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Strategical Interactions Between Transport Operators With Different Goals
Derek J. Clark, Department of Economics and Management, NFH, University of Tromsø, N-9037,
Tromsø, Norway
Finn Jørgensen, Bodø Graduate School of Business, N-8049, Bodø, Norway,
Pål A. Pedersen, Bodø Graduate School of Business, N-8049, Bodø, Norway,
1
ABSTRACT
This paper analyses competition between transport firms that maximise a weighted combination of
profits, income, consumer surplus and a quantity measure. When firms are symmetric, and produce
a symmetrically differentiated service, we compare the equilibrium prices that arise from collusion,
Cournot and Bertrand competition. The results are discussed in the light of the firms' payoff
functions and the degree of substitutability or complementarity between the services. We find that,
when firms place sufficient weight on consumer surplus, the collusive price can be below that
arising from the oligopolistic solution. Atypically for models of oligopolistic competition this case
can yield prices from Bertrand competition that are above those that arise from quantity
competition.
Article title: Strategical Interactions …
2
1. INTRODUCTION
Even though economists have long debated the objective function of firms (see for example
Baumol, 1959, Osborne, 1964, and Williamson, 1974), these ideas have so far had minimal
influence on the presentation of the classic models of imperfect competition in ordinary economic
textbooks; examples are Varian (2003), Nicholson (2005) and Frank (2006). When showing the
different outcomes from quantity and price competition in duopoly models, the firms are assumed
to be profit-maximising entities. The consequences for the consumers and the total welfare loss of
monopoly or cartel solutions compared to duopoly competition are thus also estimated from
assuming profit maximising behaviour. This is also the case when analysing the outcomes of
incomplete competition between transport operators in particular, see Pedersen (1999) for a
review.
As far as the supply of scheduled transport services in Norway is concerned, duopoly models
assuming other goals for the transport operators than profit are, however, relevant for two major
reasons: First, on many routes one, two or three suppliers are commonplace. This makes models of
imperfect competition relevant. Second, at least in Norway, local industry and local authorities
have substantial equity interests in the transport companies. A reasonable assumption is that these
groups of owners are not only concerned about the firms' profit, but also about the standard of the
transport services which they offer. Furthermore, the managers of the companies have often some
power to pursue their own goals (Williamson, 1974) and they are interested in running large
companies and being popular among the public by offering good transport services. The latter is in
particular relevant when the leadership lives and works in the companies' licence areas. This
explains why models assuming other goals for the transport firms than profit maximising are of
interest. For a more thorough discussion of objective functions for bus transport operators, the
3
interested reader is referred to Nash (1978) and Glaister and Lewis (1978). Solvoll et al (1994),
Jørgensen and Pedersen (2004) and Jørgensen and Preston (2005) focus also on transport operators'
goals with special emphasis on the Norwegian transport sector.
With this background, the aim of this paper is to analyse strategic interactions between transport
companies when their objective functions are identical but comprise a weighted sum of profits,
revenue, consumer surplus and sales. More specifically, we want to infer how the transport
operators' weights on different goals influence prices and quantities in the transport markets when
earlier rivals begin to collude. The analysis is carried out for different degrees of substitutability or
complementarity in demand between the transport services. In this work we are thus not concerned
about the internal processes within the transport companies that may lead to other objectives than
profit maximization. We regard transport companies' utility functions as exogenous and do not
discuss, as for example in Sklivas (1987) and Fershtman and Judd (1987), how different reward
contracts between owners and managers may lead to non-profit maximising firms.
The further organisation of the paper is as follows. In section 2, we solve the equilibrium prices and
quantities under Cournot (quantity) and Bertrand (price) competition and when the operators
collude. In section 3 we focus on the consequences for the travellers of different competitive
situations in the transport markets by analysing how the differences between the collusive price on
one hand and the Cournot and Bertrand prices on the other hand depend on the weights the
operators place on different goals and the degree of substitutability or complementarity in demand
between the services. In order to get comparable mathematical solutions we assume that the
transport firms have identical cost structures. Lastly, the fourth section 4 summarises the most
4
important results and discusses them briefly in the light of the Norwegian transport policy. The
most critical assumptions of the model are also emphasized.
2. THE MODEL
2.1 Assumptions on costs and demand
Suppose there are two transport operators 1 and 2 who each produce their own version of a
transport service. For simplicity, we follow Singh and Vives (1984) by assuming that a
representative consumer has the following utility function based on the levels of use of the services
q1 and q 2
U (q1 , q2 )  q1  q2 
q
2
1
 2 gq1q2  q22
2

