INTRODUCTION
With the development of the regional economy and urban expansion in China,
journey distances are lengthening and journey modes are diversifying. Alongside
the rapid development of China’s urban transportation infrastructure, traffic
congestion and environmental pollution are concurrently growing. A key challenge,
here lies in the tension between the available supply and the actual demand
for mass transit systems. The importance of public transport in alleviating
was first highlighted in France at the end of the 1960s. In his book “The
Transit Metropolis: A Global Inquiry” Cervero (1998)
introduced the concept of the “transit metropolis” a region where
a workable fit exists between transit services and urban form. He discussed
when, how and why cities can grow around new mass transit systems. Learning
from successful international examples, this study aims to construct a plan
of public transport priorities for China and to develop a public transit system
to help solve the problems of city traffic.
While many countries have increasingly been paying attention to efficient multilevel
public transportation systems, this also entails several constraints such as
limited funds, a low degree of travel satisfaction and potentially an improper
intrinsic structure of a public transportation system.
The rail and bus systems are important elements of a typical public transportation
system however, this also means that they compete for the government’s
limited investment. This study uses the Lanchester competition model to describe
the behavior characteristics of each of these systems respectively. With the
designed investment effectiveness function, the study evaluates the efficiency
of fund use under the condition of dynamic market share and variable market
scale and projects optimal investment strategies, which are then designed to
provide a basis for the government’s investment allocation decisionmaking.
There are four types of vehicle in Shanghai’s public transport system,
rail transit, bus, taxi and ferry. Given the homogeneity of market share and
services offered, this study will mainly focus on the rail and bus systems.
Comprehensive transportation reports in Shanghai show that passenger traffic
within these two systems has risen from 73.1% in 200380.7% in 2011. Later,
the proportion of passenger traffic volume that the rail and bus systems assume
together within the whole public passenger transport model is over 80%. In addition,
compared with taxis, the rail and bus systems have greater scope to benefit
society. The 12th five year plan for Shanghai’s public transportation indicates
that public transport is being prioritized in terms of the respective market
share of the rail and the bus system. While government investment in public
transport is set to increase dramatically, the distributive decision of government
investment between the rail and the bus system is, as yet, uncertain. This study
aims to obtain different results according to different investment allocations.
In order to achieve all the goals of 12th five year plan for Shanghai’s
public transportation, it is necessary to analyze the investment amount and
investment structure between the two types of transportation system. Moreover,
in order to effectively improve the performance of future public transport investment,
it is essential to evaluate the rationale for previous investment allocations.
THEORETICAL MODEL
This study investigates the investment allocation between the two types of
public transport companies (bus and rail) using the Bass diffusion process and
the Lanchester competition model. Firstly, the Bass diffusion process was used
to show new products promotion. Then it was used to map the fluctuation of company
market share under variable market scale. Over the course of the development
of Shanghai’s public transport system, the scale and proportion of public
passenger transport assumed by the rail and the bus system is set to significantly
increase. Thus, it is feasible to adopt the Bass diffusion process to display
the variable proportion of public passenger transport between the two types.
The Lanchester competition model is a duopoly competition model. The reason
for choosing this model for this study is due to the competitive relationship
between the bus and the train or subway, in passenger traffic volume and investment
allocation. If one chooses to travel by subway, it is impossible for that individual
to take a bus. In other words, the choice between subway and bus is a mutually
exclusive one. Therefore, the percentage of passengers taking the bus and the
percentage of those taking the subway are used to study changes of market share.
The percentage of passengers taking the bus can be derived by taking the bus
times divided by the sum of taking the bus times and taking the subway times.
The same is true for the percentage of those using the subway. Even though,
the bus and subway services are complementary in terms of the public transport
system itself, they act in competition in terms of attracting government investment.
In other words, the allocation of investment between these two modes are a zerosum
game under the established government’s public transport investment. It
can, therefore, be seen as helpful to use the Lanchester competition model in
order to highlight this competitive relationship.
Bass diffusion process: The Bass diffusion process created by Bass
(1969) is generally used to describe the characteristics of dynamic market
scale variation.
Suppose q(t) is passenger volume at time t and Q(t) is its cumulative passenger
volume from time 0t, then the Bass process can be characterized as shown in
Eq. 1:
where, a is the innovative parameter, b is the replaceable coefficient, S is
the potential market scale.
Building investment model: The Lanchester model (Case,
1979) is an important model that is used in differential game theories.
Specifically, it can be used to study duopoly competition under the condition
of dynamic market share.
Using the Lanchester model, the following equation can be obtained:
where, S_{i}(t) is the market share of company i at time t, m_{i}(t)
is the investment of company i at time t, ρ_{i}(m_{i},
m_{j}) is the investment effectiveness of company i. This function of
the two variables shows company i’s investment performance of m_{i}(t)
under the investment m_{j}(t) of competitor j.
Obviously, S_{i}(t)+S_{j}(t) = 1.
From Eq. 1 and 2, we obtained Eq.
3:
where, q_{i}(t) is company i’s market share at time t, because
of q_{i}(t)+q_{j}(t) = q(t) and S_{i}(t) = q_{i}(t)/q(t),
This obtain:
OPTIMAL STABLE OUTCOME AND THEORETICAL ANALYSIS
Suppose the revenue function of the bus company is as shown in Eq.
4:
where, g_{i} is contribution margin from each unit of the bus company
t, μ is discount rate. From Eq. 3 and 4,
the Hamilton function can be obtained as shown in Eq. 5:
Through the first order condition, Eq. 68
obtained:
Subsequently, there is the following partial differential equation:
Generally, there may not exist an analytical solution in a partial differential
equation. Hence, this study aims to ascertain its form via the characteristics
of the investment effectiveness function.
The first character is monotonicity, namely:
The second character is the marginal decreasing effect, namely:
From the first character, it is evident that the greater the investment, the
greater the investment performance under the condition of the competitor’s
fixed investment. Consequently, the investment effectiveness function is an
increasing function of its own investment. From the second character, it is
evident that the rise of its own investment effectiveness will decrease, if
rival parties increase investment at the same time. That is equal increments
of investment lead to a reduction in the increase of investment effectiveness
under the condition of the competitor’s increasing investment.
Therefore, the form of investment effectiveness function is assumed as shown
in Eq. 9:
where, C_{1 }and C_{2} are independent constants.
It is evident that ρ_{i}(m_{i}, m_{j}) has the
former characteristics namely, the investment effectiveness function contains
monotonicity and a marginal decreasing effect. Usually, the government budget
for public transportation investment can be expressed as L(t) limited, that
is:
m_{i}(t)+m_{j}(t)≤L(t)
At the same time, clearly m_{i} and m_{j} are greater than
zero.
Therefore, Eq. 8 can be translated to Eq.
10 as follows:
From m_{i}(t)+m_{j}(t)≤L(t), the solution of Eq.
10 can be obtained, as shown in Eq. 11:
CASE STUDY
With regard to investment effectiveness function, this study aims to analyse
existing Shanghai data of the rail and the bus system from 20032011, as well
as to discern the reasonability of market share and financial investment allocation.
Estimation of parameters C_{1}, C_{2}: From Eq.
2:
Then:
Using data relating to Shanghai from the public transportation annual report
20032011, the calculations can be obtained by using the method of regression
computation (Table 1).
Plug c_{1}, c_{2} into the investment effectiveness function,
then it obtain the following equation:
It is important here to note that:
This would initially appear to contradict the competition relationship. However,
the rail and the bus system operate in a complementary dynamic within the public
transportation system as a whole, mean that they are mutually reinforcing from
the point of view of investment effectiveness. That is to say, awhile investment
in different types of public transport enterprises is completely competitive,
their investment effectiveness is complementary.
Parameters estimation in Bass diffusion process: Using the Erickson
method to estimate the parameters in the Bass diffusion process, when j=1, 2;
i≠j, the optimal investment strategy obtained as following in Eq.
13:
where, d_{i} is an arbitrary constant.
Table 1: 
Parameters estimation of Lanchester model 

