This paper investigates the properties of two types of cost restrictions that guarantee the existence of an equilibrium in pure strategies in Bayesian spatial competition models with heterogenous firms.Localized competition; market effciency, cost heterogeneity.

Claudio A. Piga

Marco Alderighi

English

Research Papers in Economics

    
     
ISSN 1750-4171 
 
        
 
DEPARTMENT OF ECONOMICS 
 
DISCUSSION PAPER SERIES 
 
 
On cost restrictions in spatial competition models 
with heterogeneous firms 
 
Marco Alderighi 
 
 
Claudio A. Piga 
 
 
WP 2009 - 09 
 
 
 
 
 
 
 
 
Department of Economics 
Loughborough University 
Loughborough  
LE11 3TU  United Kingdom 
Tel:  + 44 (0) 1509 222701 
Fax: + 44 (0) 1509 223910 
http://www.lboro.ac.uk/departments/ec 
          On cost restrictions in spatial competition models with
heterogeneous rms
Marco Alderighia
Universit a della Valle d'Aosta, Italy.
Universit a Bocconi, Milano, Italy.
Claudio A. Pigab
Loughborough University, Loughborough, UK.
RCEA, Rimini, Italy.
July 6, 2009
Abstract
This paper investigates the properties of two types of cost restrictions that guaran-
tee the existence of an equilibrium in pure strategies in Bayesian spatial competition
models with heterogenous rms.
1 Introduction
Aghion and Schankerman (2004) and Syverson (2004) were the rst to investigate the
properties of circular city models  a la Salop (1979) with asymmetric costs. These two
articles assume a very similar Bayesian set-up (i.e. the rivals' costs are unknown) but
adopt dierent restrictions to deal with the fact that when cost asymmetry is large with
respect to the transport cost parameter, a pure-strategy price equilibrium may not exist.1
On the one hand, the restriction in Syverson (2004) is such that there always exists
an indierent consumer located between two neighboring producers, i.e. a high cost rm
facing tough competition from two low-cost neighbouring rivals has a positive market share
in any state of the world. On the other, Aghion and Schankerman (2004, AS hereafter) use
a weaker restriction, which only requires that a high-cost rm has, on average, a positive
market share.
This paper develops a theoretical analysis aimed at identifying the economic implica-
tions associated with the use of dierent restrictions on cost asymmetry in spatial competi-
tion models. To this purpose, we adopt the same theoretical set-up as in AS to investigate
the properties of the two types of cost restrictions we focus on. The rst type arises when
aE-mail: m.alderighi@univda.it.
bCorresponding Author. E-mail: c.a.g.piga@lboro.ac.uk
1Existence problems in localized competition models date back to d'Aspremont et. al. (1979), where it
is shown that in the Hotelling set-up a pure-strategy equilibrium fails to exist when rms are too close.
1the `no mill-price undercutting' condition holds (Eaton and Lipsey, 1978). This condition
implies that rms commit not to undercut their opponents, i.e. that even if protable,
rms would not choose to charge a price that may completely exclude a rival from the
market. Our analysis is further extended by deriving the second type of cost restrictions
without relying on rms' commitment, in line with the approaches in Vogel (2008) and
Alderighi and Piga (2008).
Our main result is that the second restriction type imposes a stricter limit on the
amount of cost asymmetry that is consistent with the existence of an equilibrium. Fur-
thermore, we show that the restriction in AS is too weak even relative to our rst type, so
that in that model with positive probability a high-cost rm may have (ex-post) a negative
market share. We obtain that after imposing the rst restriction type, some of their policy
recommendations are not supported by their analysis.
2 The set-up
We shall rst recall AS's model and notations. On a circumference of unitary length,
n evenly located rms produce a homogeneous good. Firms have either a high unit cost
cH or a low unit cost cL; q denotes the probability of drawing a high-cost rm; cH, cL
and q are common knowledge. Firms know their, but not their rivals', type. On the same
circumference, a continuum of uniformly distributed consumers has mass normalized to
1. A generic consumer receives a utility u = v   pi   tdi if she buys one unit of the good
from rm i, where v is the product's gross valuation, pi is the mill price charged by rm
i, t is the unit transport cost and di is the shortest distance of the consumer from rm i.
Let pi, pi 1 and pi+1 denote the prices charged by rm i and its two closest rivals. The
marginal consumer between i and i + 1 (i   1) lies at a distance xR (xL) from rm i:
xR (pi+1;pi) =
1
2n
+
1
2t
(pi+1   pi) (1)
xL (pi 1;pi) =
1
2n
+
1
2t
(pi 1   pi) (2)
The expected prot of rm i is given by: i = (pi   ci)(ExL (pi 1;pi) + ExR (pi+1;pi)).
Following AS, the expected market share of rm i, conditional on all other low- (high-)
cost rms charging the same price pL (pH), is given by:
Di (pH;pL;pi) = ExL + ExR =
1
n
+
1
t
(qpH + (1   q)pL   pi). (3)
2Symmetry and rms' prot maximizing behaviour imply that equilibrium prices and ex-
pected market shares are:2
pH =
t
n
+ cH  
1
2
(1   q)c (4)
pL =
t
n
+ cL +
1
2
qc (5)
DH =
1
n
 
