Montréal
Série Scientifique
Scientific Series
2001s-48
A Tale of Two Ports
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Les organisations-partenaires / The Partner Organizations
•École des Hautes Études Commerciales
•École Polytechnique
•Université Concordia
•Université de Montréal
•Université du Québec à Montréal
•Université Laval
•Université McGill
•Ministère des Finances du Québec
•MRST
•Alcan inc.
•AXA Canada
•Banque du Canada
•Banque Laurentienne du Canada
•Banque Nationale du Canada
•Banque Royale du Canada
•Bell Québec
•Bombardier
•Bourse de Montréal
•Développement des ressources humaines Canada (DRHC)
•Fédération des caisses Desjardins du Québec
•Hydro-Québec
•Industrie Canada
•Pratt & Whitney Canada Inc.
•Raymond Chabot Grant Thornton
•Ville de Montréal
© 2001 Ngo Van Long et Kar-yiu Wong. Tous droits réservés. All rights reserved.
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or its partners.
A Tale of Two Ports
*
Ngo Van Long
†
, Kar-yiu Wong
‡
Résumé / Abstract
Nous analysons un jeu entre deux ports, dont le premier a l’avantage de
prendre sa décision avant son adversaire. Dans chaque port, il y a un cartel qui
s'occupe de la distribution des biens produits par des établissements localisés entre
les deux ports. Les cartels prennent comme donnée l'infrastructure construite par
les gouvernements respectifs qui se font concurrence. On démontre que l'un des
deux gouvernements a intérêt à utiliser la stratégie de «préemption» en
investissant très généreusement en infrastructure.
This paper examines how two geographically separated ports compete for
a market consisting of manufacturing firms located between the two ports. There
is a firm in each port, and these two firms, taking the infrastructure provided by
their governments as given, compete in a Bertrand sense. The governments,
however, can also compete in terms of investment in infrastructure. This paper
shows that there are cases in which both the firm and the government in the port
that has a longer history in the market may have the first mover advantage. In
particular, the government can provide a credible threat by overinvesting in
infrastructure.
Mots Clés :
Géographie économique, rivalité intergouvernementale
Keywords:
Economic geography, inter-government rivalry
JEL : R12, R58
*
Corresponding Author: Ngo Van Long, CIRANO, 2020 University Street, 25
th
floor, Montréal, Qc, Canada
H3A 2A5
Tel.: (514) 985-4000
Fax: (514) 985-4039
email: longn@cirano.qc.ca
Thispaperpresentsamodelofrivalrybetweentwocitiesthataredistribution centres forgoodsproducedinalongnarrowregionconnectingthe twocities. Rivalrybetween citiesis acommonfeature ofeconomic life. Anexample that comesimmediatelytomind isthe potentialrivalrybetweenHongKong and Shanghai. There are many other examples in the world that t the present description. For example, in the United States, San Francisco and Los Angeles, and Seattle and Tacoma have long histories of competition. In Australia, Sydney and Melbourne have been known to be rivals for a long time, both in the commercial and in the cultural spheres. In Canada, Montreal and Toronto used to be (almost)equal competitors. In Germany, DresdenandLeipzighavealonghistoryofrivalry. HongKongandSingapore are possible contenders for being the principal nancialcentre for East and Southeast Asia (not includingJapan.)
When cities compete, the respective city governments may have an in-centive tointerfere, possiblytohelplocallybased rms. This mayre ectthe desire of each city tomaximize a conventional social(or, rather,city-based) welfare function. Alternatively, one may adopt the political-economy view that city oÆcials want tobe reelected, and their campaign funds can be in-creased if they help local businesses (possibly at the expenses of local tax payers.)
Our paper is an attempt to model the commercial rivalry between two cities. We abstract from considerations such as population size, agglomera-tion e ects, labor market externalities, or diversi cation of business activi-ties
1
. Our main focus isonthe investmentincityinfrastructure, such asthe road system, the lawenforcement and regulatorysystem (e.g., howe ective is the body that regulates the activities of the city stock exchange in Hong Kong,ascomparedwith Singapore?). Wemodel therivalrybetweenthe two cities as follows. We assume that in each city there is a single distributor (or a cartel of distributors). The two distributors compete in service prices that they o er tomanufacturing rms that are locatedin along, narrow re-gion joiningthe two cities. Each distributor's cost of supplyingdistribution services depends on the quality of the city's infrastructure, the provision of which is the responsibility of the city's government. The two governments compete againsteachother, becauseeachwantsitscity tobethe most pros-perous centre of commerce
In the model considered in the present paper, there are two levels of 1
Formodelsthatdealwiththeseissues,see,forexample,North(1955),Thisse(1987), Glaeseret al. (1992),Fujita and Thisse(1996), Longand Soubeyran (1998), andFujita et al. (1999)
cities compete in terms of investment in infrastructure. Then, taking the government investments as given, the two distributors compete. On the rm level, weexamine both a Bertrand-Nash equilibriumand aStackelberg leader-followerequilibriumbetween the two distributors. The latter equilib-rium is the one the distributor that develops earlier wants to achieve, but its successful achievement of this equilibrium depends on certain restrictive assumptions. On the government level,a Nash equilibriumis possible if the cities are symmetric. However, if a city is developed earlier, it has the rst moveradvantagesothatwhenarivalcitytriestoemergewithpossiblejumps in infrastructure investments, it has an incentive to increase its investment with thepurposeofdiscouragingthe investmentbyitsrival. Sucha preemp-tiveinvestmentby the rst movercould becrediblebecause investment that was made earlier isnot reversible.
