MARKET FOR TRANSPORT SERVICES: BASIC FRAMEWORK FOR
WELFARE ANALYSIS
Rostislav Staněk
Faculty of Economics, Masaryk University
rostanek@mail.muni.cz
Klíčová slova:
Produktová diferenciace, konkurence o trh, konkurence na trhu
Abstrakt:
V odvětví osobní přepravy existují dva způsoby zavádění konkurence: volný vstup a
soutěž o trh. Tento článek prezentuje základní verzi modelu, který umožňuje srovnání
obou typů konkurence na základě celkového přebytku. Poptávková strana modelu je
charakterizována heterogenními spotřebiteli, kteří se liší preferovaným časem odjezdu.
V modelu je endogenním způsobem determinována nejen rovnovážná cena, ale i počet
produktů a nabízené časy odjezdu. Výsledky modelu ukazují, že soutěž o trh je
výhodnější z hlediska celkového přebytku, pokud se rezervační cena jednotlivých
spotřebitelů příliš neliší.
Key words:
Product differentiation, open access, competitive tendering
Abstract:
There are two types of competition in the market for passenger transport services: open
access and competitive tendering. The paper presents a basic version of the model
which enables to compare these two types of competition in terms of welfare. The
demand side of the model includes heterogeneous consumers that differ according to
their preferred departure time. Equilibrium price as well as number of products and
offered departure is endogenously determined in the model. The model results show that
competitive tendering is welfare superior if consumers do not differ in their reservation
prices too much.
Introduction
The organization of passenger railways is undergoing significant changes in the past
decades. It is about 20 years since the rail reform process in Europe began with the
separation of infrastructure from operations of passenger services. Now, there is a
tendency to increase competition in the market for passenger services. There are two
ways how to open the passenger market to competition: open access and competitive
tendering (or competitive franchising). Open access competition is a competition in the
market, which means that firms are allowed to enter the market and compete in the
market by setting prices, product quality and so on. Competitive tendering is a
competition for the market. Competition for the market is characterized by the existence
of some bidding process (typically an auction organized by public authority) that
determines which firm can operate in the market. For example, public authority can run
an auction where it sells the right to operate on the market for passenger’s transport
services. If the operation on the market is expected to be profitable then the firms are
willing to pay for the right to operate in the market. In this case the firm which offers
the highest bidding price obtains the right to operate in the market. Both of these models
coexist in the member states of European Union [1].
The aim of this paper is to present a basic framework which enables to compare these
two competition regimes in terms of welfare. The model is an extension of Salop’s
model of product differentiation [4]. This model is suitable for modeling transport
markets because it captures the distribution of consumers according to their preferred
departure time. Johnson and Nash [2], [3] also present a model with similar demand
characteristics. But their model does not contain profit-maximizing firms on the supply
side and therefore it is not suitable for welfare evaluation. Unlike [2,3], the presented
model endogenizes not only a choice of product price but also a choice of offered
departure time. The model can be solved in two versions: the competitive version
corresponds to an industry with open access, the monopoly model corresponds to an
industry with competitive tendering. These regimes can then be compared according to
total surplus or consumer surplus. The paper proceeds as follows: next section describes
the model, section 2 provides a general solution of the model, section 3 shows analytical
solution and welfare comparison for special case when the consumers have the same
surplus reservation price.
1. Model
Consider a market with differentiated products. Product space can be represented by a
circle with unit perimeter. Position on the circle represents preferred departure time.
Consumers are distributed uniformly around the circle. Each consumer wish to buy one
unit of the good and have waiting costs which are increasing and convex function of the
waiting time. Consumers are willing to buy at the smallest generalized costs if it does
not exceed the gross surplus they obtain from consuming the outside good (e.g. going
by car instead of public transport). The original Salop’s model assumes that all
consumers differ only by their position on the circle, but all consumers obtain the same
surplus from the outside good [4]. This model provides a generalization of the Salop’s
model because it allows consumers to differ in their reservation prices. Concretely, we
assume that each consumer has reservation price or the gross utility obtained by
consuming the outside alternative Ri. Reservation price is random variable which is
uniformly distributed on interval ⟨0, 𝑅̅ ⟩. The utility function of each consumer is
therefore given as 𝑈𝑖 = 𝑅𝑖 − 𝑃 − 𝑇𝑥 2 , where P is the price paid x is the difference
between preferred departure time and actual departure time and T is the parameter
which measure the value of waiting costs.
The supply side of the market is formed by firms which can produce several
differentiated products. Products are represented by a point in the product space
(product can be interpreted as a train leaving in a given time of a day). The firm has to
pay fixed costs F for each product located in the product space. Once a firm locates its
products at some points in the product space, it sets prices for each product. Firm faces
zero marginal costs. Hence, firm producing M products obtains profit Π𝑖 =
∑𝑀
𝑗=1 𝑝𝑗 𝐷𝑗 − 𝐹𝑀 , where Dj is the number of consumers demanding product j. The model
is solved in two extreme versions. The monopoly version of the model assumes that
there is only one firm in the market. The competitive version of the model assumes that
there is a free entry into the industry but each firm is allowed to produce only one
product in one location. This assumption seems to be very restrictive but it does not
change equilibrium values as long as the closest neighbor of each product is not
produced by the same firm.
2. Equilibrium
This section describes the solution of the model. First consider competitive version of
the model. The model can be solved by backward induction. Suppose that N firms
entered the market in the first stage of the game. The assumption of convex waiting
costs ensures that firms choose to be located equidistant from one another on the circle.
