A DYNAMIC MODEL OF TWO COMPETING CITIES: THE EEFCTS OF
COMPETITION ON TOLLS AND LAND USE
Simon Shepherd and Chandra Balijepalli
Institute for Transport Studies, University of Leeds, England LS2 9JT
Abstract: This paper analyses the impacts of competition between two cities
when considering demand management strategies on both the optimal tolls and
upon longer-term decisions such as business and residential location choices.
In particular, this paper considers two cities of different size (in terms of area,
population, workplaces) and focuses on the asymmetric interactions between
the two. Furthermore, this paper analyses the impact of separation between the
two cities on optimal tolls.
1. INTRODUCTION
Cities compete with each other. For more than fifty years, Public Choice Theory
has explored the notion that cities compete to attract and retain residents and
businesses (Tiebout, 1956; Basolo, 2000). Likewise, the Public Finance & Tax
Competition literature identifies competition between cities on tax-and-spend
policies (Wilson, 1999; Brueckner, 2001).
Research in the transport literature, on the other hand, has focused
predominantly on intra-city issues. The strong focus in recent years has been
on road user charging, economic theory suggesting benefits will accrue to a city
from a combination of congestion relief and recycling of revenues within the city
(Walters, 1961). Beyond the theoretical benchmark of full marginal cost pricing,
the design of practical charging schemes, such as those adopted by local
authorities in recent Transport Innovation Funds (TIF) bids, have generally
focused on pricing cordons around single, mono-centric cities (Shepherd et al,
2008). As our own research has demonstrated, it is possible in such cases to
design the location and level of charges for a cordon so as to systematically
maximise the potential welfare gain to the city (Shepherd and Sumalee, 2004;
Sumalee et al, 2005), yet there is an implicit premise here that the city acts in
isolation.
This paper is part of a larger research project which aims to answer the
following policy questions:




In what ways do and could cities compete using fiscal demand
management policies?
How should cities design their policies to achieve individual and collective
„best‟ outcomes?
Should cities consider sharing revenue streams – should they compete or
co-operate?
How significant are these policies to the redistribution of business and
residents between cities?
Early work by Marsden and Mullen (2012) has looked at the motivations of
decision-makers in local government in different towns and cities of four major
city regions in England. It showed that towns and cities both compete and
collaborate to maximise their own competitive position. The major cities are
© Association for European Transport and Contributors 2012
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seen as the main powerhouses of growth, with other towns and cities trading on
particular distinctive skills sets or tourist offers and spill-over effects from the
major cities. Working together they can act as a more powerful voice to argue
for investment from central government.
Mun et al (2005) studied the optimal cordon pricing in a non-mono-centric city.
In this paper, they assume a city developed in one dimensional space such as
a linear city but with more than one CBD. The model describes the distribution
of demand and traffic congestion under various pricing schemes. They have
compared three alternative schemes: no-toll equilibrium, first-best optimum, and
optimal cordon pricing. Optimal cordon pricing is defined as the combination of
cordon location and toll amount. Their research revealed that cordon pricing is
not always effective for congestion management in non-mono-centric cities and
it tends to be effective as the urban structure is more mono-centric. Our work
significantly differs from that of Mun et al in that we analyse the optimal toll
between two cities competing with each other whereas Mun et al consider one
city with many CBDs. Moreover, our model considers cities developed in twodimensional space that is to say that the transport network represents a realworld physical network as opposed to the one-dimensional model adopted by
Mun et al (1995).
The aim of our paper is to model the impacts of competition between cities
when considering demand management strategies on both the optimal tolls and
upon longer-term decisions such as business and residential location choices.
The research uses a dynamic land use transport interaction model of two
neighbouring cities to analyse the impacts by setting up a game between the
two cities who are assumed to maximise the welfare of their residents. The
work builds on our earlier work by Koh et al (2012) who studied competition in a
small network using a static equilibrium approach for private car traffic alone.
Our research extends this by setting up a dynamic model which includes all
modes and longer term location responses.
The model is used first to study an isolated city (representative of Leeds) and a
simplified welfare function is used to determine the optimal toll around the
central area and its impacts on location decisions and other transport
indicators. A twin city is then added to the model thus introducing traffic
between the cities. This traffic may be charged to enter the central area along
with own residents, however the revenue may be retained by the city - a form
of tax exporting behaviour which would in theory increase the welfare of the
city. However as shown by Koh et al (2012), both cities will have an incentive
to charge and a game may evolve where in the long run both cities may be
worse off in terms of welfare than if none had charged in the first place.
