Munich Personal RePEc Archive

Trade through Endogenous
Intermediaries
Kilenthong, Weerachart and Qin, Cheng-Zhong
University of the Thai Chamber of Commerce, University ofCalifornia Santa Barbara

13 April 2010

Online at https://mpra.ub.uni-muenchen.de/22046/
MPRA Paper No. 22046, posted 16 Apr 2010 14:33 UTC

Trade through endogenous intermediaries
Weerachart T. Kilenthong∗
University of the Thai Chamber of Commerce
andCheng-Zhong Qin†
University of California at Santa Barbara
April 13, 2010

Abstract
We apply an intermediation game of Townsend (1983) to analyze trade in an exchange economy through endogenous intermediaries. In this game, each trader has theopportunity to become an intermediary by offering to buy or sell unlimited quantities of
the commodities at a certain price vector and for a certain group of customers subject
to feasibility constraint. An intermediary will not be active unless some of its customers
subsequently choose to trade with it. We introduce an “intermediation core” and showthat the subgame-perfect equilibrium allocations of the intermediation game are contained in the intermediation core, similar to the inclusion of competitive equilibrium
allocations in the core usually studied. We also identify, in terms of the supportingintermediary structures, intermediation core allocations which are also subgame-perfectequilibrium allocations of the intermediation game. These results provide both a characterization and welfare properties of subgame-perfect equilibrium allocations of theintermediation game.
∗†

Email: weerachart kil@utcc.ac.th.
Email: qin@econ.ucsb.edu.

11Introduction

Fundamental to the Walrasian model of exchange is the requirement that trade be governedby a uniform price system. If commodities pass between two consumers in a certain ratio,
then they cannot pass between two other consumers in a different ratio. However, prices in
the Walrasian model are given ex machina and are not responsive to the consumers’ buying
and selling decisions. A theory is therefore needed to give an account of how prices areformed.
In the literature, several approaches to price formation in general equilibrium settings
have been proposed. In this paper, we follow the approach by Townsend (1983). Under thisapproach, consumers trade through endogenous intermediaries. The approach is describedby a two-stage intermediation game. One variant of this game works as follows. In the firststage, each trader individually and simultaneously offers to buy or sell unlimited quantitiesof the commodities at a certain price vector and for a certain group of other customers
subject to feasibility constraint. A trader may be chosen to be customers of more than
one intermediaries. However, each trader must subsequently choose to trade with at mostone intermediary in the second stage. Furthermore, a trader is obligated to act as anintermediary at the announced terms should some customers of his group choose to tradewith him; otherwise, he is free to act as a customer of an intermediary that includes him.1
The subgame-perfect equilibrium (SPE in short) is chosen as the solution concept for theintermediation game. Notice that a trader’s second stage choices may depend on what theother traders choose. For this reason, the social equilibrium in Debreu (1952) is applied tothe subgames in stage 2.
The subgame-perfect equilibrium of the intermediation game has the following properties. First, each non-intermediating trader maximizes utility by choosing an intermediarywith whom to trade and the amount to trade. Second, traders partition themselves into
disjoint trading cooperatives, such that there is an active intermediating trader within eachcooperative who specifies the terms of trade. Third, there is no incentive for entry of new
1

This is one of the several variants of the model in Townsend (1983). See Townsend (1978), Boyd andPrescott (1986) and Boyd et al. (1988) for applications of the intermediation games.

2intermediaries nor change of strategies by existing intermediaries. It follows that the equilibrium has both cooperative and non-cooperative aspects. However, unlike competitive
equilibrium, a SPE allocation is not necessarily contained in the core usually studied. Thisis largely due to the restricted feasibility of the allocations as imposed by intermediation.
The reader is referred to Townsend (1983) for detailed equilibrium analysis of the intermediation game.
The core of an economy is based on coalitional improvements that depend on what
each coalition can achieve by its own members. The core usually studied is based on theassumption that any reallocation of a coalition’s total endowments among its members
is feasible for the coalition. However, it is not clear under this formulation of feasible
coalitional allocations how players organize themselves into coalitions and how they carryout the trade.
The purpose of this paper is twofold. First, we apply the idea of intermediation tomodify the core of an economy by reformulating feasible allocations for coalitions and for the
economy. Specifically, for an allocation to be feasible for a coalition, it must be achievableby having one member behave as an intermediary while the other members act as pricetaking customers of the intermediary. Thus, at any feasible allocation of a coalition, allmembers but the intermediating trader maximize utility subject to budget constraint. For
an allocation to be feasible for the economy (i.e., for the grand coalition), however, weallow for the possibility that trade is carried out in multiple disjoint intermediaries (see thediscussion below Definition 3). Hence, we require that there exist a partition of all traders
into disjoint sub-coalitions, such that the restriction of the allocation to any sub-coalitionis achievable by having some trader in the sub-coalition act as an intermediating trader
for the rest of the sub-coalition. We call the resulting core the intermediation core. Theintermediation core is explicit about how traders organize themselves into coalitions, andhow trade gets carried out.2
2

