A model of a spatial economy with trading posts
Munetomo Ando

Daisuke Oyama

Graduate School of Economics, University of Tokyo

Graduate School of Economics, University of Tokyo

Abstract
This paper develops a framework in which exchange takes place at pairwise markets in amonetary economy with locations. In the economy there are several towns, each of which has
pairwise markets characterized by trading posts. At each trading post, taking prices as given,
agents submit commodities simultaneously with no resale, as opposed to an instantaneoustrading in the Walrasian economy. Commodities can be transported across towns with costs.
In this setting, we demonstrate that a trading center emerges in equilibrium if, and only if,
heterogeneity in transportation costs among towns is large enough.

We thank Hideo Konishi, Akihiko Matsui, an anonymous referee, and the associate editor, Thijs ten Raa, for their comments
and suggestions. We are also grateful to seminar/conference participants at Kyoto University, University of Tokyo, the 2001
Annual Meeting of the Japanese Economic Association in Hiroshima, and the 15th Annual Meeting of the Applied Regional
Science Conference in Sapporo for comments. At early stages of this work, we have benefited from discussions with Akira
Oseto. The first author acknowledges Grant−in−Aid for JSPS Fellows.
Citation: Ando, Munetomo and Daisuke Oyama, (2002) "A model of a spatial economy with trading posts." Economics
Bulletin, Vol. 18, No. 1 pp. 1−11
Submitted: August 28, 2002. Accepted: December 20, 2002.
URL: http://www.economicsbulletin.com/2002/volume18/EB−02R10002A.pdf

1.

Introduction

In the real world, we can observe that a large variety of commodities aretraded in a small number of locations. We call such a location a trading
center. The purpose of this paper is to explain where and why a trading
center emerges. To this end, we develop a simple model of a spatial economy
where exchange takes place at pairwise markets called trading posts. In themodel, there are several locations, or towns, each of which has trading posts
where a single commodity and fiat money are exchanged. We formulate a
trading center as a town at which active trading posts concentrate.
This paper builds on two strands of literature: monetary economics
and urban economics. In the traditional Walrasian economy, there is an
economy-wide market, and agents are allowed to instantly resell their purchases. In such an economy, money plays no essential role, so that in orderto provide a theoretical foundation for monetary economics, one has to depart from the Walrasian economy. The trading post approach, introduced
by Shapley and Shubik (1977) and applied to a situation with fiat money by
Hayashi and Matsui (1996), postulates that there are trading posts where
prespecified pairs of objects are traded and that trade takes place simultaneously so that agents cannot use revenues from trade in one trading post
for purchase in other posts.1 Monetary exchange patterns can be explicitly
described by such a model. The present paper is an attempt to introduce theconcept of location into a model of monetary exchange with trading posts.
Spatial agglomeration in economic activities has been one of the central
issues in the field of urban economics. It is often argued that the difference intransportation costs among locations plays an important role in generating
the asymmetry of economic activities. Recently, Konishi (2000) studies theformation of transportation hub in a general equilibrium model with threelocations. He shows that when transportation costs between locations areheterogeneous enough, a hub city emerges and population agglomeration occurs in that city.2 The present paper addresses agglomeration in transaction
activities in a model of decentralized trade with trading posts.
We pursue these two lines described above by analyzing a simple spatialmodel with trading posts. There are three towns and three types of perishable commodities in the economy. In each town, there are agents endowedwith a location-specific production technology. We assume the complete lackof double coincidence of wants: residents in town i desire the commodity
produce in town i + 1 (mod 3). Commodities can be transported acrosstowns with the so-called “iceberg” transportation costs. There are three
1

In Shapley and Shubik (1977), the price at a trading post is determined by the ratioof aggregate offers of each of the two objects, while in Hayashi and Matsui (1996), there is
an auctioneer in each trading post, and each agent behaves as a price taker. The present
paper follows the latter.
2
Berliant and Konishi (2000) study a model of market places with setup costs.