In section 2 we present the basic model and determine the equilibrium when one of the ports is the only supplierof the distribution services to the production rms. In section 3, we characterize the Nash-Bertrand equilib-rium between two distributors, one located in each city. Section 4 analyzes the case in which one distributorhas the rst moveradvantage, thatis, this is a Stackelberg game between the two distributors. Section 5 studies the optimal unilateralintervention from the point of viewof one city, under the assumption that the two distributors play a Nash-Bertrand game. In sec-tion 6,wedealwith asimultaneous-movegame between thetwocities,while section 7 discusses the Stackelberg game between the two cities. In both of these sections, we maintain the assumption that the two distributors play a Nash-Bertrand game. The lastsectiono ers some concluding remarks.
2 The Model
Thetwocitiesaredenoted byNandS(North andSouth),respectively. 2
The distance between them isnormalized at unity. Sis locatedat point 0and N is located at point 1. There are two types of rms: production rms, and distribution rms. Production rms form a continuum, and are uniformly distributed between S and N. A production rm is indexed by x; where xis itsdistance fromSand 1 xis itsdistance fromN. These rmsproduce the
2
Our convention of naming the two cities comes from the fact that in many of the examplescitedearlier,thetwocitiesareroughlyinaNorth-Southpositions;forexample, HongKong-Shanghai,Seattle-Tacoma,andSanFrancisco-LosAngeles.
same good.
Forsimplicity,assumethateachproduction rmproduceseitheroneunit of output,or nothing. Thecost of producing theoutput isa given constant, which is normalized as zero. The cost to rm x of shipping its outputfrom locationxto Sisbx
2
=2, and the costof shippingthe output fromlocationx to N is(b=2)(1 x)
2 .
4
In city S, there is a single distribution rm labeled distributor S (or, equivalently, a cartel of distributors) that provides the service of sending (and marketing) the goods to a foreign market. The cost of providing this serviceisC
S
perunitofgoodsold,whichistakenasgiven bythedistributor. The cost C
S
depends onthe quality of the city's infrastructure,I S
; (suchas the quality of the road network, and of the law enforcement system), as described by the following function,
C S =C S (I S ); where C 0 S (I S )<0and C 00 S (I S )>0: 5
It should be noted that I S
is labeled as investment forsimplicity,but itis infactthe accumulatedinvestment inthe past less depreciation. It is therefore astock, not a ow.
Distributor charges a price S
C S
; the same for all production rms, for itsdistribution service.
6
The world priceofthe goodisP,assumed tobe given exogenously.
Incity N,thereisarivaldistributorlabeleddistributor N,withunit cost of distribution C N given by C N =C N (I N ); where C 0 N (I N )<0;C 00 N (I N )>0: 3 Monopoly Distributor
We begin by focusing on one distributor. Suppose that initiallythe cost of the rivaldistributor in N is so high that it cannot compete with distributor S; e.g., the quality level I
N
is so low that C N
(I N
) > P: Then the southern distributor is a monopolist.
3
Many features of this model are borrowed from the spatial competition model of Hotelling (1927).
4
Our assumption that the transport cost is quadratic in distance is in line with d'Aspremontet al. (1979).
5
This meansthat an increasein thequalityof thecity's infrastructurewill lowerthe costofthedistributionservice,buttherateofchange inC
S
with respecttoI S
increases withthelatter.
6
Charging the sameprice to all production rms implies no price discrimination by eachdistribution rm.