It follows that feasible coalitional allocations in this paper are different from those in both Mas-Colell(1975) and Qin et al. (2006). In the former, feasible allocations of a coalition are required to be allocations
in competitive equilibrium of the sub-economy composed of members of the coalition, while in the latter, no
one is required to maximize utility subject to budget constraint.

3

Second, we compare the SPE allocations of the intermediation game with allocationsin the intermediation core. We show that SPE allocations of the intermediation game arecontained in the intermediation core under general conditions. Furthermore, we identify, interms of the supporting intermediary structures, intermediation core allocations which arealso SPE allocations of the intermediation game. Our result implies that an intermediationcore allocation can be supported as a SPE allocation of the intermediation game whenever
all intermediaries in the supporting intermediary structure have at least two customers.
This stability of the intermediation core allocations resembles the contestability notion inindustrial organization (Baumol et al., 1982). In particular, the two-customers requirement
ensures that, for any active intermediary, there is always a contestable intermediary who
stands ready to serve the other customers at the same terms. These results help to characterize equilibrium allocations of the intermediation game and to analyze their welfare
properties.
The rest of the paper is organized as follows. Section 2 introduces the intermediation
game, its subgame-perfect equilibrium, and the intermediation core. Section 3 establishes
the main results and Section 4 concludes. The appendix contains an example showing anunequal treatment of at an intermediated core allocation.

2

The Economy

We consider an exchange economy with finite ℓ physical commodities. Let N be the set oftraders. A trader i ∈ N has an initial endowment ω i ∈ Rℓ++ and consumption set X i ∈ Rℓ+ .
His preferences are represented by a strictly increasing utility function U i : X i → R. Theeconomy is described by the list E ≡ X i , U i , ω i i∈N .

2.1

A Intermediation Game

Following Townsend (1983), we consider a non-cooperative intermediation game with endogenous intermediaries. A trader can try to gain market power by offering to intermediate
for a certain group of customers. However, the degree of market power is weakened by4

competition between intermediaries. Specifically, the game has the following two stages
Stage-1
Each trader i announces a subset C i ⊆ N with i ∈ C i and a price vector pi ∈ Rℓ+ . The pairsi = (pi , C i ) represents trader i’s offer to buy or sell unlimited quantities at price vector pi
i = C \ {i}. We use si = ∅ to denote the announcement with C i = ∅,for customers in C−i−iin which case, trader i forgoes the opportunity to act as an intermediary.
Stage-2
Given Stage-1 announcements s = (s1 , · · · , sn ), trader i’s feasible choices are as follows.
(i) si = ∅In this case, trader i can either choose to trade with an intermediary offered by a traderin j ∈ N : i ∈ C j or stay autarkic. Denote this choice by di (s). Here, di (s) = j
means that i chooses to trade with j while di (s) = 0 means that he chooses to stayautarkic. When di (s) = j, trader i also chooses a net-trade vector z i (s) such thatpj · z i (s) = 0 and z i (s) + ω i ∈ X i ,where j = di (s). The resulting bundle for trader i is
 z i (s) + ω i if di (s) 6= 0,
xi (s) = ωi
if di (s) = 0.

(1)

(2)

(ii) si 6= ∅i , trader i can act as if si = ∅. If dk (s) = i
In this case, if dk (s) 6= i for all k ∈ C−i
i , however, trader i must act as an intermediary characterized by his
for some k ∈ C−i

Stage-1 announcement. The resulting bundle for i is
xi (s) = ω i −X{k:dk (s)=i}
5zik (s).

(3)

If the bundle in (3) does not belong to X i , then it is not feasible for i to act as an intermediary for the group k : dk (s) = i . In that case, trader i will act as if si = ∅.
Observe that traders’ feasible choices in the second stage depend on each others’ choices

within this stage. Because of this, we apply the social equilibrium in Debreu (1952) to thisstage in our determination of subgame-perfect equilibrium of the two-stage game.
Definition 1. A subgame-perfect equilibrium is a strategy profile (s∗i , d∗i , z ∗i )i∈N such thatfor each trader i(i) (d∗i (s), z ∗i (s))i∈N is maximal for all Stage-1 choices s, given (d∗j , z ∗j )j6=i ; and(ii) s∗i is maximal, given (s∗j )j6=i and given (d∗j , z ∗j )j∈N .

2.2

Intermediation CoreWe modify the feasibility of an allocation for a coalition by requiring that the allocation
be achieved with one member behaving as an intermediary for the rest of the coalition’s
members. The intermediating member is the organizer of both the coalition and the trade
behind the allocation. Formally,Definition 2. Given C ⊆ N , a C-allocation xi i∈C is feasible for coalition C if
X

X

ωi

(4)

subjectto p · xi ≤ p · ω i

(5)

i∈C

xi =

i∈C

and there a price schedule p such that, xi solves
max U i xi
xi

for all i ∈ C except for at most one member in C.
The set of all C-feasible allocations is denoted by F (C). The trader whose bundle
does not maximize utility subject to budget constraint is the intermediating member. The
rest of the members are the customers of the intermediary. In case each member’s bundle
maximizes utility subject to budget constraint, any one of them can be considered as theintermediating trader.
6

Definition 3. An allocation x̄i i∈N is in the intermediation core if there exists a partitionki{C̄ k }mk=1 of N such that (x̄ )i∈C̄ k ∈ F (C̄ ), for k = 1, 2, · · · , m, and there does not exist any

coalition C ⊆ N such that U i (xi ) > U i (x̄i ), ∀i ∈ C, for some C-allocation (xi )i∈C ∈ F (C).
We call the collection {(p̄1 , C̄ 1 ), · · · , (p̄m , C̄ m )}, with p̄k a supporting price vector forC̄ k for all k, a supporting intermediary structure for the allocation x̄ = (x̄i )ni=1 . To be in
the intermediation core, we allow an allocation to be achievable through multiple disjoint
intermediaries in stead of just one grand intermediary. Notice also that the intermediationcore remains the same if for any coalition C, we modify F (C) by allowing trade between
members in coalition C be achievable though multiple disjoint intermediaries.
Since coalitional improvements through intermediation are more restrictive and since
competitive equilibrium allocations are in the core usually studied, it follows that competitive equilibrium allocations are in the intermediation core. We summarize this resultin the following lemma whose proof is omitted. An implication of this lemma is that theintermediation core of an economy is non-empty under general conditions.
Lemma 1. Competitive equilibrium allocations are intermediation core allocations.

3

Main Results

We begin with a result that shows that competition between endogenous intermediaries isstrong enough to make SPE allocations of the intermediation game intermediation core allocations. We then establish a partial converse of this result; namely, an intermediation core
allocation is also a SPE allocation of the intermediation game whenever all intermediaries
in the supporting intermediary structure have at least two customers.
Theorem 1. The SPE allocations of the intermediation game are intermediation core allocations.
Proof. Let (x̄i )i∈N be an allocation resulting from a SPE, (s∗i , d∗i , z ∗i )i∈N , of the intermediation game. Suppose it is not in the intermediation core. Then, there exist a coalition C

7and (xi )i∈C ∈ F (C) such that

U i xi > U i x∗i , ∀i ∈ C.

(6)

Let p be the price schedule that supports xi i∈C . Choose j ∈ C whose bundle xj does not

maximize U j subject to budget constraint.3 Now, consider sj = (p, C). By (6), each traderi ∈ C with i 6= j chooses to trade with j. That is, each i ∈ C−j choosesd∗i sj , s∗−j = j, ∀i ∈ C \ {j} .

(7)

Since (x)i∈C ∈ F (C) and since p is the supporting price vector, (7) implies that xi (sj , s∗−j ) =xi for all i ∈ C−j ; hence, xj (sj , s∗−j ) = xj . By (6), (x∗i )i∈N cannot be a SPE allocation of
the intermediation game, which is a contradiction.
We now establish a partial converse of Theorem 1.
Theorem 2. Let x∗ = (x∗i )i∈N be in the intermediation core with supporting intermediarystructure {(p∗1 , C ∗1 ), · · · , (p∗m , C ∗m )}. If |C ∗k | ≥ 3 for all k = 1, 2, · · · , m, then x∗ is a
SPE allocation.
Proof. For each k = 1, 2, · · · , m, let j1k ∈ C ∗k be the intermediating trader for the othermembers of C ∗k and j2k , j3k ∈ C ∗k be two other members. Now, consider players’ strategiesks∗j1 = p∗k , C ∗k , k = 1, 2, · · · , m,and
 p∗k , C ∗k \ {j k } , ∃k : i = j k or j k ;
1
2
3
∗i
s =

∅,
otherwise,d∗i (s∗ ) = ∅,∃k : i = j1k ; j k , ∃k : i ∈ C ∗k and i 6= j k .
1
1

(8)

(9)

(10)

Since (x∗i )i∈N is an intermediation core allocation, it follows form (8) and (9) that d∗ (s∗ )in (10) is maximal. In addition, any strategy profile that that has (s∗ , d∗ (s∗ )) as the pathof play can implement allocation x∗ . Thus, it suffices to show that (s∗ , d∗ (S ∗ )) is a SPE
path.
3

In case everyone’s bundle maximizes utility subject to budget constraint, choose j arbitrarily.

8

To this end, notice that considering (s∗ , d∗ (s∗ )) as candidate SPE path, we only need to
specify traders’ choices at the off-equilibrium path in the events, in which a single trader
deviates. For any trader j and for any sj = (p, C) 6= s∗j , define d∗i (sj , s∗−j ) for i 6= j byd∗i (sj , s∗−j ) =










0,

d∗k (sj , s∗−j ) = i,j,i ∈ C, U i (xi (p)) > U i (x∗i ), d∗k (sj , s∗−j ) 6= i, ∀k,∗k , i ∈ C ∩ C ∗k , U ( xi (p)) ≤ U i (x∗i ), d∗k′ (sj , s∗ ) 6= i, ∀k ′ ,
j k ∈ J−ij

−j



 j k ∈ J ∗k , i ∈ C ∗k \ C, d∗k′ (sj , s∗ ) 6= i, ∀k ′ ,
−ij
−j

(11)

∗k = J ∗k \ {i, j}, and define d∗j (sj , s∗ ) by
where J ∗k = {j1k , j2k , j3k }, J−ij
−j

d

∗j

(s , s∗−j ) =jif ∃k : d∗k (sj , s∗−j ) = j,0,

 d∗j (s∗ ), otherwise.

(12)

By (11) and (12), d∗i (sj , s∗−j ) is maximal for all i. Thus, it only remains to show that trader
j has no incentive to deviate from s∗ j = (p∗j , C ∗j ) to sj = (p, C).
Since x∗ is in the intermediation core, either j or some member i ∈ C−j such thatd∗i (sj , s∗−j ) = j must not be strictly better off. However, by (11), the members in C
who choose to transact with j must be strictly better off. It follows that j, the deviator,
cannot be strictly better off. This shows that trader j does not have has any incentive tounilaterally deviate in Stage 1 given the State-2 maximal choices in (11) and (12) and theothers’ Stage-1 choices in (8)-(10).
The three-members restriction ensures that, for any active intermediary, there is always
a contestable intermediary who proposes to serve all customers at the same price. If there
are at least two active intermediaries one of which has only one customer, the intermediating
trader of the one-customer intermediary could have beneficially proposed a different pricevector for perhaps a different set of customers. This is shown in Example 1 below, in which
an intermediation core allocation need not be a SPE allocation.
Example 1. Consider an exchange economy with two physical commodities and N ={1, 2, 3}. Traders’ preferences are representable by utility function U i (x1 , x2 ) = min{x1 , x2 }for i = 1, 2, 3. Traders’ endowments are given by ω 1 = (10, 0) , ω 2 = (0, 9) , ω 3 = (1, 0). We
9show that the allocation x̄ = (x̄1 , x̄2 , x̄3 ) with x̄1 = (5.5, 4.5), x̄2 = (4.5, 4.5), and x̄3 = (1, 0)
is in the intermediation core. Notice that (x̄1 , x̄2 ) ∈ F ({1, 2}) and x̄3 ∈ F ({3}). Noticealso that no single trader alone can improve upon the allocation. Since neither trader 1 nor
trader 3 is endowed with good 2, they together cannot improve upon the allocation. Trader
2 and trader 3 cannot join together and improve upon the allocation since the maximum
amount of good 1 that trader 2 can have is 1 unit. Trader 1 and trader 2 cannot together
improve upon the allocation because their bundles consist of a Pareto optimal allocation
for them.
Let (x1 , x2 , x3 ) ∈ F (N ). Observe that it is infeasible trader 3 to intermediate for theother two because of his endowment. Suppose first that trader 1 intermediates with relative
price ρ = p2 /p1 . Then,
9ρ
9ρ
1
1
x =(
,
) and x3 = (,) ⇒ x1 =1+ρ 1+ρ
1+ρ 1+ρ
2

8
10 + 2ρ
,1+ρ 1+ρ.

Thus, for U 2 (x2 ) > U 2 (x̄2 ), it must be ρ > 1. However, with ρ > 1,
U 1 (x1 ) =8< 4 < U 1 (x̄1 ).
1+ρ

Suppose now that trader 2 intermediates. Then,
10
10
1
1
x =(
,
) and x3 = (,) ⇒ x2 =1+ρ 1+ρ
1+ρ 1+ρ1

11ρ 9ρ − 2,
1+ρ 1+ρ.

Thus, for U 1 (x1 ) > U 1 (x̄1 ), it must be ρ < 11/9. However, with ρ < 11/9,
U 2 (x2 ) =9ρ − 2< 4.05 < U 2 (x̄2 ).
1+ρ

In summary, the grand coalition cannot improve upon allocation x̄.
We now show that x̄ cannot be a SPE allocation. To this end, consider any strategy
profile (si , di , z i )i∈N that results in allocation x̄. Note that the prices of the intermediating
trader must be strictly positive.4 Because of this, trader 3 cannot intermediate for both4

Suppose first that trader 1 is a customer. Then, the price of good 1 must be strictly positive for him to

be able to afford (5.5, 4.5). If the price of good 2 is zero, then trader 1 would demand for bundle (10, 10)
which is not feasible. If trader 2 is the customer, however, then the price of good 2 must be positive for him
to be able to afford bundle (4.5, 4.5). Hence, if the price of good 1 is zero, then trader 2 would demand forbundle (9, 9), which is not compatible with trader 1 receiving (5.5, 4.5).

10

trader 1 and trader 2. The reason is as follows. Trader 3 is endowed with 1 unit of good 1
only. Thus, it is not feasible for him to act as an intermediary for trader 1 or trader 2 alone.
On the other hand, if both traders 1 and 2 are customers of trader 3, then they will demandat price vector p for a total quantity of (10p1 + 9p2 )/(p1 + p2 ) > 9 units of each good. Since
total endowment of good 2 is 9 units, it is not feasible for trader 3 to intermediate for bothtraders 1 and 2. Notice also that trader 3 cannot be a customer of either trader 1 or trader2, because otherwise he would have demanded for a bundle different from x̄3 .
Consider a deviating strategy s̃1 = (p̃, C̃) with C̃ = {1, 2, 3} and ρ̃ = p̃2 /p̃1 < 1. Since
trader 2 cannot be a customer of trader 3, d2 (s̃1 , s−1 ) = 1. In that case, trader 1 is boundto intermediate. Thus, d3 (s̃1 , s−1 ) = 1 and

 
 

9ρ̃
9ρ̃
1
9
1
10 + 2ρ̃1
x = (11, 9) −
,
,
,
−
=
.
1 + ρ̃ 1 + ρ̃
1 + ρ̃ 1 + ρ̃
1 + ρ̃ 1 + ρ̃As a result, trader 1’s utility level is




9
9
10 + 2ρ̃,=> 4.5 = U 1 x̄1 .
U 1 x1 = min
1 + ρ̃ 1 + ρ̃
1 + ρ̃

4

Conclusion

In this paper we followed the approach by Townsend (1983) to consider trading in an exchange economy through endogenous intermediaries. Under this approach, each consumer
can form an intermediary by offering to buy and sell unlimited quantities of the commodities
at a certain price vector for a group of other consumers. We introduced an intermediationcore by reformulating coalitional feasible allocations. Like the inclusion of the competitive equilibrium allocations in the core, we showed that the subgame-perfect equilibrium
allocations are contained in the intermediation core. Furthermore, we showed that conditions exist with which the subgame-perfect equilibrium allocations fill up the intermediation
core. This paper contributes to the literature on intermediation by providing tools for characterizing the subgame-perfect equilibrium allocation of the intermediation game and foranalyzing their welfare properties.
11

A

Unequal Treatment Property of Intermediation Core

Consider an economy in which there are three types of traders, a, b and c, each of which consists of two identical traders. We name them as a1 , a2 , b1 , b2 , c1 , c2 . Preferences and endow
ments of type-b and type-c traders are given by U b (x1 , x2 ) = min 12 x1 , x2 , U c (x1 , x2 ) =min x1 , 12 x2 , ω b = (1, 0), ω c = (0, 1). The type a’s endowment is ω a = (10, 10) and its
preferences will be precisely described later.