1

trading posts in each town where exchange between each commodity andfiat money takes place, so that any commodity is possibly traded in eachtown. All trades occur simultaneously with no resale, so that any trade involves the exchange of fiat money for commodities. Each agent can behave
as simultaneously a buyer and a seller in one period: as a buyer, he uses his
current money holdings to purchase the desired commodity; as a seller, heproduces his home commodity and then sells it to accumulate money thatis used in later periods. A trading center is defined to be a town where all
trading posts are active. Our goal is to identify the condition under which
concentration in transaction activities occurs. We show that in the presenceof positive transportation costs, a trading center can emerge in equilibrium
if, and only if, heterogeneity in transportation costs among towns is large
enough.
Economides and Siow (1988) also relate agglomeration in economic activities to the distribution of markets, incorporating trading frictions intoa spatial economy. In their model, agents faced with endowment shocks
are located along a linear city. Each agent must select a market to participate in before his endowment is realized, unlike in the standard Walrasian
economy where agents can participate in a complete market structure without first going to any particular market location. Due to liquidity considerations, agents prefer to trade in a single location, while transportation
costs have potential to spread trading activities over locations. Economides
and Siow (1988) illustrate this tradeoff between liquidity and transportation
costs and show that in a Nash equilibrium, several markets operate. The
present paper differs from their paper in that in our model, agglomerationis generated by heterogeneity in transportation costs, while in theirs, it iscaused by participation externality.
It is important to relate our framework to the spatial impossibility theorem due to Starrett (1978), which is one of the central results in urban
economics. The theorem says that if (i) markets are complete, (ii) markets
are perfectly competitive, (iii) there are no relocation costs, (iv) there are
no externalities, and (v) space is homogeneous, then there is no equilibriumwith a positive aggregate transportation cost in a closed spatial economy.
Thus, in order to demonstrate transaction agglomeration, we must violate
at least one of these conditions. The present paper drops (i) in that no resaleis assumed, as well as (iii) and (v). Examples which violate at least (i) and(v) include Wang (1990), Berliant and Wang (1993), and Konishi (2000).
This paper is organized as follows. Section 2 presents our model. Section
3 establishes our results. Section 4 concludes.

2

2.

Model

We consider an infinite horizon economy with locations inhabited by infinitely lived agents. Time is discrete, and periods are indexed by t (t =0, 1, . . . ). There are three towns {1, 2, 3} and three types of commodities{1, 2, 3} in the economy. In addition, there is fiat money, which is distributed to the agents at time 0 in a lump-sum fashion. All commodities
completely perish within a period, while fiat money is completely durable.
In each town, there is a unit mass of identical agents. A typical agent who
lives in town i is denoted by i. Agent i can produce commodity i, i.e., eachagent has access to a location-specific production technology. Agent i derives utility only from commodity i + 1 (mod 3), so that there is a complete
lack of double coincidence of wants.3
In each town, there are three trading posts. At each trading post {0, j},
j = 1, 2, 3, fiat money and commodity j are exchanged. We assume that ineach period, trading takes place simultaneously with no resale of commodities acquired in other trading posts. Thus, since commodities are completely
perishable and there is a lack of double coincidence of wants, agents otherthan agents j − 1 and j do not participate in the transaction at trading post
{0, j}.4
Commodities can be transported across towns with costs, while fiatmoney can be transported with no cost. We assume that transportation
costs are paid by the transported commodity; in other words, the commodity melts away during the transportation. Transportation of one unit ofcommodity from town j to town k (j 6= k) requires (tjk − 1)/tjk units of thecommodity, where tjk > 1. Here we assume that tjk = tkj for all distinct
j and k. Therefore, when one unit of commodity is transported between jand k, only 1/tjk units arrive at the destination. We set tjj = 1.
Indirect transportation is allowed. If, for example, t12 × t13 < t23 holds,
commodities can be transported from town 2 to town 3 through town 1 withless cost. We define [tjk ] = min{tjk , tj` × t`k }, where ` 6= j, k.
We assume that each agent must consume commodities in his own town.
i , and z i
If agent i obtains xii+1 , yi+1
i+1 units of commodity i + 1 from trading
posts in towns 1, 2, and 3, respectively, then the amount of his consumptionof commodity i + 1 is
ci =

i
zi
xii+1 yi+1+
+ i+1 .
[ti1 ][ti2 ][ti3 ]

3This assumption, which is standard in the literature of search-theoretic monetary
economics (e.g., Kiyotaki and Wright (1989) and Green and Zhou (1998)), gives fiat money
an essential role as a medium of exchange.
4
For the same reason, no barter trading posts could be used.