The objective ofdistributor Sis tomaximizeits pro tby choosinga service price
S
and a cuto point x m
1suchthat all rms withdistance x>x m
willbeexcluded. For rmsthatarenotexcluded,theparticipationconstraint is that they shouldearn non-negative pro t. Formally,the distributor seeks to max S ;x m S = Z x m 0 ( S C S )dx=( S C S )x m ; (1) subject to P S b 2 x 2 0; 8x2[0;x m ]; (2) and 0x m 1: (3)
It is convenient tode ne the mark-up ondistribution cost as z,
z = S
C S
: (4)
It is clear that problem(1) isequivalent to
max z;x m S =zx m ; (5) subject to (3) and P C S z b 2 (x m ) 2 0: (6)
Toanalyze problem(5), we makeuse of Figure1. In the(x;z)space, we rst construct various iso-pro t contours, each of which corresponds to the same pro t. Each iso-pro tcontour, such as XZ,has aslope of
@z @x XZ = z x <0; (7)
and is convex to the origin. Condition (6) is the area bounded from above by curve PC, which has a slope of
@z @x PC = bx<0; (8)
and is concave to the origin. Constraint (3) refers to the region bounded by the vertical axis and the vertical line x = 1: The problem of the rm is to choose (x;z) that reaches the highest iso-pro t contour subject to the constraints (6) and (3). Depending onthe nature of the solution,two cases can bedistinguished.
distributor Sserves allthe production rms,is de ned by the following con-dition: P C S 3b 2 : (9)
To see why there is a corner solution, note that at x m
= 1; conditions (9) and (6) (the latter with anequality) implythat at(1; P C
S b) @z @x PC @z @x XZ ; (10)
or that contour XZ is at least as steep as curve PC. This is the case shown in Figure 1 (with a strict inequality in condition (10)). The optimal point with the highestpossible pro t is E.
At this point,condition (6) impliesthat the optimalprice of the goodis
S
=P b=2; givinga pro t tothe rm of
SM = SC (C S )=P C S b 2 : (11)
The Partial Market Case (Interior Solution) | This case, in which distributor S serves some of the production rms closer to itself, is de ned by the followingcondition:
P C S < 3b 2 : (12)
Let us suppose that starting from point E there is a drop in the price P; or a rise in the cost C
S
; or both, so that curve PC shifts down to, say, ~ P ~ C; satisfying condition (12). This produces a point of tangency,
~
E, between ~
P ~
C and iso-pro t contour ~ X ~
Z, it is optimal for distributor S to serve all the production rms within the interval (0, x):~ Using (7) and (8), with (6) holdingasanequality,wegetthe solutiontothe pro t-maximizingproblem: z =2(P C S )=3; and x m = q 2(P C S )=(3b) (13) S = 2P +C S 3 : (14)
SM = SI (C S )= 1 p b 2(P C S ) 3 3=2 ; (15)
which is what contour ~ X ~
Z represents. Notethat in these two cases with all production rmsas price takers distributor Swillset the service price
S so that the production rmbeing served and farthestaway fromportSearnsa zero pro t.
3.2 Optimal Investment by Government S
The government of port S is to choose the optimal investment in infras-tructure, I
S
, which determinesthe cost of distribution rm Stakesasgiven. The government chooses I
S
to maximize the welfarefunction V S of the port de ned as V S = SM (1+ )I S ; (16)
where is the marginalcost of public nance 7
, and SM
is the pro t of the distribution rm as a monopolist, de ned by either (11) or (15) depending on whether an interior solution exists. Problem (16) yields the rst order condition 0 SM (C S )C 0 S (I S )=(1+ ): (17) In condition (17), 0 SM (C S ) = 1 and 00 SM (C S
) = 0 in the Whole Market case, or 0 SM (C S ) = q 2(P C S )=(3b) and 00 SM (C S ) = [2=(3b)] 1=2 [P C S ] 1=2
in the Partial Marketcase.
Whengiven the marketpriceofthe good,condition(17) isillustrated by schedule ABFDE in Figure 2. This schedule has two components. Segment ABFD is a vertical line, corresponding to the Whole Market case (C
S < P 3b=2);inwhich(17)reducestoanequationwithoneunknown,I
S
:Denote the unique solutionby I
0 S
:Curve DEcorrespondstothe PartialMarketcase (C
S
>P 3b=2): Condition (17) is di erentiated to give the slope of curve DE: dC S dI S DE = 0 SM C 00 S C 0 S 00 SM <0: (18)
Also illustrated in Figure 2 is schedule CI, which represents C S = C S (I S ); which, by assumption, is downward sloping and convex to the origin. The intersection between schedules ABFDE and CI givesthe optimal infrastruc-tureinvestment. Attheinitialmarketprice,theoptimalpointoccursatFin
7
the gure, with an infrastructure investment of I S
; leading toa service cost of C
S
: DistributorS chooses toserve the whole market. 8
Suppose now that there is a signi cant decrease in P to, say, ~ P: As a result, the curve that represents condition (17) shifts to AB
~
FG, with its portionB
~
FG corresponds tothe case inwhichdistributor Sserves onlypart of the market. With the slope in (18) suÆciently large in magnitude, B
~ FG cuts schedule CIat point
~
F, leadingtoan optimalinfrastructureinvestment of
~ I
0 S
and a corresponding service cost of ~ C S : 9
Making use of Figure 2 and the previousanalysis,thefollowingpropositioncaneasilybeestablished,the proof of whichis given inthe appendix:
Proposition 1: (a) If currently C S
P 3b=2; a small rise in P will not a ect the government'sinfrastructureinvestment whiledistributor S contin-ues to serve the whole market. The service price charged by distributor S increases bythesameamount. (b)IfcurrentlyC
S
>P 3b=2;asmallrise in P willlead toanincrease inthe government'sinfrastructureinvestment and afallintheservicecost,whiledistributorSservesmore rms. Thechangein the service pricechargedby distributor Sis determinedby condition (14).