The representative consumer maximizes CS  U (q1 , q2 )  i21 pi qi . Inverse demand is then
symmetric and linear, given by
p1  1  q1  gq2
(1)
p2  1  q2  gq1
where the parameter g  (1,1) measures the degree of substitutability or complementarity
between the services. If g  1(1) then they are perfect substitutes (complements) and g  0 is the

case of independent markets. In equilibrium (denoted by ), consumer surplus will be represented
by
CS  
q12  q 22  2 gq1 q 2
2
The cost functions of the two firms are
C1 (q1 )  c1q1
C 2 ( q 2 )  c2 q 2
Profits for each firm are then simply given by
5
(2)
 1  (1  q1  gq2  c1 )q1
 2  (1  q2  gq1  c2 )q2
We allow firms to have several weighted goals:
M i  (1       ) i  Ri  CS  qi
(3)
where Ri  pi qi is the revenue accruing to firm i . The payoffs to firm i are thus a weighted sum
of profits, revenue, consumer surplus and a quantity of service measure. We assume that each firm
has the same weight on each component, and that each weight is positive. In particular,
1       0 .
Our choice of payoff functions for the transport operators is open to attack from two major sources:
first, we assume the operators have identical goal functions. In Norway, for example, the
proportions of shares held by local authorities, local businesses and outside private interests vary
considerable among the transport companies (Mathisen and Solvoll, 2005). Since these three
groups of owners are likely to put different weights on the goals specified in (3), the transport
companies' payoff functions may differ. The fact that the managers of the companies have some
power to pursue their own goals, also points in the direction of different goal functions. Second, we
assume that every transport operator is only concerned about total consumer surplus; that is the
sum of surplus which himself and his rival bring about. This is reasonable when the companies
serve the same population and when local authorities and local businesses have substantial equity
interests in them. These groups of owners are concerned about the wellbeing and travel
possibilities of all inhabitants. The more power the managers have, the more dubious is, however,
this assumption because their status and reputation among the inhabitants are dependent on how
travellers evaluate their services compared to those of rivals.
6
2.2 Market solutions for different competitive situations
2.2.1 Quantity competition
Suppose the firms maximize their profits by choice of the quantity variable. This leads to the
following reaction function for firm 1:
q1 
q 2 g (2    1)  c1 (1       )  1  
2  3  2
(4)
where the denominator is positive by the second order condition. Quantities are strategic
substitutes (complements) for 0  () g (2    1) . Simultaneous solution of the reaction
functions for each firm gives (C superscript indicates Cournot - quantity - competition)
ci 2  3  2 1         gc j 2    11       
qiC 
 (1   )2  3  2  g 2     1
3  2  2  g 2    12  3  2  g 2    1
(5)
where the stability condition makes the denominator negative.1 Equilibrium quantity is always
decreasing in own cost and in the rival's cost when quantities are strategic complements. In the case
of strategic substitutes, firm i will increase its production when the rival's marginal production
cost increases. For equilibrium quantity to be positive the firms' costs cannot be too different:
ci 
(1   )2  3  2  g 2    1  gc j 2    11       
2  3  2 1       
Given the existence conditions, it is straightforward to verify that quantity is decreasing in a firm's
own marginal cost, whilst it increases (decreases) in the marginal cost of the rival when quantities
are strategic substitutes (complements).
Equilibrium price is
7