With this simple model set up we will study the impact on the optimal tolls set
by the cities and how the game develops between cities, first of equal size and
amenity and second with cities which differ in size and amenity. This paper
focuses on cities of different sizes to study the asymmetric interactions between
the two. The impact on location decisions and other transport indicators will be
investigated as will the implications for regulation and the development of cities
within regional partnerships.
© Association for European Transport and Contributors 2012
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2. DYNAMIC LAND USE TRANSPORT INTERACTION MODEL
2.1 MARS Model
MARS is a dynamic Land Use and Transport Integrated model. The basic
underlying hypothesis of MARS is that settlements and activities within them
are self organising systems. MARS is based on the principles of systems
dynamics (Sterman 2000) and synergetics (Haken 1983). The present version
of MARS is implemented in Vensim®, a System Dynamics programming
environment. MARS is capable of analysing policy combinations at the
city/regional level and assessing their impacts over a 30 year planning period.
As MARS is not the subject of this paper readers are referred to Pfaffenbichler
et al (2010) for a detailed description.
2.2 Calibration of Small Model
2.2.1
Case Study
In order to investigate the competition between cities and the impacts of
competition on optimal tolls and other indicators, it was decided to use a simple
hypothetical case study based on an aggregate representation of an existing
more “spatially” complex model. This follows the discussion in Ghaffarzadegan
et al (2011) who highlight the benefits of using small system dynamics models
in addressing public policy issues. Thus we first of all developed a 2 zone
model of Leeds based on the full 33 zone model and then extended this to form
a two city model with 4 zones based on the idea of two cities within reach of
one another (see figure 1). The larger city is based on the population of Leeds
which is split between the inner zone 1 and outer zone 2 with more growth
predicted in the outer zones by 2030. We call this City A and the neighbouring
city we will call City B is based loosely on Bradford.
2
4
1
City A
3
City B
Figure 1: The Two City Zones
The forecast population in the 2-zone model of Leeds was validated against
that of the full sized Leeds model with 33 zones. This full sized Leeds model in
turn was calibrated to UK TEMPRO forecasts of population, jobs and workers
(DfT, 2010). Table 1 shows the population in the inner and outer areas (zone 1
and 2 in the 2 zone model) for the 2-zone and 33-zone model of Leeds. This
© Association for European Transport and Contributors 2012
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validates the 2 zone model and note that both models exhibit more growth in
the outer zones i.e. urban sprawl continues in Leeds as forecast by TEMPRO.
Table 1 Population of Leeds in Year 30
Region
2-zone
model
33-zone
model
Zone1 (1-13 of 33
zones)
342,879
343,384
Zone2 (14-33 of 33
zones)
621,780
621,801
Total
964,659
965,185
In what follows, each city may decide to charge car users to travel to the central
area (zones 1 or 3) within the peak period. The charge will be applied to their
own residents as well as those from the other city (where it exists).
2.2.2
Welfare measures
For a single city in isolation, the local authority is assumed to maximise the
welfare of their citizens. The traditional form of welfare measure within the
transport field is the Marshallian measure which sums consumer and producer
surplus. The welfare measure includes the assumption that all revenues
collected are recycled within the system i.e. shared back between the residents.
For our case the measure may be estimated by the so called “rule of a half”
(Williams, (1977)) and so the welfare measure is written as follows for the
single city case :-
(1)
where,
= travel time between each OD pair ij with road charge
= travel time between each OD pair ij without road charge
= trips between each OD pair with road charge
= trips between each OD pair without road charge
τ = toll charge to central zone
n = number of origins/destinations
It should be noted that for the isolated city, the local authority objective to
maximise welfare will coincide with the objective of a higher level regulator e.g.
© Association for European Transport and Contributors 2012
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the national government or some appointed regulator. Once we move to the
twin city case, we now have two authorities whose aim is to optimise the
welfare of their residents – including their journeys to/from the other city region.
The welfare measure for each city is similar to that used in the isolated city, but
now we need to consider transfers of revenue from city A to city B and vice
versa. The welfare for city A may be written as follows :(2)
where,
welfare of residents from city A (origin i lies in city A), based on (1)
above
= revenue collected by city A from residents of city B
= revenue paid by the residents of city A to city B
Now in the twin city case the higher level or global regulator would consider the
total welfare of all residents i.e.
.
3. NUMERICAL STUDIES
This section describes the application of the MARS model under various
charging regimes for an asymmetric case where the second city is (based on
Bradford) smaller than the first (based on Leeds) and with workplaces
distributed so that more are within City A. Table 2 shows the distribution of base
year population and work places in City A and City B.