3

Agent i’s utility is given by∞Xβ t log ci (t) − k i (t) ,t=0where ci (t) is the amount of i’s consumption of commodity i + 1 at time t,k i (t) is the amount of i’s production of commodity i at time t, and β ∈ (0, 1)
is the common discount factor.
The price of commodity j at trading post {0, j} in town 1 (town 2, town
3, respectively) at time t is denoted by pj (t) (qj (t), rj (t), respectively).5
Given a sequence of price profiles, agent i is faced with the following decision
problem:
!
!
∞
i (t)
i (t)
X
xii+1 (t) yi+1
zi+1
t
i
max
β log
+
+
− k (t) ,[ti1 ][ti2 ][ti3 ]t=0

y i (t)z i (t)xi (t)+ [ti2 ] m+ [ti3 ] m= k i (t),s.t. [ti1 ] mpi (t)qi (t)ri (t)ii
pi+1 (t)xii+1 (t) + qi+1 (t)yi+1(t) + ri+1 (t)zi+1
(t)

(1)

+ mi (t) = M i (t),ii
M i (t + 1) = xim (t) + ym(t) + zm(t) + mi (t),M i (0): given,
i (t), z i (t), respectively) is the money demand submitted towhere xim (t) (ymmpost {0, i} in town 1 (town 2, town 3, respectively), mi (t) is the nominal
money balances not used for monetary exchange, and M i (t) is the nominal
money balances at the beginning of time t. The Kuhn-Tucker conditions forthis problem are: for all i = 1, 2, 3, and all t = 0, 1, 2, . . . ,βt− µi (t)pi+1 (t) ≤ 0,
[ti1 ]ci (t)βt− µi (t)qi+1 (t) ≤ 0,
[ti2 ]ci (t)βt− µi (t)ri+1 (t) ≤ 0,
[ti3 ]ci (t)λi (t)−+ µi (t + 1) ≤ 0,pi (t)/[ti1 ]λi (t)−+ µi (t + 1) ≤ 0,
qi (t)/[ti2 ]5“=” if xii+1 (t) > 0,

(KT 1)

i (t) > 0,“=” if yi+1

(KT 2)

i (t) > 0,“=” if zi+1

(KT 3)“=” if xim (t) > 0,

(KT 4)

i (t) > 0,“=” if ym(KT 5)

We assume that the price of a trading post is always called, say, by an auctioneer,even though no amount of commodity is submitted to the trading post. Otherwise, there
could be an equilibrium where all amount of commodity j is traded in a unique {0, j}
trading post only because no other {0, j} trading post has the price called.

4λi (t)+ µi (t + 1) ≤ 0,ri (t)/[ti3 ]i (t) > 0,“=” if zm(KT 6)µi (t + 1) − µi (t) ≤ 0,“=” if mi (t) > 0,

(KT 7)− β t + λi (t) ≤ 0,“=” if k i (t) > 0.

(KT 8)−At trading post {0, i} in town 1, for example, if xii+1 units of commodity
i + 1 are to be obtained by agent i, then the amount of money that i needs
to pay is pi+1 xii+1 , which in turn is equal to agent i + 1’s money demandxi+1
m when the market clears. Equilibrium is thus defined as follows.
Definition 1. For given M (0), a monetary equilibrium is defined to be a sei (t), y i (t), z i (t), z i (t), mi (t), M i (t), k i (t); p (t),quence {xii+1 (t), xim (t), yi+1
i
m
m
i+1
∞
qi (t), ri (t), i = 1, 2, 3}t=0 such that:i (t), y i (t), z i (t), z i (t),1. for each agent i = 1, 2, 3, {xii+1 (t), xim (t), yi+1mm
i+1
mi (t), M i (t), k i (t)} satisfies the Kuhn-Tucker conditions (KT 1)–(KT 8)
with the constraints in the problem (1).

2. markets clear: for all i = 1, 2, 3,
pi+1 (t)xii+1 (t) = xi+1
m (t),ii+1
qi+1 (t)yi+1(t) = ym(t),ii+1
ri+1 (t)zi+1(t) = zm(t).

Trading post {0, j} is said to be active if a positive amount of fiat money
and commodity j are submitted to the post. We define a trading center interms of active trading posts.
Definition 2. A town is said to be a trading center if all trading posts inthis town are active.
Throughout the rest of the paper, we focus only on stationary equilibria,
in which every variable is constant over time, while our model can generaterich patterns of trading.

3.

Results

In this section, we identify the condition under which concentration in transaction activities occurs. We show that in our environment with positivetransportation costs (i.e., tjk > 1 for all distinct j and k), a trading centercan emerge in equilibrium if, and only if, heterogeneity in transportation
costs among towns is large enough.
The first proposition establishes the “only if” part: transaction concentration does not occur unless transportation costs are sufficiently heterogeneous.
5

Proposition 1. If town i is a trading center in a monetary equilibrium,thentii+1 × tii+2 ≤ ti+1i+2 .

(2)

Proof. See Appendix.
For town i to be a trading center, the residents of town i + 1 and of town
i + 2 need to exchange in town i. Inequality (2) implies that the indirecttransportation through town i yields less cost for them than the direct transportation between towns i + 1 and i + 2. If tii+1 × tii+2 > ti+1i+2 holds for
all i, there exists no trading center: for each town i, trading post {0, i + 2}
does not activate.
Since (2) can not hold simultaneously for distinct towns, we obtain thefollowing.
Corollary 1. There is at most one trading center.
Next, the following claim clarifies where and why the trading center
emerges under large enough heterogeneity. An argument analogous to theproof of Proposition 1 can be applied, so that the claim is stated without aproof.
Claim 1. If tii+1 × tii+2 ≤ ti+1i+2 , then in a monetary equilibrium,(a) trading post {0, i} is active in either town i or town i + 2, or in bothtowns;(b) trading post {0, i + 1} is active in either town i or town i + 1, or inboth towns; and(c) trading post {0, i + 2} is active in town i, i + 1, or i + 2 or some (orall) of them.
The following proposition demonstrates that if inequality (2) holds, then
town i can become the trading center in equilibrium.
Proposition 2. If tii+1 ×tii+2 ≤ ti+1i+2 , then there exists a monetary equilibrium in which town i is a trading center.
Proof. See Appendix.
We construct in Appendix a stationary equilibrium where town 1 is the
trading center, which is depicted in Figure 1. In this equilibrium, all trading
posts in town 1 are active, while each of towns 2 and 3 has only one active
trading post: trading post {0, 2} in town 2, and {0, 1} in town 3. Fractionη (∈ (0, 1)) of agents 1 sell at {0, 1} in town 1 their production of commodity
1, which fraction η of agents 3 buy. The rest of agents 1 transport theirproduction to {0, 1} in town 3 to trade with fraction 1 − η of agents 3. By
6utilizing fiat money they obtained in the previous period, fraction ξ (∈ (0, 1))of agents 1 buy commodity 2 transported from town 2 at {0, 2} in town 1,
and the rest of agents 1 buy commodity 2 at {0, 2} in town 2. Since theindirect transportation through town 1 provides less cost to agents 2 and3, commodity 3 may be traded at {0, 3} in town 1. In this equilibrium,
the whole amount of commodity 3, which is produced in town 3 and thenconsumed in town 2, is traded in town 1.6

4.

Conclusion

We have presented a simple spatial model with trading posts to demonstrateconcentration in transaction activities. The model allows us to depict thetrading center as a location where all trading posts activate and the suburbs
as locations where some of trading posts are inactive. We have shown that in
the presence of positive transportation costs, a town can become the trading
center in equilibrium if and only if the indirect transportation though thistown requires less cost than the direct transportation between the suburbs.
The present framework which includes fiat money would also allow us todiscuss various topics on monetary economics. We could examine the effectof monetary policy on the distribution of transactions by introducing thegovernment, as in Alonso (2001) and Matsui (1998). We leave these issues
for future research.

6

We also have equilibria where ξ = 0 or 1, and η = 0 or 1. In equilibria with ξ = 0
and/or η = 0, even though large enough heterogeneity exists, there is no trading center.

7

Appendix
Proof of Proposition 1. For the expositional convenience, we set i = 1. Suppose that town 1 is a trading center.
Since town 1 is a trading center, trading post {0, 3} at town 1 is active.
For the buyers of commodity 3 (the residents of town 2) to join trading post
{0, 3} at town 1, the following conditions must hold:11≥ ,p3 [t12 ]
q3
1
1
≥
.
p3 [t12 ]r3 [t23 ](3)(4)

Similarly, for the sellers of commodity 3 (the residents of town 3),
q3
p3
≥
,
[t13 ][t23 ]
p3≥ r3[t13 ](5)(6)

must hold.
For p3 , q3 , and r3 satisfying (3)–(6) to exist, we must have [t23 ] ≥ [t12 ] ×[t13 ], and hence,t23 ≥ [t12 ] × [t13 ](7)(recall that [t23 ] = min{t23 , t12 × t13 }).
Since t12 , t13 > 1, we obtain [t12 ] = min{t12 , t13 × t23 } = t12 and [t13 ] =min{t13 , t12 × t23 } = t13 from (7), so that (7) implies t23 ≥ t12 × t13 .
Proof of Proposition 2. Setting M i (0) = M for all i, we construct a stationary monetary equilibrium in which town 1 is the trading center (i.e., alltrading posts in town 1 are active), and trading post {0, 2} in town 2 andtrading post {0, 1} in town 3 are active.
Since x12 , y21 > 0, (KT 1) and (KT 2) hold with equality for i = 1. Hence
we haveq2 (t) =p2 (t).
t12

