organization minimizes transaction costs
but argues that distributive issues are better accommodated through bilateral bargaining, provided
that such bargaining produces no externalities on third parties, in which case multilateralism is
more appropriate (see also Rixen and Rohlng 2007).
Another general cause for decentralized bargaining is preference heterogeneity. The idea comes
from the literature on scal federalism, according to which decentralized provision of a public good
is more ecient than uniform provision through the central government whenever individuals have
preferences that are heterogenous and inter-jurisdictional policy externalities or scale economies are
small (Oates 1999). Central to the argument is the assumption that citizens preferences are better
known by local than central agencies.
2
However, Harstad (2007) makes information revelation
endogenous and obtains the opposite result: incomplete information makes uniformity preferable
because it deters individuals from strategically delaying the revelation of their preferences. By
making incomplete information irrelevant, uniformity minimizes negotiating costs.
Our paper also touches on the problem of membership which states to include, which to
excludeand the apparent trade-o between the depth and breadth of cooperation (Downs, Rocke,
and Barsoom 1998). The lack of credibility of the threat of punishment that would have all countries
withdraw cooperation in response to noncompliance is responsible for limiting participation to states
with an immediate interest in supplying the public goodSchellings (1978) "k group." As Barrett
(2003: 205) argues, the punishment threat is only credible if the net gain of cooperation is small, for
if it is large, the threat and the tolerance for self-inicted pain must also be large. The preference
for a broad yet shallow agreement may reect concerns for fairness or fast-increasing marginal costs
(Barrett 2003: 304).
3
The breadth-versus-depth dilemma disappears if the regime founder is able and willing to turn
the public good into a club good and discriminate against non-participants. Each members fear
of being discriminated against elicits broad participation. A good example is non-MFN-based
bilateral trade agreements. The discriminatory feature built into a club good has some economists
promote bilateralism as a surer way to reach global free trade than MFN-based multilateralism
when countries have asymmetric endowments (Aghion, Antrs, and Helpman 2007; Saggi and
Yildiz 2010). Bilateralism is an eective way to bring small countries into the fold, for they would
free ride under a multilateral bargain. Such is in part why some legal scholars viewed the use
of bilateralism in the late 1980s as a constructive complement to multilateral trade negotiations
(Hudec 1990; Sykes 1992). Endogenizing the club or public format of the good, Stone, Slantchev,
and London (2008) argue that a hegemon on the decline reaches a tipping point before which it
prefers to oer a club good (a discriminatory regime), but past which it is better o with a public
good (a non-discriminatory regime).
The breadth-versus-depth dilemma also evaporates if one suspends the requirement of unifor-
mity of obligations among members (Gilligan 2004). Allowing countries to choose the level of their
contribution makes for a regime that is both deep (members with an interest in the provision of the
good contribute to that eect) and broad (members willing to contribute less or free ride altogether
may still join in). In other words, the customization of obligationsa hybrid of the multilateral and
bilateral strategieselicits greater participation.
This quick review of existing research on lateralism and membership reveals a vibrant diversity
from which we retain and further develop the two following ideas: rst, multilateralism is a solu-
tion to high transaction costs, including the costs of negotiating and enforcing agreements; second,
bilateralism may be a solution to several aws incurred under multilateralism: free riding or exclu-
sion in the case of a public good and inecient uniformity in the face of preference heterogeneity,
asymmetric endowments, distributive concerns, or high marginal costs. The present theory and
model tackle these ideas in a way that strives to be intuitive and mathematically tractable.
2 Transaction Costs and the Member Surplus
One of the key concepts that generate our results, transaction costs, is commonly used in the regime
literature and requires little introduction. The second concept, member surplus, is new and in need
4
of lengthier introduction. The trade-o that these two factors produce and the various solutions
that are conceivable provide the basis for our theoretical claims.
Our denition of transaction costs is borrowed from Williamsons (1985: 20) work. Transaction
costs are "the costs of negotiating, drafting, and safeguarding an agreement," with safeguarding
broadly dened to include what is necessary to make the agreement enforceablethe setup and
running costs of monitoring, dispute settlement, renegotiation in the face of uncertainty, and, in
Williamsons (1985: 21) terms, "the bonding costs of eecting secure commitments." While some
safeguarding costs are only incurred after an agreement is reached, the prospect of facing such
problems complicates and prolongs matters during the negotiation phase as well (Fearon 1998).
Transaction costs have scale economies in the sense that they make the signing of n dyadic treaties
costlier than the signing of one treaty with n participants.