Table 2 Distribution of population and workplaces in two cities
Region
Population
Work places
City A: Zone 1
262,560
282,695
City A: Zone 2
453,042
172,435
City B: Zone 3
168,038
134,369
City B: Zone 4
289,947
81,961
3.1 Modelled Scenarios
This section sets out the optimal tolls and welfare implications for a number of
different cases, namely:



City A or B tolls alone
City A or B regulated
City A and City B – regulated
City A and City B Nash game
The first set of cases consider City A or City B tolling within the twin city set up
so some tax exporting behaviour is now possible. The addition of the “City A or
B regulated” test allows us to compare the solutions when a higher level
© Association for European Transport and Contributors 2012
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regulator controls the toll in one of the cities to maximise the welfare of
residents from both cities. Finally there are two cases to consider where both
cities are tolling users. The first is a regulated scenario where tolls are set to
maximise the total welfare of all residents, the second is the Nash game where
cities are maximising their own residents‟ welfare in a non-co-operative game.
In this final scenario, tax exporting behaviour is assumed and revenues are not
recycled between the cities. Note that for the symmetric case we only need
consider one city in presenting the results as the tolls will be identical.
Next, it should be noted that as the model predicts the impacts over a 30 year
period we allow the tolls to vary over time.
To avoid oscillations in the toll
schemes, we limit the tolls to adjust linearly from the year of implementation
(assumed to be year 5) until the final year (assumed to be year 30).
In order to find the optimal toll levels, the VENSIM optimisation tool was used to
maximise the appropriate welfare measure by varying the start and end values
of the relevant tolls. Note that to calculate the Nash outcome, a diagonalisation
approach was used whereby City A maximised welfare first with tolls for City B
held constant, then city B optimised with tolls for city A held at the previous
iteration values. This was repeated until convergence which was within three
rounds of the game.
3.2 Optimal tolls and the impact of separation
Table 3 shows the change in welfare for the base case of asymmetric cities.
The base case is defined as the one in which city A and city B are neighbours
to each other. Although the cities are next to each other, because of the spatial
nature of the cities, the distance between the two centres of the cities has been
set equal to 18km. It is noticeable that the optimal charges for B are higher than
for city A while the welfare gained is lower. This is due to the higher proportion
of non-residents entering the cordon and paying the fee in city B than in city A.
When a regulator controls City B alone then the tolls are now lower than for the
regulated City A scenario. From the whole system point of view there is more to
be gained from tolling in city A than in city B due to the greater population and
associated time savings. Regulation of both cities allows for slightly higher
charges in both cities to gain higher total welfare. Interestingly the Nash game
results in a positive welfare in year 30 (worse off in earlier years), though lower
than the regulated case and with higher optimal tolls applied to both cities. City
B is stronger of the two players and is able to charge more due to the tax
exporting behaviour.
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Table 3: Optimal tolls and welfare changes in year 30 for the base case
Scenario
Tolls €
Welfare A €
Year 30
Welfare B €
Welfare Total €
Year 30
Year 30
Single City A
4.00
120,000
-84,643
35,356
Single City B
5.30
-112,222
107,711
-4,511
Regulator City A
1.65
82,535
-22,816
59,719
Regulator City B
1.49
-23,504
55,522
32,018
Regulator 2-city (City A
tolls)
2.06
69,261
48,930
119,191
City B tolls
2.15
Nash Game (City A tolls)
4.77
23455
31894
55350
City B tolls
6.43
Intuition and theory would suggest that as cities become further apart then
interaction between cities would be reduced and the possibilities for tax
exporting behaviour would be similarly reduced. We now investigate what
happens as we increase the separation between the cities. We may also expect
that the smaller city may lose its status as the stronger of the two as separation
is increased. Table 4 shows the optimal tolls and the welfare changes when the
two cities are separated by 30km centre to centre.
Firstly we can note that as cities are separated further then the single optimal
tolls are lowered. This is due to the lower proportion of users from the opposing
authority and hence reduced opportunity for tax exporting behaviour. The
changes in welfare are more balanced between cities which results in a higher
overall change in total welfare with further separated cities. Secondly, however
it is noticeable that the tolls under all regulated cases are higher with the
increased separation. This is thought to be due to the greater time benefits
over the longer distance routes between cities which can be locked-in under
regulation with revenues recycled to the benefit of both cities‟ residents.
It is noted that city B is weaker than before as the optimal tolls are now lower
relative to the tolls shown earlier in Table 3. However the city B is still stronger
and is able to charge more than A when competing with A (Nash game) as well
as in the regulator controlling two cities case. However, it is interesting to note
that the welfare change in Nash game is getting closer to the global regulator
case indicating the gains due to competition (though with much higher tolls).
© Association for European Transport and Contributors 2012
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Table 4: Optimal tolls and welfare changes in year 30, when cities are
separated by 30km
Scenario
Tolls €
Welfare A €
Year 30
Welfare B €
Welfare Total €
Year 30
Year 30
Single City A
3.28
75,173
-28,281
46,891
Single City B
3.75
-29,933
56,468
26,534
Regulator City A
1.73
59,912
-6,014
53,897
Regulator City B
1.51
-8,583
33,970
25,386
Regulator 2-city (City A
tolls)
2.29
64,448
42,039
106,488
City B tolls
2.56
Nash Game (City A tolls)
4.18
61,706
36,558
98,264
City B tolls
5.23
In order to assess the impact of separation between the two cities, we have
simulated another case with the distance between the two centres set equal to
50km (Table 5). From this additional case, we have noted that city B is no
longer the stronger of the two and as we would expect the two cities are more
independent of each others‟ influences. Firstly the optimal tolls in city B were
consistently lower than city A tolls including the Nash game. The change in total
welfare in year 30 has always been positive. When the cities are even further
apart, the Nash game welfare approaches global regulator welfare and may
even exceed it in year 30 (Figure 2), it does not however exceed the regulator
case in terms of cumulative welfare over the full 30 years1. The lower charges
found in the regulated case bring higher benefits in the earlier years while the
higher charges found in the Nash game bring higher benefits in the later years.
1
Note that the cumulative welfare is still higher under the regulated case (not shown) and that
welfare in year 30 is improved with a higher charge in this case due to higher time benefits in later
years as we have a growth in population.
© Association for European Transport and Contributors 2012
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Table 5: Optimal tolls and welfare changes in year 30, when cities are
separated by 50km
Tolls €
Scenario
Welfare A €
Welfare B €
Welfare Total €
Year 30
Year 30
Year 30
Single City A
2.23
38,170
-265
37,905
Single City B
0.48
-2,690
4,651
1,961
Regulator City A
1.61
35,115
1,909
37,024
Regulator City B
0.18
-1,109
2,022
913
Regulator 2-city (City A
tolls)
1.97
36,770
12,523
49,294
City B tolls
1.34
Nash Game (City A tolls)
2.51
39,530
11,444
50,974
City B tolls
1.41
Welfare Total
60000
40000
Welfare
20000
0
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29 31
-20000
Regulated
-40000
-60000
Nash Game
Years
Figure 2 Nash Game and Regulated Welfare Totals
3.3 Long term land use response to tolls
Tables 6,7 show the response to tolling in various scenarios such as City A/B
optimising, CityA/B or both regulated and finally a Nash game. The response is
measured in terms of change in residents (Table 6) and jobs (Table 7) over the
do-nothing case, thus a negative number indicates residents/jobs moving out of
the zone and a positive number indicates the opposite effect. In general we see
that charged areas attract residents but lose out slightly on jobs moving out of
the charged area. This is intuitively sensible as the residents want to avoid
© Association for European Transport and Contributors 2012
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paying toll by moving into the area. However, a small number of workplaces
move out of the charged area in order to retain/attract customers.
Starting with the single city A charging alone, (Table 6), residents are attracted
from all other zones towards zone 1 to avoid paying the charge and city B loses
a large number of residents of over 13k to city A. This out-migration results in a
net increase in both zones within City A – this is due to the higher accessibility
to the higher number of jobs within City A. On the other hand when City B
charges alone, Zone 3 attracts some residents while zone 4 loses residents to
zone 3 and City A. The effect of charging in city B alone is to shift residents to
city A overall .
The regulated cases result in a similar pattern of changes in residents with City
B losing residents to City A but to a lesser extent than in the A/B charging alone
cases due to lower charges. Finally when the Nash game is played between
the cities, City B loses even more residents to City A (around 19,000) due to
synergy of the charging effects and to higher charges being implemented by
both cities.
The impact on jobs is relatively small – zone 1 always loses jobs which appear
to be in competition for space from the influx of residents in all cases. Zone 2
always attracts jobs while zone 4 always loses jobs – hard to explain – but we
think (a) jobs develop where there are people with good access but can be
constrained by competition for space. This indicates the attractiveness of the
larger city over the smaller one for businesses due to the effect of relative
accessibility (a greater labour or customer pool is available in City A).