1
2t
(1   q)c (6)
DL =
1
n
+
1
2t
qc, (7)
where c = cH   cL and pH   pL = c=2.
3 On cost restrictions with commitment
In this Section, we show that the condition presented in AS generally allows an excessive
amount of cost asymmetry. We start by assuming that rms commit not to undercut rivals
(Eaton and Lipsey, 1978):
Condition 1 (No mill-price undercutting) pH   pL < t=n.
To guarantee that in the AS solution both types of rms have ex-post positive market
shares, i.e. xL;xR  0, it is therefore necessary that the AS price solution satises
Condition 1. Therefore, we replace (4) and (5) into Condition 1, obtaining:
c  2
t
n
. (8)
In AS, the relevant boundary condition ensuring that (4) and (5) represent a price equi-
librium is given by DH  0 (see p. 804), which implies:
c  2
1
1   q
t
n
, (9)
i.e. the degree of cost asymmetry that is allowed in the AS set-up is proportional to the
parameter t=n, but it can be arbitrarily large when q is close to 1. Thus, (9) is weaker than
(8). We delve deeper into the possible implications of adopting (9) instead of (8) by rst
demonstrating that DH > 0 is a weighted average of positive and negative `quantities'.
Then, we show that when xL < 0 and/or xR < 0, rm i's `real' market share should be
2In AS, the expressions equivalent to (4) and (5) are mis-typed; indeed, the subsequent equations, e.g.,
(6) and (7), are identical to the ones we present here.
3null, and therefore DH > 0 in (3) is misspecied (i.e. positive and negative quantities do
not cancel out).
We begin by decomposing DH into three parts in accordance with the number of low-
cost neighbours a high-cost rm may potentially face. We focus on the parameters' range
for which (9), but not (8), is satised, i.e.: 2 t
n < c  2 t
n
1
1 q:
1. with probability (1   q)
2, rm i has two low-cost neighbours; therefore using (1) and
(2) we obtain xL (pL;pH), xR (pL;pH) < 0 and xL + xR = 1=n   c=2t < 0;
2. with probability 2(1   q)q, rm i has one low-cost (e.g. i   1) and one high-cost
neighbour (e.g. i + 1), therefore xL < 0 and xR > 0 (or viceversa), with xL + xR =
(1=n   c=4t) T 0. (This case is represented in Figure 1)
3. with probability q2, rm i has no low-cost neighbours and xL + xR = 1=n.
Adding up previous results yields the expected market share of a high-cost rm reported
in (6):
DH = (1   q)
2

1
n
 
1
2t
c

| {z }
+ 2q (1   q)

1
n
 
1
4t
c

| {z }
+ q2 1
n |{z}
 0.
< 0 T 0 > 0
(10)
Therefore, in AS, when rms' cost asymmetry is suciently large, the expected market
share of a high-cost rm is the weighted average of positive and negative market shares, i.e.,
with positive probability some rms will be ex-post selling strictly non-positive quantities.
Furthermore, when (8) is violated, (10) generally misrepresents the corrected expected
market share of the high-cost rm i. To show this, assume that (pure-strategy) equilibrium
prices exist and that they are given by (4) and (5). Moveover, consider dierent intervals
of cost asymmetry, indexed by k = 0;1;2;:::
2k
t
n
< c  2(k + 1)
t
n
,
where k represents the number of high-cost neighbours needed to shield rm i, i.e., to
guarantee that a high-cost rm is not undercut by a low-cost rm located too close.
Conditional on k high-cost rms being located at the right and k high-cost rms being
located at the left of i, a (k + 1)-step neighbour is high-cost with probability q, and it is
low-cost with probability (1   q). When a (k + 1)-step neighbour is low-cost, it undercuts
all the neighbours of rm i, and therefore it competes directly with rm i. Otherwise, if
a (k + 1)-step neighbour is high-cost, the 1-step neighbour of rm i has a positive market
share and competition takes place beween that 1-step neighbour and rm i. Hence:
41. with probability q2k, rm i is not undercut, and therefore its market share is (1   q)
 1+k
n   1
2tc