10
4 Duopoly in Distribution
Now consider the duopoly case, in which both distributors S and N may provide services to the production rms. These two distributors set service prices,
S and
N
; respectively. Tosimplifythe analysis,weassumethat the 8
Asanexample,onemaytakethespecialfunctionalform
C S (I S )= S I S ; where S
isapositiveparameter. Assumingtthatthemarketpriceishighenoughsothat rmSservesthewhole market,weobtaintheoptimalinvestmentcondition
I S = S 1+ 1=2 : 9
To satisfy the second-order condition, schedule GB is steeper than schedule CI, as showninFigure2. Seetheappendix foraproof.
10
Notethatforsimplicitywehavenotexplicitlyimposedtheconditionthat investment is irreversible, i.e., I
S
0. Under irreversibility, there is an asymmetry between an increase in P and adecrease in P: While agovernment may beinterested in increasing its investment in response to an increase in P; it doesnot destroy someof its previous investmentinthecaseofadropin P:
serves the whole market (the Whole Market case). 11
Production rms take the service prices N
and S
as given, and choose the distributor with the lower distributioncost. Denote x
c
as the rm indif-ferent between dealingwith N or S,satisfying
S + bx 2 c 2 = N + b(1 x c ) 2 2 ; or x c = 1 2 + 1 b ( N S ): (19) Given that x c
2[0;1];three cases can bedistinguished: 12 Case S: N S b=2. From (19), x c
=1; with allproduction rms served by distributor S. Case N: S N b=2, implying that x c
= 0; with all production rms served by distributor N. Case B: j N S j < b=2: In this case, 0 < x c < 1; with distributor S serving rms located at (0, x c
) while distributor N serving those at (x
c ; 1).
4.1 Reaction Functions of the Distributors
De ne y N S , and (y)x c
:Letus rst consider distributor S,which maximizes itspro t, taking
N asgiven: S = Z (y) 0 ( S C S )dx=( S C S ) (y); (20) subject to P S b[ (y)] 2 =20: (21)
Condition (21), which makes sure that all production rms served by the distributor donot have negative pro ts, is assumed to be not binding. The reactioncurveofdistributorSisderivedasfollowsandisillustratedinFigure 3:
Case N (regionA): Inthis region, (y)=0, and isbounded belowby the lineXKL,
S =
N
+(b=2),with distributorS serving no rm. 11
Thissimpli estheanalysisandguaranteesinteractionsbetweenthetwodistributors. 12
Inallthesecases,themarketpriceisassumedtobehighenoughsothatnoproduction rmsconcernedmakealoss.
line, YJGH, S
= N
(b=2);withdistributorScapturingthe wholemarket. Note that
S
=P b=2; the monopolistic service price, when N
P:
Case B (region D): In this region, (y) 2 (0;1); and is bounded by lines XKL and YJGH. The reactionfunction in this region isobtained from the solutionto problem(20) and is given by
S =R S ( N )= b 4 + 1 2 [C S + N ]: (22)
In the diagram, the line represented by (22) cuts YJGH at point G, where
N =C
S
+ (3=2)b. Combiningtheaboveresults,weconcludethatthereaction curve of distributor is given by schedule EFGHI in Figure 3. We now have the following interesting result:
Proposition 2 (Limit Pricing): Assume that condition (9) holds. If distributor N charges N so that P > N > C S +(3=2)b; then distributor S will charge a service price that captures the whole market for itself. This service priceis below the monopoly price.
Note that in Proposition 2, condition (9) implies that distributor S asa monopolist will capture the whole market. If P >
N > C
S
+(3=2)b; the distributor's reaction curve is given by GH in Figure 3. The corresponding service price charged by S is less than the monopolist price, P b=2; but S captures the whole marketwith distributor N serving no production rm. This case isinteresting as the presence of distributor N poses as a threat to distributor S, who chooses to set a price lower than the monopolist price. On the other hand, distributor S sets a price low enough to deter N from entering the market.