Table 6 Change in residents in year 30: base case
Scenario
Single City
A
Single City
B
Regulator
City A
Regulator
City B
Regulator
2-city
Nash
Game
Change in
residents
Zone 1
Change
in
residents
Zone 2
Change in
residents
Zone 3
Change in
residents
Zone 4
10,843
2,536
-2,990
-10,388
2,327
3,320
3,582
-9,228
5,293
1,593
-1,586
-5,298
9,20
1,587
1,380
-3,885
8,030
3,646
141
-11,816
13,771
5,083
1,759
-20,612
© Association for European Transport and Contributors 2012
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Table 7 Change in jobs in year 30: base case
Scenario
Single City A
Single City B
Regulator
City A
Regulator
City B
Regulator 2city
Nash Game
Change in
workplaces
Zone 1
-149
-120
Change in
workplaces
Zone 2
890
247
Change in
workplaces
Zone 3
-246
23
Change in
workplaces
Zone 4
-496
-151
-66
453
-132
-255
-54
97
7
-49
-149
-479
697
1320
-138
-182
-410
-661
4. CONCLUSIONS
The main conclusions from this research are as given below:
Firstly, when the cities are close to each other global regulator controlling the
two cities results in the best possible welfare for the two cities together.
However, the optimal tolls in this case may be higher than the case of regulator
controlling each city. If the cities are further apart (such as 50km centre to
centre), then the global regulator case and Nash game will result in a similar
total welfare. Although the total welfare is similar, the optimal tolls in global
regulator case are still lower to that of the Nash game which means some of the
gains in the larger city are passed on to the smaller one when globally
controlled.
Secondly, the tolling attracts residents to move into the charged area while a
small number of workplaces move out. If the cities are close to each other, the
smaller city loses out residents to the larger one due to the higher number of
workplaces. But if the cities are further apart, then the impact of the larger one
on the smaller is less dominant and relatively smaller number of residents move
to the bigger one.
Finally, it is concluded that for twin cities (located close to each other) it would
be beneficial to the community as a whole if the demand management policies
such as tolling are globally planned and executed rather than each city acting in
isolation.
REFERENCES
Basolo V (2000) City spending on economic development versus affordable
housing: Does inter-city competition or local politics drive decisions? Journal of
Urban Affairs 41(3), 683-696.
Brueckner JK & Saavedea LA (2001) Do local governments engage in strategic
property-tax competition? National Tax Journal 54(2), 203-229.
DfT (2010) http://www.dft.gov.uk/tempro/ web site accessed 8 March 2012.
Ghaffarzadegan, N., Lyneis, J. and Richardson, G.P. (2011) How small system
dynamics models can help the public policy process, System Dynamic Review
27(1), 22-44.
© Association for European Transport and Contributors 2012
11
Haken, H. (1983). Advanced Synergetics - Instability Hierarchies of SelfOrganizing Systems and Devices, Springer-Verlag
Koh, A., S.Shepherd and D.Watling (2012) Competition between Two Cities
using Cordon Tolls: An Exploration of Response Surfaces and Possibilities for
Collusion, Transportmetrica (Forthcoming)
Marsden, G. and Mullen, C. (2012) How does competition between cities
influence demand management? Proc. of Universities’ Transport Study Group,
January 2012, Aberdeen.
Mun, S., Konishi, K. and Yoshikawa, K. (2005) Optimal cordon pricing in a nonmonocentric city, Transportation Research Part A, 39, 723-736
Pfaffenbichler, P., Emberger, G. and Shepherd, S.P. (2010): A system dynamics
approach to land use transport interaction modelling: the strategic model MARS
and its application. System Dynamics Review vol 26, No 3 (July–September
2010): 262–282.
Shepherd, S.P. and Sumalee A. (2004). “A genetic based algorithm based
approach to optimal toll level and location problems”. Networks and Spatial
Economics, 4.
Shepherd SP, May AD & Koh A (2008). How to design effective road pricing
cordons. Proceedings of the Institution of Civil Engineers, Transport: Road
User Charging, 161, TR3, 155-165.
Sterman, J. (2000). Business dynamics : systems thinking and modelling for a
complex world, Irwin/McGraw-Hill, Boston.
Sumalee, A., May, A.D. and Shepherd, S.P. (2005). Comparison of judgmental
and optimal road pricing cordons. Transport Policy, Volume 12, Issue 5
September 2005, pp384-390.
Tiebout CM (1956). A Pure Theory of Local Expenditures. Journal of Political
Economy 64(5), 416-424.
Walters A (1961). Theory & Measurement of private & social cost of highway
congestion. Econometrica 29(4),676-699.
Williams, H.C.W.L. (1977) On the formation of travel demand models and
economic evaluation measures of user benefit. Environment and Planning A,
1977, volume 9, pages 285-344.
Wilson, J.D. (1999) Theories of tax competition, National Tax Journal, 52(2),
269-304
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