Size of Middleman in a Barter Economy Preliminary Paper

Size of Middleman in a Barter Economy
Preliminary Paper
Andrei Shevchenko∗
September 7, 1999
Abstract
This paper develops a model of intermediation in search environ-
ment with complete information and explains the optimal choice of the
size of a middleman in an economy with an arbitrary finite number
of goods. The model with linear storage cost predicts the existence
of stationary equilibrium with the same number of shelves in every
store and uniform distribution of agents over states. The efficiency
consideration shows that the market might not create enough inter-
mediaries in the economy allowing for Pareto improving government
intervention. The paper elaborates very useful framework for further
extensions of the problem.
∗I am very grateful to Randall Wright, Ruilin Zhou, Andrew Postlewaite and Neil
Wallace for many valuable suggestions and comments. All remaining mistakes are mine.
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1
Introduction
In our century a vast majority of people could not imagine their life without
money: we are paid by cash or check on our jobs, we use paper money or
credit cards shopping for goods and services taking as given well developed
financial sector. But at the same time there are countries where due to the
transition from one organization of the economy (command) to another (mar-
ket) governments while trying to suppress inflation create a situation when
common for well elaborated market economies system of payments between
different agents does not work, and these agents are forced to use barter, i.e.
exchange goods for other goods. For instance, some economists in Russia
(A.Yavlinskii) claim that 80% of all transactions use barter exchange. That
is why the study of barter economies is not only of theoretical interest but
also helps to explain real facts that some modern countries are experiencing
in everyday life. One can observe that in these economies while money is
washing away from all the transactions it is accompanied by the significant
reduction in the variety of goods produced by individual plants making it
optimal in the new barter environment. To explain this optimal behavior we
are going to concentrate our attention on the trade-off that agents face in the
exchange process: maximization of the variety of goods that increases the
probability of trade versus minimization of production costs. In our model
we will translate this situation into the optimal decision of a middleman who
makes a choice as to the number of shelves in the store that she owns.
As it was pointed out by A.Rubinstein and Wolynsky (1987): ”despite
the important role played by intermediation in most markets, it is largely
ignored by the standard theoretical literature... because the study of inter-
mediation requires a basic model that describes explicitly the trade frictions
that give rise to the function of intermediation”. That is why we are going
to use search theoretical methods to emphasize the role of middlemen in the
exchange process. In search literature there are several different approaches
in which intermediaries emerge endogenously. Usually authors assume that
middlemen have some type of an advantage with respect to other agents in
the economy, for example, an advantage in terms of more efficient search pro-
cess (A.Rubinstein and Wolynsky, 1987) or an advantage in terms of private
information (Yiting Li, 1996). In our model we do not give any privileges of
described types to middlemen: all information is available to all participants
and search process is completely symmetric ex-anti. The main difference of
this model is the possibility for middlemen to store more than one good and
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this feature drives all the results of the exercise.
The paper proceeds as follows. After describing the model we will solve
it for the equilibrium in the simplest case of 2 goods in the economy. Having
in mind the logic for this simple case it will be easier to proceed with the
general case of K goods. After that we will analyze the welfare features of the
model. At the end we will discuss some possible extensions of this exercise
and summarize the results.
2
The Model
The model is similar to the economies considered in Kiyotaki and Wright