+ q 1
n, and
2. with probability 1   q2k, rm i is undercut and its market share is 0.
Thus, the expected market share of a high-cost rm is q2k  
(1   q)
 1+k
n   1
2tc

+ q 1
n

,
which coincides with (10) when k = 0, but generally diers from it, when k  1. Accord-
ingly, for k  1, (3) cannot be used to derive the expected market shares presented in (6)
and (7). Therefore (4) and (5) do not necessarily represent the equilibrium prices of the
game.
4 On some economic implications
Some of the economic policy implications in AS derived under (9) fail to hold when
(8) is imposed.3 The main point is given by the fact that AS analysis strongly relies on
the fact that with sucient heterogeneity, aggregate prot is increasing in the degree of
competition (i.e. 1=t). In particular, in Proposition 2.(iii), AS state that competition-
enhancing policies increase aggregate prots if
 c
2t
2
> 1
n2q(1 q) (see page 820 in AS).
Because 1
q(1 q) > 1, this condition violates (8), which if taken into account implies that
aggregate prot is always decreasing in the degree of competition.
Other propositions are aected by the erroneous boundary conditions. The argu-
ment parallels that for Proposition 2.(iii). An important implication of Proposition 3
hinges around the assumed divergence of interests between high- and low-cost rms,
i.e. @H=@t > 0 and @L=@t < 0. However, the latter requirement is equivalent to
c > 2t=(nq), a condition which never holds under (8). Therefore, the possibility of a
low-competition political economy trap is not sustained by the analysis. Moreover, in the
analysis of entry carried out in Section 4 of AS and summarised in Proposition 6, increas-
ing the degree of competition increases the probability of entry by a low-cost rm and
reduces that of entry by a high-cost one if @L=@t < 0, which, as discussed above, does
not hold under (8). Finally, in the simulation of welfare eects in Section 5, AS consider
some values that are out of the boundary conditions.
3To prove that AS results continue to hold when (8) is violated entails complex computations for
deriving a new price equilibrium, that due to the discontinuity and non quasi-concavity of the pay-o
functions is likely to be in mixed strategies (Dasgupta and Maskin, 1986). This is beyond the goals of this
work.
55 On the boundary conditions without commitment
Previous considerations have been derived while assuming that rms are committed to
Condition 1. As discussed in Alderighi and Piga (2008), if costs satisfy (8), but rms are
free to deviate from Condition 1, then computed prices are not necessarily an equilibrium,
since for some parameter ranges, a low-cost rm could protably deviate by undercutting
their closest rivals.
The following proposition guarantees the equilibrium existence without relying on the
`no mill-price undercutting' assumption.
Proposition 1 When q 2 (0;1), equations (4) and (5) represent a symmetric Bayesian
(pure-strategy) Nash price equilibrium if:
c  (q)
t
n
, (11)
where (q) =

q + q2 + 2   q
p
6q + q2 + 5

=
 
q   q3 + 1

2
 
4   2
p
3;2

and 0 (q) < 0.
Proof See Appendix.
Proposition 1 states that it is more likely to have an equilibrium when q is small. The
intuition follows from the fact that since protable deviations mainly occur when low-cost
rms try to undercut high-cost ones, when q is small a low-cost rm has a lower incentive
to deviate because: rst, the low-cost equilibrium price is smaller (with respect to the
case where q is large) and therefore undercutting yields a lower unit margin; second, the
chance of undercutting a high-cost rm and the size of its market share are smaller, so that
undercutting yields a smaller rise in expected market share. This argument also claries
why there are no existence problems in the standard circular model (Salop, 1979). In fact,
when costs are symmetric (c = 0), a pure-strategy equilibrium always exists as there are
no less-ecient rms to be undercut.
References
[1] Aghion, P. and Schankerman, M. (2004). `On the welfare eects and political economy
of competition-enhancing policies', Economic Journal, vol.114, pp. 800-24.
[2] Alderighi, M. and Piga, C.,The Circular City with Heterogeneous Firms (July, 13
2008). Available at SSRN: http://ssrn.com/abstract=1159381
[3] d'Aspremont C., Gabszewicz, J.J. and Thisse, J.-F. (1979). `On Hotelling's \Stability
in Competition"', Econometrica, vol. 47, pp. 1145-50.
6[4] Dasgupta, P and Maskin, E. (1986). `The existence of equilibrium in discontinuous
economic games, I: Theory', Review of Economic Studies, vol. 53, pp. 1-26.
[5] Eaton, C. B. and Lipsey, R. G. (1978). `Freedom of entry and the existence of pure
prot', Economic Journal, vol. 88, pp. 455-69.
[6] Salop, S. C. (1979). `Monopolistic competition with outside goods', Bell Journal of
Economics, vol. 10, pp. 141-156.
[7] Syverson, C. (2004). `Market structure and productivity: a concrete example', Journal
of Political Economy, vol. 112, pp. 1181-1222.
[8] Vogel, J. (2008). `Spatial competition with heterogeneous rms', Journal of Political
Economy, vol. 116, pp. 423-466.
Appendix
Proof of Proposition 1. To be an equilibrium, we check whether a player has no incentive
to deviate from its strategy. We focus on the situation in which the boundary condition (8) holds,
and we start analyzing the incentive for a rm to undercut its neighbours. Note that condition
(8) implies that by gradually lowering its price, a deviating rm rst undercuts 1-step high-cost
neighbours, then 1-step low-cost neighbours, afterwards, 2-step high-cost neighbours, and so on
and so forth.
A) We start by analysing the case in which rm i charges pi 2
 