ci 3  2  2  g 2 2    1 1       
piC 
 gc j     11       
 2  3  2  g 2    12  2  g  g  1
3  2  2  g 2    12  3  2  g 2    1
(6)
We see that a firm's own price is increasing (decreasing) in the marginal cost of the rival for
substitutes (complements) in demand since 1        0 implies that 0      1 . The
effect of a firm's own marginal cost on its price in equilibrium is determined to be
sign pci i  sign (2  3  2  g 2 2    1)
C
 0 since 2  3  2  0 by the second order
condition and g 2 2    1  0 if the bracket is positive. If on the other hand 2    1  0
then the whole expression is least likely to be positive for
g  {1,1} . Then
2  3  2  g 2 2    1  1      0 .
2.2.2 Price competition
Suppose now that the firms compete by setting prices strategically in the product market. The
reaction function of firm 1 can now be written as
p1 
(1  2 )(1  g )   (2  g )  c1 (1       )  gp2 (2    1)
2  3  2
(7)
where the denominator is positive by the second-order condition for a maximum.2 Solving the
reaction functions simultaneously yields the equilibrium prices as
ci 3  2  21         gc j 2    11       
piB 
 3  2  2  g 2    11  2  2  g 2    1
3  2  2  g 2    12  3  2  g 2    1
(8)
Here the B superscript refers to Bertrand (price) competition. The denominator is negative by the
stability condition.3 The derivative
pi B
ci
 0 is then straightforward to determine. Also
for strategic complements (substitutes).
8
pi B
c j
 ()0
Equilibrium quantities are given by
(1       )(ci (3  2  2  g 2 (2     1))
 gc j (1     ))
 (1   )(1  g )(3  2  2  g (2     1))
g  1g  1
qiB 
(9)
3  2  2  g 2    12  3  2  g 2    1
We see that
qi B
ci
 0 and
qi B
c j


0 for g 0 .


2.2.3 Collusion
In this case the firms maximize
M1  M 2  (1      )( 1   2 )   ( R1  R2 )  2CS   (q1  q2 )
Firm i would then optimally produce 4
qiCOLL 
(1       )(ci  gc j )  (1   )(1  g )
2(2    1)(1  g )(1  g )
(10)
at a price of
piCOLL 
3  2  1  ci (1       )
2(2    1)
(11)
Positive quantity (first inequality) and price (second inequality) are assured by
(1   )(1  g )  gc j 1       
1     
 ci 
3  2  1
(1       )
When 2    1  0 it follows from (10) that equilibrium quantity is always decreasing in own
cost and in the colluder's cost when they produce complementary services ( g  0 ). In the case of
substitutable services, the operator's own quantity decreases when the colluder's cost increase. The
equilibrium price is always increasing in own cost but uninfluenced by the colluder's cost.
9
3. COMPARATIVE ANALYSIS WITH SYMMETRIC FIRMS
3.1 Stability conditions
Having solved the general model for equilibrium prices and quantities, we proceed here to analyse
the effects of different goals on the behaviour of the firms in the transport sector. To isolate these
effects we assume in this section that the firms are identical, i.e. ci  c j  c . For the case of
symmetry, it is well known that a profit and income goal can give the same results as maximizing a
combination of profit and quantity for a judicious choice of the weighting parameters.5 Hence we
set   0 for the remainder of this section. Table 1 gives an overview of the conditions under which
the interior equilibria exist for the cases of Cournot and Bertrand competition and collusion.
Insert Table 1 here
The stability condition for the case of Cournot and Bertrand is shown in Figure 1 as the area under
the lower envelope of the curves. Note that the second order condition does not bind here. From
Figure 1 it is also clear that the second order condition on  is the strictest for the case of collusion.
In any comparison of these cases, this is the condition on  that will bind. Note that the maximum
level of cost that is commensurate with equilibrium is the same in each case, whereas the lowest
level varies.
Insert Figure 1 here
3.2 Comparing the collusive and the Cournot cases
The difference in price between the collusive and the Cournot cases can be written as
10
  p COLL  p C 
(   g (2  1))1    c(1     )
21  2 (3  2  g (2  1))
(12)
Given the existence conditions in Table 1, the denominator of  is negative. The conditions that
limit cost dictate that the second term in the numerator is positive and hence
sign   sign (   g (1  2 )) . If gˆ (  ) 