(1993) and Burdett, Coles, Kiyotaki and Wright (1995) except that money is
not introduced at this stage. There is a continuum of infinitely lived agents
with total population normalized to one who produce and consume goods at
discrete periods of time. There are K different consumption goods produced
in the economy. Production side is modeled exactly in the same way as in
Burdett, Coles, Kiyotaki and Wright (1995): there are K types of producers
in equal proportions where type i produces commodity i. Cost of produc-
tion is normalized to zero. Every period produces’s desire for consumption
good is a random draw from the uniform distribution. A producer can not
store more than one good and produces her specific commodity immediately
after consumption. If a random desire coincides with production specializa-
tion the producer consumes her own good, otherwise she enters a trading
sector. Trade is characterized by bilateral random matching process when a
producer can meet either a middleman or another producer. In the last case
if both parties desire each others’ goods one-for-one swap occurs followed
by immediate consumption and production. In the case of a meeting with a
middleman there is no problem of double coincidence of wants due to the spe-
cific assumption on preferences of middlemen: we assume that a middleman
satisfies her random desire (to keep balanced nutrition) automatically having
random flow of buyers into the store1. Every store has k shelves and it takes
exactly one shelve to store every good. Storage cost in terms of disutility is
an increasing function in the number of shelves: c(k). We assume that only
1We could have assumed the same preferences for a middleman as for a producer but it
would make our computations much more difficult without giving any additional insights
to the model. To justify these preferences we assume that a middleman has a technology
that allows her to transform any good into a desired consumption good.
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buyers move searching for their consumption goods while sellers stay2.
A buyer walks into a random store and if she finds the desired good
bargaining process over the price occurs where the final price depends on
the outside options for both parties (we assume complete information in the
sense that the buyer observes all inventories in the store). Obviously, trade
requires mutual consent. After the price is set both agents consume their
shares of the good and as a result the seller has a new configuration of goods
in the store (total utility derived from consuming one good will be normalized
to one). A seller chooses the optimal size of the store at the beginning of
time. After the choice has been made the seller is not allowed to change
the number of shelves: she is committed to the original choice. It might be
possible to show that ex-anti optimal decision is optimal choice ex-post, but
at this stage of the research it has not been done so we have to assume the
existence of the commitment technology. At least in the case of very patient
agents that choice is endogenous both ex-anti and ex-post.
Now, when we already know how the model works, it is easy to understand
the main trade-off for a middleman in the economy: the bigger the size of
the store the higher the probability that a random buyer will find her desired
good and consumption will take place. But this increase in the number of
shelves requires additional cost which can offset possible gains.
3
Equilibrium for K = 2 Case
At the beginning of this section we will study the optimization problem for
a middleman.
Let V (k1, k2) denotes the value function for middlemen where k1 is the
number of the first good in the store and k2 is the number of the second
good. If the number of shelves in the store is k then equality k1 + k2 = k
should hold3. The arrival rate for middlemen is proportional to the number
of buyers, i.e. it is equal to βnb, where nb is the fraction of buyers in the
economy. Respectively, let ns be the total fraction of middlemen, s.t. nb +
2The justification for this assumption could be found in Burdett, Coles, Kiyotaki and
Wright (1995), taking into account an additional assumption that moving cost for middle-
men is much higher than for buyers.