pL   t
n;pH   t
n

, so that it
is able to undercut a 1-step neighbour only when the latter is a high-cost rm. In this case, it
is necessary to extend (3) to account for undercutting. Thus, when i + 1 is a low-cost rm, the
marginal consumer between i and i + 1 is located at the distance given by (1), and when i + 1 is
a high-cost rm, the marginal consumer is located between i and i + 2 at the distance of:
xRR (pi+2;pi) =
1
n
+
1
2t
(pi+2   pi).
The marginal consumer between i and i   2 is dened similarly: xLL (pj;pi) = xRR (pj;pi),
with j = L;H. By symmetry, the expected market share of rm i is therefore given by:
Di (pH;pL;pi) = q (q  2xRR (pH;pi) + (1   q)  2xRR (pL;pi)) + (1   q)  2xR (pL;pi)
= q

q

2
n
+
pH   pi
t

+ (1   q)

2
n
+
pL   pi
t

+ (1   q)

1
n
+
pL   pi
t

.
7We focus on the case that i is a low-cost rm, since in this case the incentive to deviate is
larger. Note that the prot function of i, (pi   cL)Di (pH;pL;pi), is increasing in pi, so that the
maximum deviation prot is reached when the rm sets pD = pH   t
n, and obtains a unit margin
pD   cL =
q+1
2 c. Thus, the expected demand is:
Di =
q + 2
n
  (1   q)(q + 1)
c
2t
.
And the expected prot is:
D =
q + 1
2
c

q + 2
n
  (1   q)(q + 1)
c
2t

. (12)
In order to have no incentive to deviate, ID = D   L < 0. Noting that L = t(DL)
2 and after
some simplications we obtain: ID =  1
t
  t
n
2
+ 1
2t
  t
n
 
q + q2 + 2

c   1
4t
 
1 + q   q3
(c)
2.
Therefore:
c <

1
q   q3 + 1

q + q2 + 2   q
p
6q + q2 + 5

t
n
. (13)
B) We now consider a case where a low-cost rm charges pDD 2
 
pH   2 t
n;pL   t
n

, so that
it is able to undercut both types of 1-step neighbour. Again, the prot function in the interval
is increasing in pDD, so that the maximum deviation prot is reached when the rm sets pDD =
pL   t
n, and obtains a unit margin pDD   cL =
q
2c. The expected market share is:
Di =
2
n
+
1
t

qpH + (1   q)pL   pL +
t
n

=
3
n
+
q
t
c
2
.
The incentive to deviate is hence IDD = DD   L =
q
2c
  3
n +
q
t
c
2

  t
  1
n + 1
2tqc
2
.
Therefore, there is no incentive to deviate when c  2
q
t
n, condition that is always satised when
(13) holds.
C) By assuming lower deviation prices, unit margins reduce so that rms have less incentive
to deviate. Conversely, it is easy to show that rms have no unilateral incentive to increase their
prices.
8Figure 1: Market shares as computed in AS and considering undercutting.
9

and Piga, C.,The Circular City with Heterogeneous Firms

Freedom of entry and the existence of pure pro

Market structure and productivity: a concrete example',

Monopolistic competition with outside goods',

On Hotelling's \Stability in Competition&quot;',

On the welfare eects and political economy of competition-enhancing policies',

Spatial competition with heterogeneous

On cost restrictions in spatial competition models with heterogeneous firms

http://www.lboro.ac.uk/departments/sbe/RePEc/lbo/lbowps/WP_cost_restrictions_2009_09.pdf

On cost restrictions in spatial competition models with heterogeneous firms

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics