Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

SPATIAL EQUILIBRIUM OF TAXI SPOT
MARKETS AND SOCIAL WELFARE
Kakuya Matsushima (Department of Urban Management, Graduate School ofEngineering, Kyoto University, JAPAN), kakuya@psa.mbox.media.kyoto-u.ac.jp
Kiyoshi Kobayashi (Graduate School of Management, Kyoto University, JAPAN)

ABSTRACTIn this paper, how local behavior of agents affects global spatial equilibrium pattern isanalyzed, in a matching model with taxies and passengers. There exist thick market
externalities in taxi spot markets: the more taxies gather at a market, the more customers willvisit the market, and vice versa. There always works the positive feedback mechanism insuch ways that as more taxies and customers visit the market to transact the service,transaction costs can be reduced and the market functions more efficiently. In this paper, anequilibrium model of multi spot markets is presented considering both agglomeration
mechanism caused by thick market externalities and dispersion mechanism caused bytransaction costs. We have shown that there is a possibility of multiple equilibria, andanalyzed the social efficiency of the respective equilibria.
Keywords: thick market externality, transaction cost, taxi market

1. INTRODUCTION
Transactions for taxi services occur in spot markets, such as at taxi bays settled in cities.
Deregulation has accelerated price competition in setting taxi fees, but has thus far produced
no notable differences between taxi companies, with taxies still offering customers rather
homogeneous service. In spot markets, transactions for services are realized by matching
the suppliers (taxis) with the consumers (passengers) of taxi services. The structure of taxi
spot markets has heretofore not been studied. While macroscopic analysis containing about
the deregulation of taxi markets, and theoretical and practical studies have appeared abouttaxi market equilibrium in a whole city, there have been few microscopic studies about spot
markets. This paper proposes a market equilibrium model for analyzing the structure of a taxi
spot market, in an attempt to remedy the dearth of studies explaining the self-organizing
mechanisms of spot taxi markets.
In taxi markets, both the suppliers and the consumers are required to visit the spot market
places to transact the services. Neither suppliers nor consumers are privy to full information
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1Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

about the state of supply and demand to be found at the spot, and both are required todecide whether they should visit the market place or not, based upon their imperfect guess
about the current volume of market transactions. When suppliers and consumers of services
are matched with each other at the market place, transactions for services are realized, while
if neither appears in the market, no transaction takes place. If the agent finds no match at thespot, he must wait for the arrival of a match or leave the spot.
Both consumers and suppliers must cover transaction costs, which include travel costs tovisit the market and the waiting costs necessary for finding a match at the spot market.
Because of `imperfect guesses' and `transaction costs,' pecuniary externalities function torealize market transactions. For instance, if more suppliers visit the market and wait for thearrival of consumers, consumers can more easily find matching suppliers, and vice versa.
More frequent visits to the spot market by both agents will give further payoffs to both. Thisphenomenon, called `thick-market externality,' means that the expectations of both suppliersand consumers that supply and demand will increase in fact bring about these increases.
Likewise, expectations of lower supply and demand are also self-fulfilling. In a matchingmarket with imperfect information and transaction costs, strategic complementarity caused
by market interactions brings thick-market (thin-market) externality and causes multipliereffects in the market. In such markets with positive (negative) feedback effects, there is apossibility that multiple equilibria exist in the market.
On the other hand, in order to transact taxi service in a spot market, both passengers andtaxies actually have to visit there. As travel cost to a market is key function for passengers todecide which market they use, the market with too much travel cost will not be chosen, even
if it is the market which can serve a transaction with fewer waiting time. Consequently, allcustomers and taxies do not concentrate on one specific market, but service may betransacted in several markets. The spatial structure of spot markets in a certain area will be
determined by correlation with the travel cost and the thick market externality. In a matchingmarket with the mechanism of the centralization thorough thick market externality anddecentralization with travel cost, there might exist multiple equilibria.
Both consumers and suppliers must cover transaction costs, which include travel costs tovisit the market and the waiting costs necessary for finding a match at the spot market.
Because of `imperfect guesses' and `transaction costs,' pecuniary externalities function torealize market transactions (Howitt and McAfee, 1987; Howitt, 1990; Kobayashi et al.), 1998).
For instance, if more suppliers visit the market and wait for the arrival of consumers,
consumers can more easily find matching suppliers, and vice versa. More frequent visits tothe spot market by both agents will give further payoffs to both. This phenomenon, called`thick-market externality,' means that the expectations of both suppliers and consumers that
supply and demand will increase in fact bring about these increases. Likewise, expectationsof lower supply and demand are also self-fulfilling (Farmer, 1993). In a matching market withimperfect information and transaction costs, strategic complementarity (Bulow et al., 1985;
Cooper and Andrew, 1988) caused by market interactions brings thick-market (thin-market)
externality and causes multiplier effects in the market. In such markets with positive(negative) feedback effects, there is a possibility that multiple equilibria exist in the market.
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Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