The multilateral strategy has the advantage of saving on transaction costs (Keohane 1984;
Ruggie 1992; Rixen 2010). This is true because only one set of negotiations is required, and
because most multilateral treaties incorporate forums that facilitate further decision-making and
provide economies of scale in monitoring and dispute resolution. The multilateral strategy saves on
transaction costs also because coordinated negotiations allow more bargaining possibilities to be on
the table at once, promoting issue-linkage and chances that every oer nds a corresponding match
(Martin 1992). The bilateral strategy, in contrast, multiplies transaction costs, since a new contract
has to be negotiated, drafted, and safeguarded for each participant, while some participants bids
may be left with no attractive counterpart. Last, by treating members identically, the multilateral
strategy helps avoid the posturing and costly delays that beset bilateral negotiations in the presence
of information asymmetry (Harstad 2007).
However, the multilateral strategy will be expensive for the founder in another sensethis is
where the member surplus comes in. Multilateralism in its pure form oers only one deal and this
deal is the same for everyone (there are exceptions to this rule, which we address later). As a result,
participants are oered an incentive that is calculated to elicit the participation of the state that
is burdened with the highest cost of compliance. The problem is similar to the one that occurs
in competitive markets, where the law of one price for a particular good confers a surplus on all
producers who would have been willing to sell for less. This surplus is known in economics as the
"producer surplus."
3
In direct analogy, we call it the "member surplus." Unlike the multilateral
5
approach, the bilateral approach is immune to the member surplus because it gives to each state
the incentive it needs to participate and no more. Bilateralism corresponds in market economics to
an extreme case of market fragmentation, where a monopolist oers her reservation value to each
consumer.
To see this, imagine a situation with no transaction costs. Then the founder would always
prefer the bilateral approach to the multilateral approach. This is easily seen in Figure 1, featuring
on the horizontal axis an ordering of all members according to their compliance cost z
i
and on the
vertical axis the cost to the founder t(z
i
). The slope of the curve is positive, since the higher a
members compliance cost, the higher its participation price. Assume that the founder wants to
include country N and all those to its left. By construction, the N
th
country is the most expensive
member to be included in the regime. The cost of wooing countries with compliance cost less than
or equal to N by means of a multilateral contract is equal to the rectangle, because the multilateral
approach forces the founder to pay the transfer it pays to the N
th
member to all other members. In
contrast, by means of bilateral contracts alone, this same cost would be no more than the triangle
situated below the curve, only half the size of the rectangle. The triangle above the curve is the
member surplus that results from the multilateral approach.
[Figure 1]
Assume for the sake of simplicity that any single deal, bilateral or multilateral, costs constant T
to process, with T positive. There are now two possibilities. One possibility is that the N bilateral
contracts could be costlier than one multilateral contract for everyone. This case is represented in
Figure 2A, where the cost of the multilateral contract is the same as before plus the transaction
cost incurred once (z
N
t (z
N
) + T), while the cost of the bilateral contracts is the former triangle
augmented with the transaction cost incurred N times
P
N
i=1
(t (z
i
) + T)
s
i
P
j6=i
s
j
and c (s
i
) = cs
i
,
with c the marginal cost. Variable s
i
may be thought of as an investment in a polluting technology
9
or any activity producing an externality, for instance shing in the high seas, setting protective
taris, or curbing human rights. In each case, an investment s
i
generates a negative externality s
i
inicted upon every other member. Parameter is the externality index; it is greater than or equal
to zero, with a value of zero indicating no externalities and a value greater than zero indicating
their presence. Variable a
i
scales member is marginal gain for engaging in the activity based on
the activity. (Note that the model would equally work and yield the same comparative statics if
we had opted for a dierent mix of marginals, i.e., constant gains and marginally increasing costs,
or a dierent sign on the externality.)
In the absence of a founder, each member maximizes u
i
(s
i
, s
j
) with respect to her choice vari-
able s
i
such that s
i
, s
j
0. This version of the game yields a competitive equilibrium in which
every member produces s
#
i
=
a
i
2c
2
. This quantity is greater than the individual production level
that would maximize the social optimum,
P
N
i=1
u
i
, which is equal to s
i
=
a
i
2(c+(N1))
2
(see the
appendix for the demonstration of both results). As one would expect, the presence of a nega-
tive externality yields a competitive equilibrium that is economically inecient because members
overinvest in the activity that causes the negative externality.
Enter the founder, intent on designing a regime that would lead a large number of members to
reduce their excessive investment level. We use the social optimum to operationalize the founders
optimum, yet it is important to note that the model and results can accommodate any notion of
optimum as long as it is socially more desirable than the competitive equilibrium. The founder
achieves this result by oering an incentive to each member. For the sake of simplicity, we assume
the incentive to be a positive transfera payment t (s
i
) . We posit the following functional form
for transfers: t (s
i
) = ts
i
, with ts
i
the transfer given to member i and t a positive variable standing
for the subsidy rate.
6
As shown in Figure 5, the transfer can be given in several ways: through a multilateral instru-
ment in which members are treated uniformly (they are given the same transfer); through a series
of bilateral agreements by which the founder is able to customize transfers to each members need;
or through a combination of multilateral and bilateral instruments, where a subset of members is
treated identically and another is treated based on individual need. We model the distribution of
instruments according to the template of Figure 5. As above, we assume that any single instrument,
bilateral or multilateral, costs constant T to process.
10
On the founders side, we assume that the founder values at constant V any member i
0
s invest-
ment that conforms with the founders notion of what is optimal. For the sake of convenience, we
also assume that V is suciently large for the founder not to run into a budget constraint.
The founder moves rst, oering a contract to all members simultaneously. Then the members
simultaneously decide to reject or accept the oer. No subset of members has the capacity to
organize a coordinated response to the founders oer. If a member rejects, there is no contract with
that member. If a member accepts, the contract is executed as written; we are not giving the founder
the capacity to make the signing of a contract with one (or more) member(s) contingent on the
acceptance of all contracts by all other members.
7
There is no room for shirking once the member
has accepted the founders oersigning an international instrument makes the commitment credible
for both sides. Credibility is the result of a costly signaling game or a reputation game that is not
modelled here.
A strategy for the founder species the (t, x, y) regime she proposes, that is, choosing cutpoints
x and y Z
+
and subsidy rate t R
+
that maximize her aggregate utility while simultaneously
oering transfer levels suciently high to induce the y members to invest optimally. A strategy
for any member i is a mapping specifying for every combination of institution and subsidy rate an
investment level in the activity causing the externality that maximizes her individual utility. The
solution concept is the subgame perfect Nash equilibrium. Formally, it means for the founder and
the members to simultaneously solve the program
P =
1. max U
P
1xyN,t0
= xg (x) T +
P
y
z=x+1
(g (z) T) ,
2. with g (i) = V t
s
#
i
s
i
, and
= 1 if s
i
= s
i
= 0 if s
i
6= s
i
,
3. max u
i
s
i
0
= a
i
s
i
P
j6=i
s
j
cs
i
+ t
s
#
i
s
i
2
for all i [1, N] , and s
i
=
a
i
2(c+(N1))
2
,
subject to:
5. a
i
s
i
P
j6=i
s
j
cs
i
+ t
s
#
i
s
i
a
i
q
s
#
i
P
j6=i
s
j
cs
#
i
,
for all i, j [1, N] .
The rst clause formalizes the founders maximization problem, choosing subsidy rate t and
11
cutpoints x and y so as to oer a single multilateral treaty to members 1 to x, and bilateral
contracts to members x +1 to y.
8
Clause 2 species the founders utility function, earning positive
constant V for every member who cuts activity down to the level required to implement the social
optimum, at the cost of transfer t
s
#
i
s
i
s
#
i
s
i
, calculated to give her an incentive to reduce activity below the competitive equilibrium,
s
#
i
, whose value is reported in clause 4, along with that for the socially optimum value s
i
.
Clause 5 species the incentive constraint for each included member, insuring that none of them
has an interest in unilaterally deviating from the founder-induced optimum equilibrium.
The program is solved in the appendix. The equilibrium value of the subsidy rate, t
, is equal
to (N 1) , which can be interpreted as the externality rate, since each member causes N 1
externalities, each time with marginal impact .
Assuming, to arrive at an explicit solution, the following functional form for marginal gains,
a
i
= ia, with i [1, N] and a > 0, we are ready to state the solution.
Proposition 1 There exists a subgame perfect equilibrium in which the founder oers
(1) transfer t
x
to members i indexed 1 to x
i
;
(2) transfer t
i
to members i indexed x
+ 1 to y
i
;
(3) no transfer to members i indexed y
= (N 1) , s
i
=
a
i
2(c+(N1))
2
, s
#
i
=
a
i
2c
2
, x
[x, x], x = x + 1, y
[y, y],
y = y + 1, x =
1
4
(a
2
3
(N1)
3
+32Tc
2
(c+(N1))
2
+2ca
2
2
(N1)
2
)
a(N1)
2c+(N1)
1
4
, y = 2
c
a
V T
2c+(N1)
c+(N1)
N1
1.
Note the convexity of the founders cost curve of Figure 5, t (z
i
) = t
s
#
i
s
o
i
, conrming, to
the extent that the model builds on realistic assumptions, the choice of the model in Figure 4A as
more plausible than the one in Figure 4B.
12
4 Asymmetric Information
The model assumes complete information. What if, instead, the founder were ignorant of each
members compliance cost and dependent on their declarations? We argue that in such conditions
it is in the interest of each member to claim a cost greater than his actual one and thereby ex-
tract a higher payment. Asymmetric information operates like a transaction cost, disqualifying
customization in favor of uniformity (Harstad 2007).
To see this, imagine that the founder has a good sense of the overall distribution of marginal-
gain scalar a
i
. We further assume that the founder does not know where any given member i is
located on that distribution. Hence, even if the founder has a good sense of the nature and scope
of the instruments that she should use to build the regime, she does not know which instrument to
oer to which member. A rational strategy for any member under such circumstances is to claim
to be the high marginal gain type a
y
, with y being the presumed highest-marginal-gain member to
be included in the regime. As a result, all agents with an actual marginal gain located on and to
the left of cutpoint y would claim to be at that very cutpoint.
The founders best response to such misrepresentation is to give up on bilateralism and oer a
multilateral deal calculated to include the member with marginal gain a
y
. In response to such an
oer, all agents with marginal gains inferior or equal to a
y
accept the regime and all those with
marginal gains above that threshold stay out. The multilateral instrument is optimal here because
it functions like a partial information revelation mechanism. It leads each member to truthfully
sort themselves out into a camp of members and a camp of nonmembers. The revelation is partial,
however, because nothing is revealed on how agents are distributed within each camp. But note
that such information is unnecessary in the context of the multilateral instrument, which treats
everyone the same way and thus can be used successfully in the absence of any information on
members individual characteristics.
The situation is very dierent for bilateral instruments. Individualized information is needed
in order to tailor the bilateral incentive to each members type. In the case where all agents claim
to be the a
y
type, the bilateral strategy is suboptimal: it does not enable the founder to save
on the member surplus, since all agents are given the transfer that corresponds to the a
y
claim,
thus multiplying transaction costs for no osetting benets. Bilateralism fails in the presence of
13
asymmetric information. Our static model reproduces Harstads dynamic result, according to which
information asymmetry leads to posturing and delaying tactics during negotiations, thereby raising
transaction costs.
5 Comparative Statics and Predictions
The model produces three sets of comparative statics that are relevant to the mix of lateralisms
and exclusion:
(1) Transaction costs (information asymmetry and T): We have modelled two sources of trans-
action costs, information asymmetry and the negotiating and governance costs (T). We saw that
information asymmetry raises the cost of the bilateral approach, making a multilateral treaty a
more ecient method. Similarly, a rise in T causes both an increase in x
s
#
i
s
o
i
s
i
P
j6=i
s
j
cs
i
, a function that is twice-dierentiable and concave. Assuming
i
0 to be the Lagrangian
parameters, the optimal level of s
i
, s
#
i
, satises the necessary and sucient rst-order conditions
a
i
1
2
s
1
2
i
c +
i
= 0 and the Kuhn-Tucker conditions s
i
i
= 0 for any i [1, N], thus forming
a system of 2N equations and 2N variables (s
i
and
i
). There is no solution possible in which,
for any member i [1, N],
i
> 0, because it would imply s
#
i
= 0, making the corresponding
rst-order condition indeterminate. Therefore, the only possible determinate solution has
i
= 0
and s
#
i
=
a
i
2c
2
for all i [1, N] .
10 Social optimum
In any Pareto optimal allocation, the optimal level of s
i
, s
i
, must maximize the joint surplus of the
N members and so must solve max
s
i
0,i[1,N]
P
N
i=1
a
i
s
i
cs
i
P
N
i=1
P
j6=i
s
j
. This problem gives
the necessary and sucient rst-order conditions a
i
1
2
s
1
2
i
c (N 1) +
i
= 0, with
i
0 the
Lagrangian parameters, and the Kuhn-Tucker conditions s
i
i
= 0 for all i [1, N]. The problem
is solved like the precedent, yielding interior solution s
i
=
a
i
2(c+(N1))
2
for all i [1, N] .
11 Solving program P
11.1 The subsidy rate
We start by determining the optimal subsidy rate, t
(N 1) .
27
To meet the second condition, the incentive constraint in program P must be met for s
i
= s
i
.
This means that a
i
p
s
i
P
j6=i
s
j
cs
i
+ t
s
#
i
s
a
i
q
s
#
i
P
j6=i
s
j
cs
#
i
. Substituting
the values of s
#
i
and s
i
into the constraint yields t
c(N1)
2c+(N1)
. Since the right hand side term is
smaller than (N 1) , it follows that this second constraint is not binding, only the rst is, and
thus t
= (N 1) .
11.2 Convexity
To show that program P is convex with respect to x and thus has a xed-point solution, one needs
to show that the founders utility function, in which we have substituted the values for s
#
i
, s
i
,
and t
, is concave with respect to variables x and y. Concavity requires that for any pair of
distinct points (x
1
, y
1
) and (x
2
, y
2
) in the domain of U
P
, and for 0 < < 1, the following weak
inequality holds: U
P
(x
1
, y
1
) + (1 ) U
P
(x
2
, y
2
) U
P
( (x
1
, y
1
) + (1 ) (x
2
, y
2
)) . Developing
U
P
and rearranging yields U
P
= Ax
3
+Bx
2
+Cx+Dy
3
+Ey
2
+Fy +G with A =
1
6
R, B =
1
8
R,
C = T +
1
24
R, D =
1
12
R, E = B, F = V C, G = T, and R =
2
(N 1)
2
a
2
2c+(N1)
c
2
(c+(N1))
2
.
This and all subsequent calculations use the functional form for an members marginal gain
a
i
= ai.
Concavity thus requires
Ax
3
1
+ Bx
2
1
+ Cx
1
+ Dy
3
1
+ Ey
2
1
+Fy
1
+ G
+ (1 )
Ax
3
2
+ Bx
2
2
+ Cx
2
+ Dy
3
2
+ Ey
2
2
+ Fy
2
+ G
A(x
1
+ (1 ) x
2
)
3
+B (x
1
+ (1 ) x
2
)
2
+C (x
1
+ (1 ) x
2
)
+D(y
1
+ (1 ) y
2
)
3
+E (y
1
+ (1 ) y
2
)
2
+F (y
1
+ (1 ) y
2
)+G. Rearranging and simplify-
ing, one obtains (x
1
x
2
)
2
((x
1
(1 + ) +x
2
(2 )) A + B)+(y
1
y
2
)
2
((y
1
(1 + ) + y
2
(2 )) DB)
0, which is true since both components of the addition are negative. The rst term is negative be-
cause A+B < 0 and As coecient is greater than one, while the second term is negative because
D < 0, and both D
0
s coecient and B are positive. It follows that U
P
is concave with respect to
x and y and that there exists a unique internal maximum (x
, y
).
11.3 Lower and Upper Bounds of x
Since x
is the unique maximum over the relevant domain, it yields a utility to the founder that is
greater than the utility yielded either by x
1 or by x
(a
2
3
(N1)
3
+32Tc
2
(c+(N1))
2
+2ca
2
2
(N1)
2
)
a(N1)
2c+(N1)
28
1
4
, x =
1
4
(a
2
3
(N1)
3
+32Tc
2
(c+(N1))
2
+2ca
2
2
(N1)
2
)
a(N1)
2c+(N1)
+
3
4
. Given that x + 1 = x and that x
is an
integer, the value of x
The equilibrium value is what makes the founder indierent between extending the oer to y
th
member and earning V t
s
#
y
s
V T
2c+(N1)
c+(N1)
N1
, and thus the lower bound
value y = 2
c
a
V T
2c+(N1)
c+(N1)
N1
1. The value of y
y, y
.
11.5 Domain
Since x
must fall in interval [1, N] , we infer the domain of the function for which this result is
veried. x 1 yields condition T
3
4
a
2
2
(N 1)
2 2c+(N1)
c
2
(c+(N1))
2
T, while x N yields condition
T
1
4
a
2
2
(2N 1) (N 1)
3 2c+(N1)
c
2
(c+(N1))
2
T. y 1 yields T V
1
4
N
2
a
2
2
(N 1)
2 2c+(N1)
c
2
(c+(N1))
2
T, while y N yields T V a
2
2
(N 1)
2 2c+(N1)
c
2
(c+(N1))
2
T. Also, x
N if T > T
1 if T < T
while y
N if T < T
1 if T > T
. One last condition must be met: T = arg solve x y
b
T. Too
long to be reported here, this condition is available from the authors.
29
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