R^{2} = 0.951 
Table 2: 
Parameters estimation of Bass diffusion process 

R_{2} = 0.993 
Table 3: 
Other parameters of estimation 

Clearly, there is no solution for the Eq. 13 under the condition
of a positive discount rate. When the discount rate is zero, there is a closedloop
solution for this Eq. 13 (Case, 1979;
Chintagunta and Vilcassim, 1992; Erickson,
1992).
If i, j = 1, 2; i ≠j, then Eq. 14 can be obtained:
where, ξ_{i} is a random error. The assumption is that it is independent
and normally distributed. Firstly, it is necessary to estimate S, a and b. Then
linear regression is used to estimate parameters (Table 2).
Finally, (ba) and 2b/S need to be considered. For convenience’s sake,
they are considered as a whole, that is: k_{i} = ba, k_{2}
= 2b/S, k_{3} = aS_{0}.
If we plug k_{1}, k_{2}, k_{3} into Eq.
2, we obtain the following equation:
If we plug the former parameters into Eq. 13, g_{i}
and d_{i} can be estimated as:
Nguyen and Shi (2006) offer similar solutions, such
as: That is d_{1} = 0 and d_{2} = 0 (Table 3).
Optimal closeloop solution: Figure 1 and 2
shows the optimal closeloop solution and real investment in the rail and bus
systems respectively, under the condition of dynamic market scale.