There are several papers related to taxi markets. Among of all, Arnott presented a paper
which showed the necessity of subsidy in urban taxi markets (Arnott 1996). Matsushima et al.
formulated the model which explains endogenous market formation of taxi spot market. But
these papers did not analyze the spatial equilibrium of taxi spot markets.

2. THE MODEL(1) Assumptions
Let us take up the case where there are 2 spots market in a certain area and model the
spatial equilibrium of spot markets from the behavior by customers and taxies. Assume alinear market with the length 2, which consider a business district along a street, as shown in
Figure-1. Homogeneous visitors are uniformly distributed over the linear market. Spotmarkets are politically set at two places -x and x, by the traffic administrator with 0  x  1 . The
only taxi spot market exists in the origin when x=0. Let us assume that x>0 is satisfied, andthat the market with –x is indicated as i=1 and that with x as i=2. All customers can consume
taxi services only at spot markets, and they do not have any other transportation service
beside taxies. Passengers who arrive at spot markets will not leave until service is consumed,
while there is maximum length for taxies queue with exogenous maximum length M*(M*=0,1,･･･,2). A taxi which arrive at the market when its queue length is M* may leave fromthere. The mechanism which defines the maximum queue length of taxies will be expressed
at 2. (3).

Uniformly distributed

-x

k

0

Market 1x
Market 2

Figure 1 – Linear market
Passengers choose a spot market which maximizes their utility. Assume that the boundary ofeach market’s territory exists at k ( 1  k  k ). All passengers may use market 2 when k=-1,
that shows market 1 disappeared, while only market 1 exists when k=1. Passengers arrive at  1  k 
  1 k both market subject to a Poisson process with arrival rate 1 and 2 ,
respectively, where shows exogenous average arrival rate of passengers per unit lengthand unit time at the linear market. On the other hand, average arrival rate of taxies will be ex i  1, 2 pressed as i . Though arrival rates of taxies are endogenously derived from market
equilibrium, it is temporarily considered as exogenous (will be explained at 3.(1).

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Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi(2) Queuing model
Queuing phenomena, which are most typically observed in spot markets for taxi services,can be expressed by a double-queuing model first set forth by Kendall (1951) and further
developed by Sasieni (1961). Though the double-queuing system proposed by Kendall ishighly limited due to a failure to account for congestion, this system is adequate forinvestigating how spot taxi markets are autonomously organized via thick (thin)-market
externalities.
Consider a consumer who arrives at a spot market. If a supplier has been waiting for acustomer there, the consumer can immediately purchase service from the supplier. But if nosupplier is there, he is forced to join the consumers' queue that has already been formed.
Assume that once the consumer arrives at the market, he dose not leave until he makes atransaction with his supplier. In other words, there is no limit to the length of the consumers'
queues. On the other hand, the length of the suppliers' queues is assumed to have a limit,
due to the fact that suppliers are assumed to leave the market immediately if they find thatthe length of the suppliers' queue has already reached the upper bound M. The arrivals ofboth agents to the spot market are assumed to be purely stochastic and subject to a Poissonprocess with average arrival rates of the consumers λ and of the suppliers μ. For amoment, both λ and μ are assumed to be exogenous. This assumption will later be relaxed.
It is also assumed that the coupled agents leave the market immediately after the service
transaction has been completed between them.
Let us first consider a situation where only consumers are forming queues. Let the probability
that the length of the consumers' queues is nn  1 be Pn  . The transition mechanism of
the system is described asPn t  t   1  t 1  t Pn t   1  t tPn1 t (1)