1  2
 () g then   ()0 . Note that gˆ (0)  0 so
that the collusive price is lower for complements in demand, and higher when the products are
substitutes when no weight is placed on consumer surplus in the firms' maximization. This is
intuitively reasonable since Cournot competitors produce too much in the case of substitutes and
too little for complements compared to the collusive solution. This is exacerbated when some of
the emphasis is placed on a revenue goal in addition to profit. Note that the sign of  is
independent of  .
Insert Figure 2 here
Figure 2 shows the areas in which the collusive price is below and above that set in the Cournot
1
1
case. Since gˆ ( )  1 , if firms place sufficient weight (over ) on consumer surplus then the price
3
3
falls under the collusive outcome whatever the value of g . When the products are sufficiently
substitutable, then a sufficiently low weight on consumer surplus leads to collusive firms
increasing their price above the Cournot case. This means that even in the case of substitutes in
demand, collusive behaviour may give rise to lower prices than Cournot competition. Such a case
appears when the weight on consumer surplus,  , is relatively high and/or the degree of
substitutability in demand, g , is relatively low. Note that the above results apply for any value of
.
11
Regarding the marginal effect on the price difference,  , when the weight on revenue,  ,
increases slightly, one can calculate

(   g (2  1))c

 21  2 (3  2  g (2  1))
(13)
which has the same sign as  . This implies that in all cases, irrespective of the sign of  , the
difference in price between the collusive and the Cournot case increases in absolute terms when
more weight is put on revenues. In other words the consequences for travellers when previous
rivals try to collude are larger the more weight the operators put on revenues. Furthermore, it
follows from (12) that

(1     )(   g (2  1))

c
21  2 (3  2  g (2  1))
(14)
implying that an increase in marginal costs will have the opposite effect to an increase in  , and an
increase in these costs will moderate the price consequences of collusion.
When g is marginally increased, one can calculate that
 (1   )(1    c(1     ))

0
g
g  3  2 g  22
(15)
This implies that the difference between the collusive price and the Cournot price is decreasing
when services become less complimentary, and increasing in g when   0 .
Since  is included both in the numerator and denominator in (12), the effect on  of changes in
 is more difficult to ascertain. If we, for simplicity assume that no weight is placed on revenue


(   0 ) it follows that sign  | 0  sign 4 g 2  2  4 g 2   g 2  4 g 2  2 g   2  4  2 . This
12



 

expression can be written as 4 2  4  1 g 2  4 2  2 g  4   2  2  ( g ,  ) which is
a strictly convex function in g since 4 2  4  1  0 for 0.5    0 . It is easily verified that
 (1,  )  0 and  (1,  )  0 for  in this interval, and hence

  0
|
 0 . This means that putting
more weight on consumer surplus makes the price difference  even more negative if it is negative
originally, and less positive if it is positive originally. Collusion between transport operators is thus
more beneficial for the travellers when the operators place more weight on consumer surplus and
provide complementary services at the outset. When they produce substitutable services, placing
more weight on consumer surplus will moderate the negative consequences to the consumers
provided that g is sufficiently high so that   0 .
3.3 Comparing the collusive and the Bertrand cases
In order to compare prices in the collusive and the Bertrand cases, we calculate the price difference
 , defined as
  p COLL  p B 
( g (1  2 )   )1    c(1     )
21  2 (2  3  g (2  1))
(16)
where the denominator is positive. Again   0 for g  gˆ ( ) as before. This means that we have
the same conclusions as above. Only in the case where the services are sufficiently substitutable,
combined with relatively low weight on consumer surplus will the collusive price be higher than
the Bertrand price.
The comparative static results can be determined as follows

( g (1  2 )   )c



0 for  0
 21  2 (2  3  g (2  1)) 

13
(17)
Hence a larger weight on revenue will always increase the absolute value of the price difference
between the collusive and the Bertrand price. Furthermore
  (1     )( g (1  2 )   )

c 21  2 (2  3  g (2  1))
which again has the opposite sign to


(18)
. One can also calculate that
 1  2 1    c(1     )