The reaction function for distributor N can be constructed in a similar way. IfithasasegmentintheinteriorofregionD,thenthatsegmentsatis es the following equation
N = b 4 + 1 2 [C N + S ] (23)
Wenowderivetheequilibrium. Weassumethatbothdistributorstakethe infrastructure ineachcity (and thusits service cost andthat of its competi-tor)asgiven,andcompeteinaBertrandfashionsothatthey simultaneously choose a service price. The Nash equilibrium is described by the following proposition:
Proposition3 (Nash equilibrium): Assumethatcondition(9)holds:(a) IfC
N C
S
N S =C N b 2 : (24) If C N
P; distributor S willcharge the monopoly price S =P (b=2). (b) If C N < C S +(3=2)b, and C S < C N +(3=2)b, or more concisely, jC N C S
j < (3=2)b; then we have an interior Nash equilibrium, satisfying both conditions (22) and (23), and the equilibriumcharges are
S = b 2 + 1 3 (2C S +C N ); (25) and N = b 2 + 1 3 (2C N +C S ); (26)
provided that P is suÆciently great so that the indi erent production rm x
c
earns non-negative pro t. 13
(c)IfC S
C N
+(3=2)b; distributorNcaptures thewholemarket:Inthe subcase in which C
S
<P; distributor N charges a service price of
N =C S b 2 : (27) If C S
P; distributor N willcharge the monopolyprice N
=P (b=2).
Remarks: The above proposition indicates that (i) distributor S will use the limit pricing strategy if C
N
is below P but is suÆciently greater than C S (precisely, C N C S + (3=2)b). If C N
falls, then the limit price S
, given by (24) will also be adjusted downwards, and (ii) in the case where both distributors have positive market shares, a fall in C
N
will cause both equilibrium charges tofall, but
S
fallsby less than N
.
4.2 Services Costs and Nash Equilibrium
Construct a diagram(Figure 4)in the (C N
;C S
) space, for a given P; which we assume to be greater than (3=2)b. We rst examine how Case S is a ected by the distributors'service costs. In region W
S ;which isde ned by C N >P >C S
+(3=2)b, rmS isthemonopolythatservesthewholemarket, and its pro t is
SM =P C S (b=2)= SM (C S ); (28) 13
This conditionrequiresthat P S +(b=2)x 2 c where x c =(1=2)+(1=3)b(C N C S ) andwhere S isasgivenabove.
nopolist. In triangle T N ; which is de ned by (3=2)b < C N < P, 0 < C S < P (3=2)b, and C S <C N
3b=2; both distributors o er aservice price less than P, but distributor S captures the whole market by setting the limit price
SL =C
N
(b=2),thus earningthe pro t
SL = SL C S =C N C S (b=2); (29)
wherethe subscript Lindicates thatthedistributor ischargingalimitprice. We now turn to case N. In region W
N de ned by C S >P > C N +(3=2)b, rm N is the monopoly that serves the whole market, and its pro t is
NM =P C S (b=2)= NM (C S ): (30) Similarly,in triangleT S de ned by (3=2)b<C S <P, 0<C N <P (3=2)b, and C N <C S
3b=2;both distributors o era service priceless than P, but distributor N captures the whole market by setting the limit price
NL = C
S
(b=2),thus earningthe pro t
NL = NL C N =C S C N (b=2)>b: (31)
Finally,for case B, if the cost con guration (C N
;C S
)is inthe regionX de ned by the square f(C
N ;C S ) :0C N P,0 C S Pg excluding the two triangles T N and T S
and a small region Q in the north-east corner of this square,
14
then wehaveauniqueandinteriorNashequilibrium,i.e.,both distributors have positive marketshares. The market share of distributor S at this equilibriumis x S = 1 2 + 1 3b [C N C S ];
and its Nashequilibrium pro tis
nash SI =[ S C S ]x S = 1 b " b 2 + 1 3 (C N C S ) # 2 ; (32)
where the subscript I indicatesthat the equilibriumis aninterior one. 14
InthesmallregionQ;thecostsofbothdistributorsaresoclosetoP thatsomesubset ofproduction rmsarenotservedbyeitherdistributor. ThelowerboundaryofQ isgiven by the condition that the marginal production rm earns zero pro t and is indi erent betweenthetwodistributors.
In the preceding sections, it was assumed that the two distributors play a Nash-Bertrandgame,choosingtheirservicepricessimultaneously. Thereare, however, some cases inwhich one of them can play as aStackelberg leader. One possiblecase isnow described.
To describe such a possibility, let us rst investigate more properties of the distributors'reaction curves. Consider Figure 5,whichis obtained from Figure3,withsomeaddeddetails. DistributorS'sreactionfunctioninFigure 3is reproducedinFigure5,and islabelled JNKH,whereKisthe kink. The co-ordinatesofpointsJ;KandHarerespectively(0,C
S =2+b=4),(C S +3b=2, C S
+b)and (P,P b=2). Inthe regioninwhichbothdistributorsservesome production rms (region D in Figure 3), the pro t of distributor S is given by S =[ S C S ]x c =[ S C S ] 1 2 + 1 b ( N S ) : (33) The lociof ( N ; S
) that givethe same pro t of distributor S, which can be called aniso-pro t contour, has the slope
@ S @ N = @ S =@ N @ S =@ S = 1 1 k ; (34) where k = b+2( N S ) 2[ S C S ] :
Note that the slope of an iso-pro t contour at a point on line JK, part of the distributor's reaction curve, is in nity, implying that k = 1. This further implies that the slope of the contour is positive (negative) above (below) JK. In Figure 5, a Stackelberg equilibrium is depicted at point S, where distributor N's reaction curve LNM is tangent to distributor S's iso-pro t curve
s S
, which is higher that what distributor S can get at a Nash equilibrium. Since theslopeof distributorN's reactioncurve NMis equalto 2, k =1=2 at point S.