 t 1  t Pn1 t   ot !
ot ! is the higher order term and ot ! t  0 as t  0 . On the contrary, consider the
opposite situation where only suppliers are forming queues. Denote the probability that m
suppliers are waiting for the consumers' arrivals by Qm t  . The transition of the system isgiven byQM t  t   1  t QM t   1  t tPM 1 t   ot !
Qm t  t   1  t 1  t Qm t   1  t tQm1 t  t 1  t Qm1 t   ot !,

(2)
(3)

where ot ! t  0 as t  0 . Defining the probability with no queues by P0 t   Q0 t  ,the state equation is given byQ0 t  t   1  t 1  t Q0 t   t 1  t Q1 t (4)

 1  t tP1 t   ot !,
Dividing both sides of eqs. (1), (2), (3), and (4) by t and considering the t  0 limit, inthe long-run steady states, we see that    Pn  Pn1  Pn1  0

 QM  QM 1  0
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(5)
(6)

Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

    Qm  Qm1  Qm1  0
    Q0  Q1  P1  0

n  1,, ; m  1, M  .

(7)
(8)

For the stability of the steady states, it should hold that    . From eqs. (5)-(8), thestationary probabilities of Pn and Qm are respectively given byn  1,,  m  0,, M  ,Pn   M  n QM
Qm   M  m Q M

(9)
(10)

where     . From this definition, we see that


M

n 1

m 0

 Pn   Qm  1 .

(11)

Substituting eqs (9) and (10) to (11), we see thatQM 1   .
Accordingly, the stationary probabilities Pn and Qm are given byPn  1    M  nn  1
Qm  1    M m M  m  0  .

(12)
(13)
(14)

Let M M  1,2, be the maximum length of the suppliers' queues. Given the averagearrival rates of the customers and the suppliers  ,   , the average lengths of the suppliers'

queue and the consumers' queue are given by

E n :  ,  , M   M 11 E m :  ,  , M   M 

(15)



1    ,
1 
M

(16)

respectively. The average waiting time of the consumers and that of the suppliers, denotedby T  ,  , M  and S  ,  , M  respectively, with the arrival rates  ,   , are given byT  ,  , M   E n :  ,  , M  
S  ,  , M   E m :  ,  , M  

(17)

(18)
The probability  that a newly arrived supplier leaves the market without joining the
suppliers' queue is defined by  QM  1   .

(19)

If M  0 , the suppliers immediately leave the market if they find that no consumers are
waiting for the arrivals of the suppliers. Then, the average waiting times of the consumers
and of the suppliers are given byT  ,  ,0  1
 1   
S  ,  ,0  0 ,respectively.

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(20)
(21)

Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi(3) Maximum length of the queue
So far, the maximum length of the suppliers' queue is assumed to be exogenous. In what
follows, these values are supposed to be endogenously determined through interactions
between suppliers' and consumers' behavior in the market. Suppose that there is no physicallimit on the length of the suppliers' queues. Each supplier who arrives at the spot market
observes the current length of the suppliers' queue and decides to join the queue or to leave.
The average waiting time of the mth suppliers in the suppliers' queues, denoted by W m  , isgiven byW m 

m



.

(22)

Then, the expected profit of the mth supplier in the suppliers' queues, denoted by  m  , isdefined by m  q 

m



d ,

(23)

where q is the expected profit per unit service transaction (measured in terms of time value).
Though a supplier does not know precisely the actual revenue before he is matched with his
customer, he can estimate the expected revenue through his past experiences. Suppose thatthe suppliers must pay transaction cost c for visiting the spot market. In order for the
suppliers to have intentions to visit the market, the condition(24)
qdshould be satisfied. If it holds that q  c , no suppliers visit the market. In turn, the spot
market disappears. Suppliers will join the queue as long as they can expect positive average
profits from the spot market. The transaction cost for visiting the market has already been
sunk by the time the supplier arrives at the spot market. In a competitive market, themaximum length of the suppliers' queues is determined in such a way that the maximum
number of the suppliers waiting in the queues is a number that can guarantee nonnegative
expected profits. From the non-negativity condition of the profits,  m   0 , the maximum
length of the suppliers' queues, M   , is defined byM      q  d    ,(25)