We consider an allocation achievable by having the two type a’s traders as the intermediating traders. In particular, we let 21 ≤ p ≤ 32 be the price ratio and {bi , ci } be the set ofcustomers for the intermediary trader ai is willing to offer. i = 1, 2. See Figure 1(a). The
dash-curve is the utility frontier of the two customers in each intermediary, which is obtained as the price ratio of the intermediary changes within the range of 12 ≤ p ≤ 23 . when
they are customers in the same intermediary, while the bold-curve is the utility frontierobtainable by themselves.5
Notice that each type a trader receives the following bundle from intermediating withprice ratio 12 ≤ p ≤ 23 and customer set {b, c}:p(p − 1)1−p
10 +, 10 +(1 + 2p)(2 + p)
(1 + 2p)(2 + p)

(13)

The locus of these bundles is presented by the G − H curve in Figure 1(b).
We now demonstrate that the allocations achieved with the preceding range of priceratios and intermediary structure are intermediation core allocations. Let C ⊆ N be any
coalition.
(i) C = {bi , cj }: The utility level of b in this coalition is at most equal to 52 , which islower than 37 , the utility level of b at the candidate allocation.
(ii) C = {ai , b1 , b2 , cj }: In order to make both traders of type b strictly better off theprice ratio must be smaller than p. This will certainly make an trader type-c worse
5

c

1−2V
This utility frontier when both of them are customers is represented by V b (p) = 2−3Vc , while the frontiercwhen both of them form an intermediary is represented by V b = 21 − V2 when 0 ≤ p ≤ 1, and V b = 1 − 2V cotherwise.

12

Good-2

VB
p=0
1/2

E
3/7
2/5

F
10.100

1/3

GH

10.054

p=
1/3

1/2

∞

10
9.950

VC

(a)

9.964

Good-1(b)

Figure 1: (a) A candidate allocation is on E − F arc, with 12 ≤ p ≤ 32 . (b) An indifference
curve of a trader type-a. The portion G − H corresponds to the consumption function of
an intermediary type A in (13) given that 12 ≤ p ≤ 23 .

off. Hence, C cannot improve upon the candidate allocation. The similar argument
applies to coalitions: {a, b, c1 , c2 }, {a, b1 , b2 , c1 , c2 }, {a1 , a2 , b1 , b2 , c1 , c2 }, {b1 , b2 , c} or{b, c1 , c2 }.
(iii) C = {ai , bj }: As Figure 1(b) shows, trader ai must receive at least 10.054 units of
good-2 (see the indifference curve of an ai in Figure 1(b)). This will leave bj with at
most 10 − 10.054 = −0.054, which is not feasible.
(iv) C = {a1 , a2 , bi }: To make bi better off, the net endowments left after satisfying bi isp1
at most 20 + 2+p, 20 − 2+p. Hence, the maximum consumption level of good-2 fora type a trader is strictly less than 10.054. This implies that none of the type a canbe better off.
(v) C = {a1 , a2 , bi , cj }: Suppose a1 is the intermediating trader. To induce one trader oftype b and one trader of type c, the proposed price must be 12 ≤ p ≤ 32 . See Figure2(a)-2b. Let point K in Figure 2(a) denote the net endowment of type a traders afterbi and cj have completed their trade, and line K-O represents the budget line for a.
13

The customer type a will choose allocation M. To achieve such allocation, her nettrade is given by segment O-M. Consequently, the opposite trade position relative toan allocation K will be the net trade of the intermediating trader (see Figure 2(b)).
Using a simple geometric argument, the segment O-M is always longer than segment
K-M. As a result, the consumption bundle of the intermediating trader (allocationN) is always below the original indifference curve of a (shown in Figure 2(b)). Thisimplies that the intermediating trader is worse off. The similar argument applies toGood-2Good-2cases where bi or cj is the intermediating trader.

10.1

10.1
K

K
M

10.050M
L

10.050

L

N
O

10

10

O

10

10

Good-1(a)Good-1(b)

Figure 2: (a) The budget line and optimal consumption allocation for a customer type a.
(b) The final allocation of the intermediary type a.

References
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Boyd, J. and E. Prescott (1986). Financial Intermediary-Coalitions. Journal of Economic
Theory 38, 211–232.

14

Boyd, J., E. Prescott, and B. Smith (1988). Organizations in economic analysis. Canadian
Journal of Economics, 477–491.
Debreu, G. (1952). A social equilibrium existence theorem. Proceedings of the National
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