Nowsupposethatcurrentlytheinfrastructureinvestmentsbygovernment N is very low, making the service cost of distributor N very high, so that distributor N chooses not to serve any production rms. So distributor S is a monopolist and charges a service price of
m S
= P b=2: Suppose now that government invests signi cantly in infrastructures, lowering distributor N's service cost so that its reaction curve is now represented by LNSM in Figure 5. If both distributors play Bertrand, the equilibrium is at point N. DistributorSwillobserve aconsiderabledrop initsservice price(to
n S
)and pro t.
S lowers its service price to s S
; which is lower than m S
but higher than n S : If distributorN isconvinced that this iswhat distributorS iscommitted to, it willchoose the Stackelberg point S.
There is, however, another possible case which is less intuitive. Suppose that the service cost faced by distributor N is in fact higher so that its reaction curve is
~ L
~
M in Figure 5. With this reaction curve, the Stackelberg equilibriumshiftstoapointlike
~
S.Inthe caseshowninthediagram,point ~ S isabovethehorizontallineatP b=2;meaningthatanticipatingtheentrance of a rivaldistributor S raises itsservice price.
Suchacasemightseemcounter-intuitive,butcanbeexplainedintuitively. In the present model, while distributor S captures only part of the market after the entrance of distributor N, when it raises its service price it antici-patesthat distributorNwillreactwith apriceatalevelhigh enoughsothat distributorSwillnotloseabigmarketshare. Thiscasecanalsobeexplained in an alternative way. When a distributor is ina monopoly situation,there isatrade-o between pro tperunit ofservicesold, and thenumberofunits of service sold. When it has a rival as a follower, the trade-o is stillthere, but it is somewhat altered: under monopoly,when a rm changes its price, itismovingalonga given demandcurvedescribedby x=[(2=b)(P
S )]
1=2 , but with the existence of afollower, when distributor S changes itsprice, it is movingalongaquitedi erentdemand curve: x
c =(1=2)+(1=b)[ N S ], where N = N ( S ).
Of course, the Stackelberg equilibrium depends on the assumption that distributor N is convinced that
s S
is irreversible. This would be the case if distributor S chooses a service price and spends resources on xing and announcing it. The more resources distributor S spends on xing the price, the more costly it is tochange itlater, and the more convincing the pricing policy is.
15
Usually the existing rm is in a better position in choosing the service price rst, meaning that distributor S, being the rst in business, is more likely to have the rst mover advantage. If, however, distributor N believes that distributor S's chosen service price is reversible at low costs, the morelikely equilibriumis the Nashequilibrium.
A Numerical Example: This numerical example shows the possibility that distributor S charges more in the duopoly case than in the monopoly case. Let P = 2, C
S
= 0:5, C N
= 1:2, and b = 1. The monopoly price for distributor S is 1:5. At that price, it serves the whole market, and its monopolypro tis1. Now,considerthe entryofdistributorN.IfSmaintains
15
Ofcourse,thecostoftheresourceson xingthepricewillcomefromthedistributor's pro t.
S N and S's marketshare willfallto0:6, and itspro twill be
S
=0:6. Clearly, S can do better by maximizing (33) where
N = N ( S ) as given by (23): This yields S =1:60, and N
=1:65,resultingina smallermarketsharefor S, x
c
=0:55 and a higherpro t, S
=0:605.
6 Optimal Unilateral Intervention
Assume that initially the costs are C 0 S
and C 0 N
; and the equilibrium is an interior one, i.e., both distributors serve some production rms. Suppose that the government of city Swants tomaximize the welfarefunction V
S by choosing investment level I
S
, taking as given the amount I N
. We assume that government S rst chooses a new (possibly higher) investment level, withgovernmentNremainingpassive. Bothdistributorstaketheinvestment levelsas given and compete ina Bertrand fashion.