where the notation [･] means the maximum natural number that does not exceed q , and 
is the average arrival rate of the consumers. If the capacity of the spot market is physicallylimited, the maximum length of the suppliers' queues is conditional upon physical capacity,
denoted by M * W ,   . Then, the maximum length is given byM * W ,    minW , M   ,(26)

where M   is the unconditional maximum length of the suppliers' queues (25) and W is thecapacity of the market.

3. SPATIAL MARKET EQUILIBRIUM
So far, the arrival rates of the consumers and the suppliers i , i are assumed to be
exogenous. In the long run, the arrival rates of both agents are endogenously determinedthrough interactions between the suppliers and the consumers at the spot market. The
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6Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

consumer will visit the market as long as the expected utility he can derive from a transaction
exceeds his reserved utility level. The consumer is assumed to remain at the market until histransaction is completed. On the other hand, the suppliers will remain at the market to wait
for the arrival of his customer only if the length of the suppliers' queue is below its upperbound. Suppliers are expected to cover the transaction costs of visiting the market, and they
will visit the market as long as they can anticipate gaining non-negative expected profit. As aresult of the free entry of agents into the market, the average arrival rates both of consumers
and suppliers are endogenously determined in the long run.

(1) Supplier (Taxis)
Let us temporarily assume that the consumers' arrival rate  is exogenously given. When the
suppliers arrive at the market with the arrival rate  , the probability that the consumers
make queues is given by  from eq. (19). Let us denote the physical market capacity in the
spot market by W W  0,1, . If it holds that W  M , we see that M * W ,    M    from eq.
(26). Given this fact, from now on let us focus exclusively upon the case where it holds that
M  W  M    . A supplier newly arrived at the market immediately leaves, if he finds thequeuing length has already reached its upper bound. From eq. (19), the probability that the
newly arrived supplier leaves the market is given by   1   . Suppliers who have left themarket can only gain a profit of -c. They can remain in the market with the probability
1     and get the expected profit of  q  S '   ,  ,W   c ,(27)

where S  ,  ,W   S  ,    is the conditional average waiting time of the suppliers when'

they can enter the market. Note that the profit is measured in terms of time value. From eq.
(18), we see1 1  W  .
W 
 
1 

If no queuing is allowed for the suppliers W  0 , it holds thatS '  ,  ,0   0 .



S '   ,  ,W  The suppliers visiting the market can gain an expected profit of E ,W  W  0,1,2, :E  ,W    q  S '   ,  , W   d   q  S   ,  ,W   d .

(28)

(29)
(30)

The suppliers will visit the market as long as they can anticipate a non-negative profit.
Provided the arrival rate of the consumers  is exogenously fixed, the long-term arrival rateof the suppliers, denoted by  * , is determined to levels that satisfyq  S  ,  * ,W  d  0 .
*





(31)

(2) Consumers (Passengers)
Let us indicate the utility of passengers who locate at y ( 1  y  1 ) acquire from consuming
taxi services with y, waiting time at market i with ti, and travel cost to the market i with ci(y),respectively. Assume a linear utility function:
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Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

Vi  y   v  ti  ci  y  .

(32)

The utility function is measured in terms of time value. The first term corresponds to theidiosyncratic utility gained by transaction of services, which includes disutility from travel time,
fare and travel cost. The utility is deterministic for each individual consumer, but a randomvariable for anybody else. ci(y) indicates the cost for passengers to move from y to market i,
and is expressed as follows.
c1  y     x  y(33)

c2  y    x  y(34)

  0 indicates a parameter. The expected utility of the representative consumer whopurchases services at the market is given byEVi  y   v  Ti  i , i , M i   ci  y  .

(35)Market i*(y) which a passenger located at point y chooses is expressed asi*  y   arg max EV1  y  , EV2 ( y ) .