The position of the consumer that is indifferent between purchasing from firm i and
1
purchasing from i’s closest neighbor 𝑥̃ is given by the following condition 𝑥̃ = 2𝑁 +
𝑁(𝑃−𝑃𝑖 )
2𝑇
, where P denotes the price of firm’s closest neighbor. Expected profit of each
firm can then be written as a price times expected demand minus fixed costs.
𝑥̃
Π𝑖 = 2𝑃𝑖 ∫ 1 −
0
𝑃𝑖 + 𝑇𝑥 2
𝑑𝑥 − 𝐹
𝑅̅
By maximizing expected profit with respect to price Pi, we obtain the best response
function 𝑃𝑖 = 𝑃𝑖 (𝑃). Because firms are located symmetrically on the circle, it is
reasonable to look for symmetric Nash equilibrium. The equilibrium price is therefore
given by the following condition 𝑃𝑖∗ = 𝑃𝑖 (𝑃𝑖∗ ). In the first stage of the model each firm
chooses whether to enter the market or not. Equilibrium number of firms is determined
by the zero profit condition for existing firms.
Π𝑖 = 𝑃𝑖∗
1
−𝐹 =0
𝑁
Now consider monopoly version of the model. Because waiting cost are convex
monopole maximizes its profit by locating the products equidistantly on the circle.
Prices of all products will be the same because the situation is symmetric. Therefore the
1
indifferent consumer lies in the middle of each two products, i. e. 𝑥̃ = 2𝑁. The expected
profit of monopoly firm is given by the following expression.
Π𝑖 = 2𝑁𝑃 ∫
1
2𝑁
0
1−
𝑃 + 𝑇𝑥 2
𝑑𝑥 − 𝑁𝐹
𝑅̅
Monopoly firm maximizes its expected profit with respect to its price P and number of
firms N.
3. Welfare analysis
Generally, it is possible to obtain only numerical but not analytical solution of the
model. Therefore, this section provides analytical solution and welfare comparison in
the simple case when the consumers do not differ in their reservation prices R. In order
to do this, the competitive version and monopoly version of the model are compared
with the allocation selected by social planner. Benevolent social planner chooses such
allocation which maximizes total surplus. Total surplus is given as follows.
𝑇𝑆 = 2𝑁 ∫
1
2𝑁
𝑅 − 𝑃 − 𝑥 2 𝑇 𝑑𝑥 + 2𝑁 (𝑃
0
1
𝑇
) − 𝑁𝐹 = 𝑅 −
− 𝑁𝐹
2𝑁
12𝑁 2
As we can see total surplus does not depend on the product price. In this case, when
consumers have the same reservation price and each consumes only one unit, the price
determines only the distribution between consumer and producer surplus but it
introduces no distortion. Total surplus depends on the sum of the fixed costs and
consumers’ waiting costs. The social planner thus chooses optimal number products N*.
Moreover total surplus is increasing in number of products if 𝑁 < 𝑁 ∗ and decreasing in
the number of products if 𝑁 > 𝑁 ∗ . Table 1 shows equilibrium values for all versions of
the model.
TAB. 1: Equilibrium values
Competitive version Monopoly version Social optimum
Price
Number of firms
𝑝∗ =
𝑇
𝑁2
3 𝑇
𝑁∗ = √
𝐹
𝑝∗ = 𝑅̅ −
3
𝑁∗ = √
𝑇
4𝑁 2
𝑝∗ ∈ (0, 𝑅̅ )
𝑇
2𝐹
𝑁∗ = √
3
𝑇
6𝐹
Source: Author’s own calculations
The results show that market allocation generates too many products in both versions of
the model. This implies that excessive entry is the main problem for an industry
regulator in the case where consumers do not differ in their reservation prices. The
results also show that number of firs is smaller for monopoly version of the model
compared to competitive version. If the regulator’s only instrument is the choice of the
type of competition, then it is optimal to choose competitive tendering instead of open
access. But this conclusion does not have to be true if the consumers differ in their
reservation prices. In that case product price excludes some consumers from the market
and causes deadweight lost. The deadweight lost is smaller in competitive version of the
model. So, generally it is ambiguous which type of competition is welfare superior.
Conclusion
The paper presents a basic framework for welfare evaluation of different types of
competition on market for passenger transport services. The framework is based on
game theoretical models of product differentiation. The modeling framework has some
important advantages. First, it incorporates some realistic demand characteristics.
Second, prices and number of products are endogenously determined within the model
by Nash equilibrium conditions. The model was used to compare open access and
competitive tendering when consumers have the same surplus from outside option. The
model shows that both regimes generate too many firms in relation to social optimum
but competitive tendering is closer to social optimum than open access.
References:
[1] CER. Public service rail transport in the European Union: An overview. Brusel,
2011, 185 p.
[2] JOHNSON, D., NASH, C. Competition and the provision of rail passenger services:
A simulation exercise. Journal of Rail Transport Planning & Management, 2012,
vol. 2, no. 1, pp. 14-22, ISSN 2210-9706
[3] JOHNSON, D., NASH, C., Charging for scarce rail capacity in Britain: a case
study. Review of Network Economics, 2008, vol. 7, no.1, pp. 53-76, ISSN 14469022
[4] SALOP, S. C. Monopolistic competition with outside good. The Bell Journal of
Economics, 1979, vol. 10, no. 1, pp. 141–156, ISSN 0361-915X
Ing. Rostislav Staněk
Ekonomicko-správní fakulta MU
Lipová 41a
63900, Brno