Let us assume that by spending I S
> 0, the cost C S
will fall to a level below C
0 S
. This fall is represented by a function F S (I S ), where F S (0) = 0, F 0 S (:)<0, and F 00 S (:)>0, and C S =C 0 S F S (I S ). 16
The government of city S maximizes V S = 1 b " b 2 + 1 3 (C N C S ) # 2 (1+ )I S : (35)
The rst-order condition is dV S dI S = !(I N ;I S )F 0 S (I S ) (1+ )0, =0 if I S >0; (36) where !(I N ;I S )= 1 3 + 1 9b C 0 N F S (I N ) C 0 S +F S (I S ) >0; (37)
and at an interior Nash equilibrium jC N
C S
j < (3=2)b. It follows that condition (36) is satis ed atsome I
S >0 if jF 0 (0)j>(1+ )=!(I 0 N ;0), that is, if the rst dollar spent on investment has a very substantial marginal e ect oncost reduction. In what follows we assume that this is the case, so that, given I
N
, atthe optimal investment level I S ,we have F 0 (I S )= (1+ )=!(I N ;I S ): (38)
The second-order condition is
d 2 V S dI 2 S = !(I N ;I S )F 00 (I S ) 1 9b [F 0 (I S )] 2 S <0; (39) 16
Notice that it isassumed that thefall in costis independentof theexisting levelof cost. A moregeneralformulationwould haveC
0 S
appearasaparameterin the function F
S :
Proposition 4: Assume that jF 0 S
(0)j > (1+ )=!(I N
;0). Then the opti-mal policy for the government of S is to raise taxes to invest in the city's infrastructure, untilcondition (38) issatis ed.
Remark: Becauseofupwardslopingreactionfunctions,ifweareconsidering optimal output tax, we would get a result similar to that of Eaton and Grossman (1986),who found thatif aforeign rmand ahome rmcompete in prices (and their reactionfunctions have positive slope)then the optimal policy for the home government is to increase the cost of the home rm by taxing its exports. In our model, we deal withmodifyingreal costs, and the optimalpolicyistoreducethecostofthehome rm,iftheinitialcostishigh. Thereare twofeatures worthnoting. Firstly,since theproduction rmsthat are located onthe long narrow regionlinking N and S are the purchasers of the services, they are the analogof theconsumers inthe third marketinthe Eaton-Grossman model. However, in our model, the \demand" for service of S, given by (19), remains unchanged if both distributors slightly raise their service prices by the same amount. This is not the case in the Eaton-Grossman model. The second feature is that while in the Eaton-Grossman model a dollar of subsidy reduces the cost of the home rm by a dollar, in our model a dollar of spending in infrastructure reduces the cost of the distributor by F
0 S
whichis not a constant (recallthat F 00 S
>0).
7 Simultaneous Government Policies
Now suppose that both cities are engaged in the infrastructure-investment game. The governments rst choose an investment level simultaneously. Then the distributors, taking these investment levels as given, compete in a Bertrand fashion.
Foragiven initialpair (C 0 S
;C 0 N
),letus nd the reactionfunction ofeach city. From (36), we obtain city S's reaction function I
S = r S (I N ), with a negative slopegiven by
dI S dI N S r 0 S (I N )= 1 S " F 0 N F 0 S 9b # <0; (40) where S
is de ned in (39): By condition (40), the magnitude of the slope is less than unity if jF
0 S (I S )jjF 0 N (I N
)j. The reaction function I S =r S (I N ) is illustrated in Figure 6. Note that there is a maximum value of city S's investment, as represented by the at part AB of the reaction schedule.
monopolist. 17
Similarly, for city N, the reaction function of government N is I N = r N (I S ); which satis es dI N dI S N r 0 N (I S )= 1 N " F 0 N F 0 S 9b # <0; (41) where N
is de ned ina similar way to S
above:The reactionfunction is illustrated in Figure6,with CD representing the maximum value of govern-ment N'sinvestment.
In general,there is the possibility of multipleequilibria, asillustrated in Figure6:IftheinitialcostsC
0 S
andC 0 N
areidentical,andthefunctionsF N
(:) and F
S
(:)are thesame, thenthereexists asymmetricNashequilibriumthat is stable, see pointE
3
in Figure6: However, there may existnon-symmetric stable equilibria, such as points E
1
and E 5
.
In the case of multiple equilibria, it can be shown, by construction of iso-welfarecurves, V
i
= constant (i=S;N),that city N prefers E 5 toE 3 to E 1
, and similarly,city Sprefers E 1 to E 3 toE 5 .
In whatfollows, wewillrestrict attention to the case wherethere is only oneNashequilibrium. Weassumethatsuchequilibriumisstable. SeeFigure 7, where the unique Nashequilibriumis E.
8 Stackelberg Game between Cities
As explainedearlier,itispossiblethat oneof thecities,sayS,develops rst. Suppose that initially city N has a very low investment in infrastructure, such asI
0 N
sothatdistributor Ncannot compete withdistributor S. 18
In the absence of activeinvestment by city N, the governmentof S and distributor SreachthemonopolyequilibriumasdescribedinSection3:Thisequilibrium can be represented by a point in between A and B in Figure 7. Note that the investment by city N that corresponds to point B, denoted by I
B N
in Figure7,isthe onethatwillyieldaservicecosttodistributorNequaltothe market price of the good, C
N
=P: Withthis service cost and marketprice, distributor N willnot wantto provide any distributionservice.