(36)

Boundary of market share as shown in Figure-1 can be defined as k    x, x  which satisfies
EV1(k)=EV2(k). Passengers located at k can acquire same utility by using either market.
Passengers’ arrival rates i to each market are1  1  k  

 2  1  k   ,(37)respectively. Taxies’ arrival rate is i which satisfies equation (31). Let us assume thefollowing equations can be satisfied at the market boundary.

dEV1  y 
dy












y k*



dEV2  y 
dy



(38)
y k*

Then EV1 k *    EV2 k *   and EV1 k *    EV2 k *   are satisfied for small  .

Following equations are satisfied from the definition of ci(y) and market equilibrium withboundary k is stable.
(39)
EV1  y   EV2  y  y   1, k 

EV1  y   EV2  y 

y   k ,1

(40)

When equation (38) is not satisfied on the other hand, it is not guaranteed that equations (39)and (40) are satisfied. This equilibrium is not stable. Other market equilibrium makes only
one market appear. 1  2 ,  2  0 , and  2  0 when k=1, and 1 is calculated fromequation (31). Following equation is satisfied at k=1.

EV1 1  EV2 1  

(41)

As EV1  y   EV2  y    for any y   1,1 , this equilibrium is also stable. Anotherequilibrium where all passengers use market 2 is also stable. From the mentioned above,
equilibrium solutions can be expressed as follows.
EV 1  EV 2
(42)

 









EV i  EVi k * (i  1, 2) 1*  1  k *  *2  1  k * 

i q  Si *i , *i , M i  d  0 when k  1 and k  1

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Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

1*  0 1*  0 *2  2

EV 2  EV2  k 

2 q  S2 *2 , *2 , M 2  d  0 when k  1
*2  0 *2  0 1*  2





(44)

EV 1  EV1  k 

1q  S1 1* , 1* , M1  d  0 when k  1

(45)





Expected utility for a passenger located at k is expressed as W  y   max EV1  y  , EV2  y  .
Arrival rate of passengers may be the level with W(y)=0 in the long term equilibrium.

(3) Market externalities and travel cost
From eqs. (15) and (16), we see that the average length functions of the consumers' and the
suppliers queues, denoted by E  ni : i , i , M i  and E  mi : i , i , M i  , are homogeneouswith respect to i and i , respectively. For arbitrary i  i  0 and   0 , it holds thatE  ni : i , i , M i   E  ni : i , i , M i E  mi : i , i , M i   E  mi : i , i , M i  ,

(46)
(47)

which means that the average lengths of the consumers' and the suppliers' queues remainunchanged, even though the arrival rates of both agents simultaneously increase with thesame rate  . For arbitrary i  i  0 and   1 , we see, from eqs (17) and (18), thatTi  i , i , M i   Ti  i , i , M i 

Si  i , i , M i   Si  i , i , M i  .

(48)
(49)

As the average arrival rates of both agents increase, the average waiting time of both agentswill decrease. Thus, thick-market externality works in such a manner that as the waiting timeof both agents becomes smaller, the number of both agents who enter the market becomesmarket. On the other hand, if either of two parameters, i or i is changed with the rate  ,the other parameter being unchanged, we see thatTi  i , i , M i   Ti  i , i , M i 

Si  i , i , M i   Si  i , i , M i  .

(50)
(51)

which means that as the arrival rate of either of the two agents increases, the average
waiting time of the concerned agent increases. Congestion occurs if the arrival rate of eitherof the two agents increases, that of the other agent being unchanged.
From eqs. (48) and (49), it is apparent that the expected waiting time will decrease if more
suppliers and consumers enter the spot market. There exists an increasing-return-to-scale
externality in market transactions at the spot market. When both agents, the consumers andthe suppliers, expect more counterparts to visit the market, this reduces the thinness of the
field of matches and induces a further round of increased transactions. This will encouragehigh levels of transactions, which in equilibrium will result in high aggregate demand andsupply, thereby fulfilling the original expectations. By the same token, the expectation of lowtransactions can also be self-fulfilling, because the associated prospect of thin markets and12th WCTR, July 11-15, 2010 – Lisbon, Portugal

9Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

longer waiting costs will discourage the arrivals of both agents, there being strategic
complementarity arising from market interactions, specifically from thick-market externality.
AS total number of passengers is limited, this positive feedback mechanism works until one
market covers all passengers, as far as there is no power to control that. However, theexistence of travel cost works as a key factor to control feedback mechanism. As number ofmarkets decreases, number of passengers covered by one market increases to makepassengers located around market border travel longer. These passengers may use closer
market even if waiting time there is longer than that of a market far from her. The structure of
spot market is endogenously determined thorough interaction between thick market
externality and travel cost.

(4) Numerical examples
Let us analyze the impact of potential number of passengers on market equilibrium throughnumerical examples. Let us design the benchmark case (Case a) where the passengers’
appearance rate  is 1.5, passengers expected utility v=10, taxies’ utility profit q=10, taxies’

transaction cost d=5, and a parameter of travel cost   1 . Set two spot markets at the edgeof linear market (x=-1.0, x=1.0). The relation between the point of spot markets and social
welfare will be analyzed in section 5. In order to make the discussion clear, queuing length oftaxies are defined thorough market equilibrium, that is, we do not set any physical maximum
length for taxies’ queuing.
Let us assume that there exists boundary point y in the linear market, and that those who arelocated in the range z (z<y) visit market 1, while those who are located in the range z’ (z’ < y)visit market 2. Figure-2 shows the expected utility EVi(y) of passengers who visit either
market 1 and 2. Passengers located at y may choose either market which makes their
expected utility maximum. For those passengers who are located at the position whereexpected utility for both markets are the same (EV1(y*)=EV2(y*)), both markets are identical.
Such a point y* shows the market boundary. Figure-2 shows 5 market boundaries from A to
E. Each market boundary corresponds to one market equilibrium. Let us consider equilibriumwith market boundary B. Assume that the market boundary moves to a point slightly righthand side of B accidentally. As expected utilities of passengers located at y’ are EV1(y’) andEV2(y’) with EV1(y’) > EVU2(y’), they may use market 1. Point y’ cannot be a market boundary,while point C will be. On the other hand, if market boundary move to left-hand side of B,
EV1(y’’) < EV2(y’’) is satisfied and equilibrium A will be new market boundary, that is, allpassengers use market 2. Therefore, equilibrium B is unstable. Equilibrium D is also unstablein the same manner. There are 3 stable equilibrium, one with center market boundary, andother two where all passengers visit either market.

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10Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

E Vi
10
9

C

B

A

D

E

8
7
6
5

E V1

E V2

F

G

4
3
2
1
0
-1.0

y
-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Figure-2 Market boundary and expected utility (Case a)
Figure-3 shows the same relation for a case with   0.5 (Case b). There are 2 stableequilibria where all passengers visit both market and unstable equilibrium with central market
border in this case. There is no equilibrium where both markets are visited. This means thatthick market externality relatively conquer travel cost with low passengers density to makeonly one market survive.

E Vi
9
8
7
6
5

E V1

E V2

4
3
2
1
0
-1.0

y
-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Figure-3 Market boundary and expected utility (Case b)
Figure-4 shows the relation between a parameter of travel cost  and density of passengers
 . There are 2 areas in the figure: 1) 3 stable equilibria like case a, and 2) 2 stable equilibria
like case b. There are possibly 2 equilibria as passengers’ density  decreases, that is, all

12th WCTR, July 11-15, 2010 – Lisbon, Portugal

11Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

passengers visit either spot market. As travel cost  increases, equilibrium where bothmarkets are visited like case a is available. Because the difference of travel cost increases
as  increases, passengers tend to visit closer market.
10.0
9.0

Travel cost α

8.0
7.0

Case a

6.0
5.0
4.0
3.0

Case b
2.0
1.0
0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Passengers density ζ
Figure-4 Passengers’ density and market equilibrium
Numerical examples above shows that structure of market spatial equilibrium is different
according to transaction cost and passengers’ density. This means that multiple markets can
co-exist at downtown areas with high dense of passengers, while in suburban are with low
density it is difficult to have more than 1 spot market. Moreover, passengers and taxies visit
the same market as travel cost increases to make another market disappear.