3Note, that the number of shelves may be bigger than the number of goods in the
economy: there is a positive probability that several buyers in a raw would ask for the
same commodity which may make it profitable to keep more than K number of shelves.
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ns = 1. These numbers will be determined endogenously in the equilibrium.
Let r denote the rate of time preference.
Then using standard Bellman’s equations from dynamic programming we
are ready to write down the expected value in terms of flow return for any
configuration of goods in a store:
1
rV0 = βnb max u1 + V1 − V0, 0 − c
2
0
1
1
rV1 = βnb
max u0 + V0 − V1, 0 +
max u2 + V2 − V1, 0
− c
2
1
2
1
· · ·
(1)
1
1
rVi = βnb
max ui−1 + Vi
max ui+1 + Vi+1 − Vi, 0
− c
2
i
−1 − Vi, 0 + 2
i
· · · 1
rVk = βnb max uk−1 + Vk
2
k
−1 − Vk, 0 − c
where Vi ≡ V (k − i, i), c ≡ c(2) and uj is the consumption share of a
i
middleman after the bargaining process when configuration of goods in the
store changes from i to j. The maximization operator reflects the fact that
the seller accepts the terms of trade only if her gains from exchange are
nonnegative4. We have got the system of k + 1 equations and k + 1 unknown
value functions Vi. Every equation shows that the flow return to a middleman
with given configuration of goods is proportional to the arrival rate βnb,
multiplied by the probability that a random buyer finds her favorite good ( 12
for (k, 0) configuration and 1 for (k1, k2), k1, k2 = 0) and multiplied by the
gains from trade. We have ruled out the possibility of meetings between the
middlemen which is justified by relatively high moving costs for sellers.
Let’s proceed by considering the price determination process. As it was
mentioned before we will assume that the prices are set in the generalized
Nash bargaining process where the current states are taken as the threat
points. It means that after mutual investigation of each others’ inventories
both agents can see how the configuration of goods in the store changes
from Vi to Vj. Therefore, the price is determined from the following simple
maximization problem:
uj
= arg max(u + V
i
j − Vi)θ(1 − u + Vb − Vb)1−θ
u
4We will assume that if a seller is indifferent between two options she would prefer to
trade.
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s.t. 0 ≤ uji ≤ 1
(2)
since the total utility derived from consumption of one good was normalized
to one. As a result we get:
uj
= θ
i
− (1 − θ)(Vj − Vi)
(3)
s.t. 0 ≤ uji ≤ 1
The logic of decision making process for an agent in the economy goes as
follows. A person evaluates the expected value functions of being a producer
or a middleman with k number of shelves. It would be easy to do so if she
could know the probability distribution over configurations of goods in the
stores and if she were able to compute all value functions V j. Moreover, at
i
this moment it is not obvious that there exists the only one optimal size of
a store. Let’s see if we can answer all these questions.
First and interesting result that we are going to show describes the be-
havior of a middleman in the trade process.
Proposition 1 A trade occurs whenever a random buyer finds her desired
good in the store5.
Proof.
This proposition claims that every time when a buyer wants to trade the
middleman would agree to trade which means that we can get rid of max
operators in the system (1):
max uj + V
i
j − Vi, 0
= max [θ(1 + Vj − Vi), 0] = θ(1 + Vj − Vi)
To prove the proposition we first assume that the statement is true and
then will show that θ(1 + Vj − Vi) > 0 or |Vj − Vi| < 1 meaning that our
original assumption was right. If the statement is true then system (1) would
look like:
1
c
V0 = βnbθ
[1 + V1 − V0] −
2r
r
1
c
V1 = βnbθ
{2 + V0 + V2 − 2V1} −
2r
r
5In other words, the distribution of middlemen over different configurations of goods is
ergodic.
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· · ·
(4)
1
c
Vi = βnbθ
{2 + Vi
2r
−1 + Vi+1 − 2Vi} − r
· · ·
1
c
Vk = βnbθ
[1 + Vk
2r
−1 − Vk] − r
Introducing notations β = βnbθ 1 , c = c , taking into account the sym-
2r
r
metry of the problem and assuming that k = 2N 6 the system (3) can be
rewritten:
V0 = β [1 + V1 − V0] − c
· · ·
(5)
Vi = β {2 + Vi−1 + Vi+1 − 2Vi} − c
· · ·
VN = 2β [1 + VN−1 − VN ] − c
or
 Vn+1 
 Vn 
V
V
 n
n−1
1
 = Υ 1