Fig. 1: 
Comparison of optimal and real investment in the rail system
under the dynamic market scale 

Fig. 2: 
Comparison of optimal and real investment in the bus system
under the dynamic market scale 
It can also be seen that real investment in the bus system is higher than optimal
investment in most cases.
Overall, it can be said that, during its earlier construction, investment in
the rail system met the optimal investment strategy, while during later construction,
its investment became insufficient. Meanwhile, the bus system had consistently
proper investment. From 20082009, its investment was excessive. This phenomenon
can also be observed from Shanghai’s comprehensive transportation report
data which shows that the volume of passenger traffic in the bus system has
been at 7 million every day since 2003.
There is another key issue warranting attention. This concerns the four types
of investment projects in bus system, as highlighted in Shanghai’s recent
comprehensive transportation reports; urban roadway, bridge, tunnel and highway,
respectively. The issue is that the ground road system is not used solely by
buses, in fact, different kinds of ground vehicles use this. In light of this,
the bus system should arguably receive more investment than the optimal investment
indicated by the former model.
Not withstanding this issue, the current study still proposes prioritising
the development of the rail system, according to Fig. 1.

Fig. 3: 
Comparison of the optimal and real investment proportion in
the rail system 
Optimal investment proportion: According to the current study, it is
evident that the ground road system is used by all kinds of ground vehicles,
not just buses. Therefore, this study aims to estimate the optimal investment
proportion from the optimal investment closeloop solution of the rail system
(Table 4).
Figure 3 shows changes in the proportion of investment in
the rail system.
It can be observed that the government’s investment in the rail system
is lower than the optimal investment.
It can be observed that the government’s investment in the rail system
is lower than the optimal investment demand. Therefore, it can be concluded
that it is necessary to inject further government investment into the rail system.
In addition, alongside adjusting these investment proportions, the ratio of
public transit and the ratio between bus ride and rail travel are proceeding
steadily.
If the ratio between bus travel and rail travel is c_{0}, that is c_{0}
= q_{1}/ q_{2}. If the ratio between bus travel and rail travel
is stable, ds/dt = 0. Using Eq. 2 and 9,
the Eq. 15 can be obtained:
If we plug c_{0} = q_{1}/q_{2} into Eq.
15, we obtain Eq. 16 as follows:
The white paper on the development of Shanghai’s transportation system
indicates that rail transport will be a key player in urban public transport
in the future. The specific objectives include quadrupling rail passenger traffic.
Table 4: 
Optimal investment proportion 

The passenger traffic volume that orbit traffic assumes within the whole system
of public passenger transport is 50%, reaching 12 million each day. Shanghai’s
12th five year plan for public transportation requires the achievement of the
following target: 50% of trips are taken by public transit and that the passenger
traffic volume that orbit traffic assumes is 50% in the urban center. An additional
factor is that passenger traffic volume that orbit and bus traffic together
assume within the whole public passenger transport had risen from 73.1% in 2003
to 80.7% in 2011. If this proportion is to reach 85% in the future, that is
c_{0} = 10/7. Then, we may obtain m_{i}(t) = 1.87 m_{2}
(t), that is the optimal proportion of the rail system investment which will
be 65%.
CONCLUSION
This study uses the investment effectiveness function to analyze the optimal
investment allocation within the public transport system under the conditions
of dynamic market scale and dynamic market share. There are three main conclusions:
• 
The public transport investment plan should be made according
to the effectiveness of investment in different development phases 
Using the investment effectiveness function, this study highlights that its
own investment effectiveness is positively related to its investment. Namely,
under the closeloop assumption, its own investment efficiency increases with
its investment, if the competitor’s investment remains changeless. Whether
investment is increased in the rail or the bus system their benefits will upgrade.
This is because both of them belong to the holistic public services landscape
and have special inherent attributes. The difference between them is that they
show variant increases in investment efficiency, a phenomenon that conforms
to the real situation of public transportation. Given that the combined use
of bus and rail transit is common for most of people travelling long distances,
either party’s increased investment will, therefore, ultimately ameliorate
citizens’ traffic conditions:
• 
At present, the passenger traffic volume of the rail system
is lower than that of the bus system. According to Shanghai’s 12th
five year plan for public transportation, future passenger traffic volume
of the rail system is predicted to comprise half of overall public passenger
numbers. Therefore, it can be seen as necessary to increase investment in
the rail system 
• 
The optimal investment strategy should be based on maximum profit 
• 
If the real investment exceeds the optimal, its own market share will
still enlarge under the condition of the closeloop solution, however, its
profit will decrease. While, clearly increased investment in the rail system
is necessary, the proportion of investment in the rail system should not
exceed the optimal ratio, given that excessive investment will lead to decreased
in total profit 
• 
Indicators for the government to select optimal investment strategies 
The optimal investment proportion based on the previous analysis can provide
clues as to the investment strategy that the government can adopt, it can also
be used to evaluate the performance of expenditure input from state finance.
This study contains several assumptions which can be tested mathematically,
such as the functional form of investment effectiveness being assumed according
to its characteristics. Infact, there are many functional forms meeting its
characteristics, this study has simply selected the special form of all. Thus
far, it has been difficult to prove that this functional form is more closely
aligned with the actual, given that the partial differential equation outlined
in this study is not the most common form. This can usefully be the subject
of further study.