4. ARRANGEMENT OF SPOT MARKETS AND SOCIALWELFARE(1) Spatial pattern of spot markets
There possibly exist multiple spatial equilibria for taxi spot markets. 2 kinds of spatial
equilibrium are possible; 1) 2 spot markets coexist, 2) only one market exits. Because thickmarket externality works as size of market becomes large, average waiting time for bothtaxies and passengers decreases. From the point of that view, it might be better that all
passengers and taxies use only one market. However, that also brings about the increase of
travel cost. Administrators should increase the number of market from the viewpoint of travelcost. Let us analyze spatial pattern of spot markets and social welfare in the following section.

12th WCTR, July 11-15, 2010 – Lisbon, Portugal

12Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi(2) Generalization of market equilibrium model
Assume that there are R (R>0) spot markets on the linear market as shown in Figure-5. Let
us indicate points of each market xi (i=1, 2, …, R) and area covered by market i [ki-1, ki]. ki1=ki means that market i is not realized, that is, nobody visits the market though it is set at a
certain point. Taxies’ behavior can be represented in the same way as 2 market model.
Expected profit of taxies who visit spot market i is expressed as follows.
E   i , M i   i q  Si  i , i , M i   d(52)
As taxies enter market until when profit equals 0, arrival rate of taxies at market i (i=1, 2,… ,R) is i which satisfies the following condition.

i q  Si  i , i , M i   d  0(53)Define passengers’ utility at point y derived form visiting market i as equation (32). However,travel cost from point y to market i is written as follows.
ci  xi , y    xi  y(54)
Expected utility of passenger y to visit market i is defined as follows.

EVi  xi , y   v  Ti  i , i , M i   ci  xi , y (55)Generalizing equilibrium conditions (42)-(45), spatial equilibrium condition with R markets areequilibrium arrival rates *i , *i and market boundaries ki*  i  0, , R  which satisfy thefollowing equations.


EV i 1  EV i

 i  1, 2, , R 

k0  1  k1  k2   k R  1(56)

 EV =EV  x , k     k  k   q  S   ,  , M   d  0 when k  k (i  1, 2, , R)(57)

*i  0, *i  0 when ki  ki 1  i  1, 2, , R 

(58)




i

EV i =EVi xi , ki*1
i

i

*
i

*
i

i

*
i

i

i

*
i

i

*
i

*
i 1

i 1

Spot Market

k0

k1

ki-1

k2

ki

kR-1

kR

1 x

-1
Figure-5 Location and boundaries for several markets

All markets in spatial equilibrium do not always exist. Assume that ki-1<ki=…kj-1<kj. In thiscase, markets i+1, … ,j-1 are not realized. As  s  0,  s  0  s=i+1, ,j-1 ,EVs  xs , y     s  i  1, , j  1




for


passengers



at

any

positiony.

EVi  xi , ki   EV i  EV i 1    EV j  EV j x j , k j 1 from equations (56)-(58). Boundary
points ki=…kj-1 indicate same position. Stability conditions of each equilibrium can be written
as the same with former sections.

12th WCTR, July 11-15, 2010 – Lisbon, Portugal

13

Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi(3) Evaluation of social welfare
Transportation administrator can control social welfare with arrange spatial locations of spot
markets. Consider a linear market with spot markets x R   x1 , , xR  . Social welfare W(y)for passenger y   ki 1 , ki  with equilibrium *i , *i ,  i  1, , R  ; ki* ,  i  0, , R  is defined
as follows.

W  y   max  EVi  xi , y  : i  1, , R
i



(59)



EVi  xi , y   v  Ti *i , *i , M i  ci  xi , y Collective expected social welfare CS is expressed as follows.

CS  x     W  y  dy
R

(60)

0

Expected profit of taxies is 0 in the long-term equilibrium, so producers’ surplus is 0. Optimal
spatial arrangement of spot markets is defined as combination of R and x R which maximize
equation (60) with constraints (56)-(58). If we consider more than 3 spot markets, there are
too much spatial equilibrium which satisfies equations (56)-(58). It is very difficult to cover all
possibilities. As this paper tries to analyze the mechanism of spatial equilibrium of spot
markets, we stick to the cases with 2 markets.