(6)
where transformation Υ is described by the following matrix:
 2 + 1 −1 c − 2
β
β

Υ =  1
0
0
0
0
1

(7)
Eigenvalues and eigenvectors corresponding to the transformation Υ are
calculated in a standard way:
1
λ1 = 1, λ2,3 =
1 + 2β ±
1 + 4β
2β
 2β − c 
 λ2 
 λ3 
e1 =
1
1
 2β − c
1
,e2 =  0 ,e3 =  0 
(8)
6using the same type of arguments it is easy to prove the statement for the case k =
2N + 1
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Note that 0 < λ2 < 1 and λ3 > 1. Then the general solution for (6) looks
like:
 Vn+1 
V
= γ e
λne
λne
 n
1 1 + γ2
2 2 + γ3
3 3
1

or
Vn = 2β − c + γ λn + γ λn
(9)
2
2
3
3
since γ
= 1. Using the formulas for V
1
0 and VN we can solve (9) for the
coefficients γ and γ . Finally we have got the analytical solution for our
1
2
original system with respect to Vi:
2β λN−n + λN−n
V
2
3
n = 2β − c −
(10)
1 + 4β + 1
λN
3 −
1 + 4β − 1 λN
2
With this close form solution it is just a matter of simple calculations to
show that Vn is increasing and concave function in n and that Vn − Vn−1 < 1.
It is interesting to note that in addition to the statement of the propo-
sition we have shown that the best configuration of the goods in the store
corresponds to the biggest variety of goods which is rather intuitive.
As usually, we are going to study only stationary equilibria, in which the
fractions of middlemen in each configuration do not change over time. To
find this stationary distribution we have to equate inflows and outflows for
every possible states taking into account that for the case K = 2 there are
at most two ways to go from any state:
ns0
ns1
· · ·
nsi
· · ·
nsk
(11)
k
nsi
=
ns
i=0
where nsi is a fraction of sellers in (k − i, i) state.
Proposition 2 For the case of 2 goods in the economy there always exists a
stationary equilibrium with uniform distribution of sellers over configurations
of goods in the stores; moreover, the equilibrium is stable.
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Proof.
For an equilibrium to be stationary nsi should satisfy the following system
of k + 1 equations with k + 1 unknowns:
1
1
βnb ns0 = βnb ns1
2
2
1
1
βnbns1 = βnb ns0 + βnb ns2
2
2
· · · 1
1
βnbnsi = βnb ns(i
ns(i+1)
(12)
2
−1) + βnb 2
· · ·
1
1
βnb nsk = βnb ns(k
2
2
−1)
where on the left hand side of every equation we have the fraction of sellers
leaving nsi state and on the right hand side there is the fraction of sellers
coming to the state.
The existence of an equilibrium with uniform distribution over states
follows immediately after observing that the above system (12) has a solution
ns0 = ns1 = · · · = nsi = · · · = nsk
We are left to show that this is the only stationary equilibrium which is
stable in the sense that the distribution over states converges to this uniform
distribution from any initial point in the space of states. To show this we
are going to reinterpret our problem in terms of Markov transition matrix.
There are k + 1 states and we know all the probabilities of moving from each
state to all others. Therefore, Markov matrix describing transformation of
the system over time looks as follows:
 1 − b
b
0
0
· · ·
0
2
2

b

1 − b
b
0
· · ·
0
2
2
0
b
1 − b
b
· · ·
0

B = 

2
2



(13)
 ···
· · ·
· · ·
· · ·
· · ·
· · · 


 0
· · ·
0
b
1 − b
b
2
2

0
· · ·
0
0
b
1 − b 
2
2
where b ≡ βnb and elements of matrix B are bij = the probability that a
k
k
system in state i at time t will be in state j at time t + 1,
bij =
bij = 17.
i=0
j=0
7Note, that matrix B looks almost like the transition matrix for a symmetric random
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Stationary equilibria of our model correspond to the solutions of the sys-
tem:
Bn = n
(14)
where n is a (k + 1) ∗ 1 vector of states. Thus if we show that matrix B has
only one eigenvalue λ = 1 while for all others |λi| < 1 then it proves that
there exists the only stationary equilibrium which is stable8.
To compute the eigenvalues of B matrix let us rewrite it (to simplify
algebra) in the following form:
 c + d d 0 0 ··· 0 
 d
c
d
0
· · ·
0
0
d
c
d
· · ·
0

D = 




 ··· ··· ··· ··· ··· ··· 


 0 ··· 0 d c
d

0
· · ·
0
0
d
c + d 
Then B = D if c = 1 − b, d = b .
2
Eigenvalues of D can be determined from the characteristic equation:
Xm ≡ det (D − λI) = 0
(15)
where I is the identity matrix: I = (δij), δij is the Kronecker delta symbol
and m is the dimension of matrix D. Expanding the determinant Xm in the
elements of the first and the last rows we obtain:
Xm = (c + d − λ)2 Ym−2 + 2d2 (c + d − λ) Ym−3 + d4Ym−4 = 0
(16)
where
 c − λ d
0
· · · · · ·
0

d
c − λ
d
0
· · ·
0
Y


m = det 
 0
d
c − λ
d
· · ·
0



 ···
· · ·
· · ·
· · · · · ·
· · · 
0
· · ·
· · ·
· · ·
d
c − λ 
walk with reflecting barriers except that diagonal elements of it bii = 0.
8Any path n(t) can be represented as a linear combination of eigenvectors generated
by matrix B and associated with eigenvalues λi. Over time all the terms with |λi| < 1
will converge to 0 leaving nonzero the only term with λ = 1.
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