Suppose nowthat cityN invests ininfrastructure. Whatisthe new equi-librium? The reactioncurve of cityS isABEF.Since cityS develops earlier, it has the rst mover advantage. Note that ABis a horizontalline, showing
17
PointBgivesthevalueofI N
thatgivesaservicecostC N
=P: 18
Inotherwords,weassumethatC N
(I 0 N
given.
If city N decides to increase its infrastructure investment, we have to consider itsreactioncurve. Suppose that atalowerservice cost, itsreaction curveisrepresentedbyKLMECDasshown inthe diagram. Theintersecting pointbetween the two reactioncurves, point E, gives the Nash equilibrium. To reach this point, city S has to invest I
E S < I 0 S ; yielding a welfare of V S to city S:Since infrastructure investmentconsidered here is anaccumulated investment, it makes no sense to destroy some of the investment made pre-viously. Socity Swillmore likelykeep the existinginvestment level. Taking thisasgiven,cityNwillreactwithaninvestmentofI
M N
;withtheequilibrium given by pointM. City S willget a welfarelevelhigher than
V S
:
City S,however, can ingeneral dobetter than point M.In Figure7, one of its iso-welfare contour, denoted by
^ V S
; touches city N's reactioncurve at L. This is the Stackelberg point,with city S acting asthe leader and city N asthe follower. SincecityShas the rstmoveradvantage,itcan increase its investment levelto I
L S
atthe timewhen city Nis about toenter the market. Reacting to this level, city N willchoose the investment level I
L N
: Thus city S successfully achieves the welfare level
^ V S
; which ishigher than V S
:
ItisoftensaidthataStackelbergequilibriumisnotalikelyoutcomeinan oligopoly market because it is diÆcult for agents to make credible threats. In this case, the threat made by city S is credible because the increase in investment is not reversible.
In the above case with the Stackelberg equilibrium at L, city N enters the market, makessomeinvestment, andcaptures partof themarket. Other cases can be considered. Consider the case in which an iso-welfare contour of cityStouches N'sreactioncurveatpointKorhigher(insteadofpointL). StartingfromthemonopolycaseandanticipatingtheentranceofcityN,city Smakesapreemptivemove,increasingitsinvestmenttoI
K S
:Inthiscase,city N will have no incentive to invest because it does not make sense to make distributor N competitive, recalling that with its investment corresponding topointB orK,distributorN'sservice cost isatleast ashigh asthemarket price, P: This result issummarized inthe following proposition:
Proposition 5 (Limit Investing): IfcityShas the rst moveradvantage, it can increase its investment to a level at which city N responds with zero investment. Therearecases inwhichcitySwillchoose toincrease its invest-ment toa level todiscourage city N fromhelpingdistributor N to enter the market.
Wehavemodelledthe rivalry between two cities,focussingoninfrastructure investment. We have shown that at the rm level, equilibriummay involve limitpricingbyadistributor. Atthecitylevel,preemptiveinvestmentbythe citythatexists rstispossible. Suchpreemptiveinvestmentcoulddiscourage the latecomer fromentering the market.
Thereare a fewissues that couldbetaken up infuture research. Firstly, what is the optimal cooperative outcome, where the cooperation may be between the two cities, at the expense of the production rms in the long narrowstripthatjointhem? Onecanalsoinvolvethewelfareoftheownersof these rmsaswell. Secondly,citiesmaycompeteinmorethanonedimension. When the model is extended to allow for this, is it possible that the non-cooperative equilibriummaymaximize, ratherthan minimize,the di erence betweenthetwocities? Coulditbethecasethatonecitybecomesa nancial centre while the other becomesa goods distribution centre?
Proof of Proposition 1: The proof of part (a) is simple and is omitted. To prove (b), write the rst-order condition (17) as
G(P;I S ) 0 SM (C S (I S ))C 0 S (I S ) (1+ )=0: (42)
The second-order condition is
@G @I S =(C 0 S ) 2 00 SM + 0 SM C 00 S ;
is assumed to be negative, i.e.,
0 SM C 00 S C 0 S 00 SM <C 0 S ;
orinFigure2,curve CI isless steep thanthe GB curve. Theresponse ofI S to anincrease inP isobtained from(42):
dI S dP = @G=@P @G=@I S = 00 SM C 0 S (C 0 S ) 2 00 SM + 0 SM C 00 S >0: (43) From (13), (14) and (43), dx m dP = s 3b 8(P C S ) 1 C 0 S dI S dP ! >0 d S d P = 2 3 + 1 3 C 0 S d I S dP : The sign of d S =dP is ambiguous.
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