(4) Numerical examples
Let us set two market with points x (0<x≤1) and –x. All parameters are set as the same withthe case a in section 3. Though there possibly exists multi stable spatial equilibrium in thiscase, let us focus upon equilibrium where 2 spatial equilibrium coexists.Figure-6 shows therelation between market point x (0<x≤1) and social welfare. At x=0 in this figure, social
welfare for the case with single spot market is also written. In both cases, spot market(s)should be located at the center of linear market(s). This case shows that social welfare with 2markets is lager than that with single market.

CS
28.8
28.6
28.4
28.2
28.0
27.8
27.6
27.4
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

12th WCTR, July 11-15, 2010 – Lisbon, Portugal

14

0.9

1.0

x

Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

Figure-6 Market arraignment and social welfare (   1.5 )Let us consider next the case with   1.0 . Though there are spatial equilibria with 2 markets,
social welfare of that is smaller than that with single market (See Figure-7). This figure can
explain the following things. Arrangement of 2 markets makes gross travel cost smaller thanthat with single market. However, average arrival rate at each market decreases as
passengers and taxies are divided to 2 markets, which make average waiting time increase.
When passengers’ density is small enough in this case, disutility from the increase in thenumber of markets conquers utility from the decrease in gross travel cost. This result showsthat social welfare is affected by passengers’ density and travel cost to each market.

CS
19.0
18.8
18.6
18.4
18.2
18.0
17.8
17.6
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x

Figure-7 Market arraignment and social welfare (   1.0 )Table-1 Passengers density and number of markets

Number ofmarkets12
3
45

  0.5

  1.0

  2.0

  3.0

  4.0

  5.0

9.05*
—
—
—
—

18.77
18.89*———

37.92
38.54*
38.28
—
—

56.96
58.24
58.31*
57.99
—

75.98
77.84
78.19*
78.08
77.33

94.99
97.39
98.00
98.06*
97.86

Until now number of markets is set as 2 for simplicity. From now on, let us set spot markets
as shown in Figure-5 in order to analyze the relation between the number of spot markets
and social welfare. Table-1 shows the relation between number of markets and social
welfare for several value of passengers’ density. * indicates that the number of markets at
this position maximizes social welfare for a certain passengers’ density. – shows that thistype of equilibrium cannot sustain no longer. Number of markets with stable spatial markets
increases as passengers density  increases. Number of markets which maximizes social
welfare also increases. Therefore, each market covers smaller range of market andpassengers’ arrival rate decreases to increase in expected waiting time, as the number of
12th WCTR, July 11-15, 2010 – Lisbon, Portugal

15

Spatial Equilibrium of Taxi Spot Markets and Social Welfare
Kakuya Matsushima and Kiyoshi Kobayashi

markets is increased. Travel cost to each market becomes smaller, on the other hand. As
these 2 effects work to expected utility for passengers interactively, number of markets may
affect social welfare complicatedly.

5. CONCLUSION
In taxi spot market where passengers and taxies are matched each other, thick market
externality works, that is, service transaction becomes efficient thorough gathering many
passengers and taxies. Because of scale economy related to this externality, more taxies
and passengers use a certain market among several markets. On the other hand,passengers and taxies have to bear transaction cost in order to enjoy the service. If a certain
market covers wider area, transaction cost of passengers located at the edge of linear
market increases. Increase in transaction cost may decentralize markets. We had alreadyanalyzes the mechanism of single market formation in a city thorough thick market externality.
This paper proposes the spatial equilibrium model where spatial equilibrium of spot market isformed through the mechanism of both co centralization and decentralization. Furthermore,
the relation between spatial arrangement of spot markets and social welfare is analyzed.
This study should be extended in various directions. First, it is necessary to formulate optimal
allocation model of spot market with this spatial equilibrium model. Second, the mechanism
where passengers’ density is endogenously determined should be analyzed. New equilibrium
model which includes spatial equilibrium model should be formulated for the purpose. Third,
information to both passengers and taxies should be considered. Giving proper information
may make transaction more efficient.

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