Similarly, we haver1 (t) = t13 p1 (t).
That k i > 0 and xim > 0 implies thatλi (t) = β t

8andµi (t) =β t−1.
pi (t − 1)/ti1

Then, from (KT 1)–(KT 3), we havep1 (t − 1),p2 (t)
q2 (t − 1),c2 (t) = β
t12 p3 (t)p3 (t − 1)/t13
c3 (t) = β.
r1 (t)

c1 (t) = β

From the constraints in (1), we have1x1m (t) + zm(t) = β p1 (t),2x2m (t) + ym(t) = β q2 (t),x3m (t) = β p3 (t)/t13 .
The above conditions together with market clearing conditions imply thatfor any ξ, η ∈ (0, 1), a profile such that
M
t12 M
t13 M
, p2 =, p3 =
,
β
β
β


t13 M
M
M
M
q1 ∈
, t12
, q2 =, q3 = t12 t13 ,
t23 β
β
β
β


t13 M
t12 M
M
M
r1 =
, r2 ∈, t23, r3 =
,
β
t13 β
β
β
β
x12 = ξ , y21 = (1 − ξ)β, z21 = 0,
t12
1
1
1
xm = ηM, ym= 0, zm= (1 − η)M,β
x23 =, y 2 = 0, z32 = 0,
t13 3
2
2
x2m = ξM, ym= (1 − ξ)M, zm= 0,βx31 = ηβ, y13 = 0, z13 = (1 − η) ,
t13
3
3
3
xm = M, ym = 0, zm = 0,
p1 =k 1 = k 2 = k 3 = β,m1 = m2 = m3 = 0,
constitutes a stationary monetary equilibrium, in which all trading posts intown 1 are active, and trading post {0, 2} in town 2, and trading post {0, 1}
in town 3 are active.
9References
Alonso, I. (2001) “Patterns of Exchange, Fiat Money, and Coordination,”
Advances in Macroeconomics 1, Issue 2, Article 2.
Berliant, M., and H. Konishi (2000) “The Endogenous Formation of a City:
Population Agglomeration and Marketplaces in a Location-Specific Production Economy,” Regional Science and Urban Economics 30, 289-324.
Berliant, M., and P. Wang (1993) “Endogenous Formation of a City without
Agglomerative Externalities or Market Imperfections: Marketplaces in a
Regional Economy,” Regional Science and Urban Economics 23, 121-144.
Economides, N., and A. Siow (1988) “The Division of Markets Is Limitedby the Extent of Liquidity (Spatial Competition with Externalities),”American Economic Review 78, 108-121.
Green, E.J., and R. Zhou (1998) “A Rudimentary Random-Matching Modelwith Divisible Money and Prices,” Journal of Economic Theory 81, 252271.
Hayashi, F., and A. Matsui (1996) “A Model of Fiat Money and Barter,”
Journal of Economic Theory 68, 111-132.
Kiyotaki, N., and R. Wright (1989) “On Money as a Medium of Exchange,”
Journal of Political Economy 97, 927-954.
Konishi, H. (2000) “Formation of Hub Cities: Transportation Cost Advantage and Population Agglomeration,” Journal of Urban Economics 48,
1-28.
Matsui, A. (1998) “Strong Currency and Weak Currency,” Journal of theJapanese and International Economies 12, 305-333.
Shapley, L., and M. Shubik (1977) “Trade Using One Commodity as a Meansof Payment,” Journal of Political Economy 85, 937-968.
Starrett, D.A. (1978) “Market Allocation of Location Choice in a Modelwith Free Mobility,” Journal of Economic Theory 19, 21-37.
Wang, P. (1990) “Competitive Equilibrium Formation of Marketplaces with
Heterogeneous Consumers,” Regional Science and Urban Economics 20,
295-304.

10

Town 1

Agent 1ξβ /t12

ηM
{0,1}
{0,2}

(1-ξ) β(1-η) M
{0,3}

β/t13

ξM

ηβ

M

Agent 2Agent 3(1-ξ) M(1-η)β /t13

{0,1}

{0,1}

{0,2}

{0,2}

{0,3}

{0,3}

Town 2

Town 3

Selling
BuyingFigure 1.

11