0
g
g  3  2 g  22
(19)
implying that sign g  sign g . The results here parallel those for the comparison of the collusive
and the Cournot cases.
When   0 ,
the

partial

effect
of


on
 

can
be
signed
unambiguously:

sign  | 0  sign 4  4 2  1 g 2  4 2  6  2 g  4   2  2 which is a strictly convex
function of g that takes negative values on the permitted intervals for g and  . Hence

  0
|
 0.
To sum up, the collusive price is lower than that for Bertrand competitors when the services are
complements and when the operators place sufficient weight on consumer surplus and produce
services with a limited degree of substitutability in demand. For purely profit maximizing firms
that run substitutable services, the collusive price is, however, always higher than that set by
Bertrand competitors. Moreover, the weights that the transport companies place on revenues (  ),
consumer surplus (  ), their marginal costs ( c ) and the degree of substitutability or
complementarity between the services ( g ) influences the difference in the collusion price and the
Bertrand price in the same directions as they influence the difference between the Cournot and
collusive prices.
14
3.4 Comparison of prices
Using (12) and (16), the difference in prices between the two types of competition can be
calculated to be
g    g (1  2 (1    c(1     ))
 p B  p C
2  3  g (2  1) 3  2  g (2  1)
(20)
The denominator is negative by the stability conditions. The third term in the numerator of (20) is
positive by the cost conditions in Table 1. When g  0 , we find that p C  () p B when
g  () gˆ ( ) , and when g  0 it is always the case that p C  p B . When the goal of the operators
is pure profit maximization, then we get the usual result that competition is more intense when
prices are chosen strategically (Bertrand). Introducing an additional goal of profit and revenue
maximization (   0,   0 ) does not change this qualitative conclusion, but the magnitude of the
difference between the Cournot and the Bertrand prices shrinks. When an additional goal involving
consumer surplus is taken into account (   0 ), then the traditional conclusion can be reversed
when the operators run substitute services to a limited degree, and sufficient weight is placed on
consumer surplus in the operators' goal functions. Prices will be expected to be higher in this case
when price is chosen strategically.
4. CONCLUDING REMARKS
The above results give rise to several comments as far as Norwegian transport policy is concerned.
Even though Norwegian politicians obviously believe in economies of scale and economies of
scope, they often react negatively when earlier rivals want to collude or merge. They fear that the
eventual positive scale and scope effects will be outweighed by abuse of monopoly power. This in
turn leads to higher prices, welfare loss for the travellers and less external efficiency of transport
services because the difference between actual fares and marginal cost increases. These points of
15
view are usually present regardless of whether the operators produce complementary or
substitutable services. Even though we can not lean on empirical evidence, our impression is that
Norwegian politicians are sceptical to collusion between transport companies no matter their
ownership structure.6
Some of the above views are not supported in our analysis. When the transport services are
complements in demand ( g  0 ), the prices will go down when operators start to collude, no
matter whether they compete in quantities or prices beforehand. The consequences on prices and
thereby the benefit to the consumers will be larger the more weight the operator puts on revenue,
the more complementary the services are and the lower the marginal cost. These conclusions hold
for any weights that the operators put on profit, revenue and consumer surplus. Furthermore, when
the firms place no weight on revenue (   0 ), an increase in their weights on consumer surplus ( ),
will strengthen the positive effects to the consumers of collusion. To encourage transport firms to
collude when they produce complementary services is thus sensible advice, in particular when the
degree of complementarity between the services is high and the firms' weight on consumers'
surplus is high. From the latter it follows that it is particularly beneficial to the consumers when
publicly owned firms start to collude. When the firms also focus on revenues (   0 ), it is,
however, ambiguous whether the benefits to the travellers of collusion are larger when the
companies place higher weight on consumer surplus then when they place low weight on this
factor.
When the transport firms produce substitute services ( g  0 ) and maximize profit (     0 ), the
collusion prices are always higher than when the firms compete in quantities and prices. The closer
substitutes the services are and the lower the production costs, the higher are the collusion prices
16
compared to the duopoly prices. The politicians' fears of purer transport services when earlier rivals
start to collude are thus, in this case, justified by our model's results.
In the case when the transport firms place weights on revenue and/or consumer surplus and still
produce substitute services, the results of collusion are, however, ambiguous. Whether the
collusion prices are lower or higher than the Cournot and Bertrand prices, depends on the degree of
substitutability between the services and the weight the operators put on consumer surplus (  ). If
the competition between the operators is not very high such that g is low and the operators put
sufficiently weight on consumer surplus (  is sufficiently high), the prices will also here be lower
in the collusion case than when they compete in quantities or prices. Collusion between firms that
are concerned about consumers' well-being (publicly-owned firms) and that do not compete
fiercely may, thus, increase travellers' welfare. On the other hand, when the firms compete strongly
beforehand and place minimal weights on consumer surplus, the travellers will lose when the firms
start to collude. Note that the above conclusions hold for any weight the transport operators put on
revenue, but the magnitude of changes in prices will depend on both  and marginal cost.
The analysis also shows the unusual result that the price set by Bertrand competitors can be higher
than Cournot prices when the operators run services that are demand substitutes to a limited degree
and place sufficient weight on consumer surplus. When the operators are pure profit maximizers or
produce purely complementary services, we obtain the usual result that the Bertrand price is lowest.
The above discussion regarding the relative magnitude of the Cournot and Bertrand prices suggests
that the welfare effects of collusion on consumers are ambiguous in general, and are dependent
upon how operators compete in advance of any collusion.
17
In summary when evaluating the possible effects on prices when firms start to collude, the
following information is important: (1) the degree of substitutability and complementarity in
demand between the services they produce; (2) the firms' payoff functions; that is the weights they
put on actual goals such as profits, revenue, consumer surplus etc (3) the firms' cost structure. In
our opinion none of these issues are seriously considered when Norwegian politicians and transport
planners evaluate the effects of collusion between transport firms.
Finally, we want to emphasise the most critical assumptions in the analysis: first we have restricted
our analysis to one particular utility function implying that inverse demand functions are
symmetric and linear. Second, we have assumed that each operator is concerned about total
consumer surplus. Third, our discussions in section 3 and 4 are based on the firms having identical
costs functions, even though they produce different transport services.
Despite these limitations, the paper has nevertheless established the feasibility of extending the
payoff function in duopoly models when aiming to analyse the effects of collusion in transport
markets under different competitive situations.
18
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Jørgensen, F. and Pedersen, P.A. (2004): Travel distance and optimal transport policy.
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19
Seade, J. (1980): On the stability of Cournot revisited, Journal of Economic Theory, 48, pp.
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20
Table 1: Existence conditions for the symmetric interior equilibrium
Stability
Cournot
Bertrand
Collusion
  min( 322gg , 322gg )
  min( 322gg , 322gg )
none
  23
  23
  12
Concavity
p  0 , q 0
1 
1 
 c  (1(2g))(11g )
1
1 
21
 c  (12 1 )
1
1 
 c  (13 1 )
FOOTNOTES
More specifically, we can determine that 3  2  2  g 2    1  0 and
2  3  2  g 2    1  0 . See Seade (1980) for a discussion of the stability conditions in
oligopoly models
2
Note that the conditions defining strategic complements and substitutes are the opposite of those
for quantity competition.
3
The stability conditions are the same as in the case of quantity competition.
4
Second order condition is 2    1  0 . The maximand is strictly concave except in the cases
g  {1,1} .
5
6
To be specific the equilibria will be the same in these cases for   1  c (1 ) .
Many Norwegian politicians, were, for example, sceptical, when the Scandinavian Airline
Systems (SAS) became the majority share holder in the rural air company (Widerøe) in the 1990s,
even though SAS and Widerøe to a large extent produce complementary services.
22
2 g
3  2g
β 1.0
2 g
3  2g
0.9
0.8
0.7
0.6
0.5
0.4
β=0.5
0.3
0.2
0.1
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Figure 1: Stability condition for Cournot and Bertrand competition
23
1.0
g
g
1.0
^g(β)
Δ>0
0.8
0.6
Δ<0
0.4
0.2
0.0
0.1
0.2
0.3
0.4
-0.2
-0.4
0.5
β
Δ<0
-0.6
-0.8
-1.0
Figure 2: Relationship between g and  yielding equal prices under Cournot and collusion
24