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Review of Economic Studies (1987) LIV, 4762
? 1987 The Society for Economic Analysis Limited
00346527/87/00040047$02.00
Approximate Bertrand
Replicated
Equilibria
in
a
Industry
HUW DIXON
Birkbeck College
First version received October 1984; final version accepted June 1986 (eds)
This paper considers the existence and properties of approximate Bertrand equilibria in a
replicated industry. Price setting firms produce a homogeneous product with weakly convex costs.
The main results are that: (a) Given E > 0, an Eequilibrium exists if the industry is large enough;
(b) If the e is small enough, and the industry large enough, any Eequilibrium is approximately
competitive. These results depend on how contingent demand is specified.
During a price war between two petrol stations in Winnipeg, Mr Hafy Carnet
reduced his gasoline price from 50 cents to 10 cents a litre, whereupon Mrs Sharon
Willard, his neighbour, cut her gasoline to 1.6 cents a litre. Police were called when,
having lost three hundred customers, Mr Carnet, "who completely forgot the rules
of the market", announced through a loudhailer that he would pay 3 cents to
anyone who filled their tank at his pumps.
Toronto Star 30.8.1983
Pricesetting is the institutional form of pricing in many sectors of industrialised
economies. This paper develops a framework which generalises the models of price
competition associated with Bertrand (1883) and Edgeworth (1897). Essentially such
models describe a trading process in an industry producing a homogeneous product where
firms set prices which perfectly informed households treat parametrically (see Allen and
Hellwig (1982,1983), d'Aspremont and Gabszewicz (1980), Brock and Scheinkman (1985),
Dasgupta and Maskin (1986), Dixon (1984a), Kreps and Scheinkman (1983), Levitan
and Shubik (1972), Osborne and Pitchik (1985), Shapley (1957), and Shubik (1955,1959)).
These papers have retained one of Edgeworth's basic assumptions, namely that firms
have constant average costs up to capacity (exceptions being Shapley (1957), Shubik
(1958), Dixon (1984a)). The main purpose of this paper is to show that this simple
BertrandEdgeworth framework can be considerably generalised.
The model presented in this paper has two main features of interest. Firstly, we
make a very general assumption about firms' cost functions, namely that firms have
continuous and (weakly) convex total costs. This assumption embraces the Bertrand case
(constant average costs), the Edgeworthian case, and the more orthodox case of strictly
convex costs. Secondly, we assume that there are strictly positive lumpsum costs of
decision and priceadjustment. The presence of such costs is certainly very plausible. It
also solves a major conceptual difficulty within such pricesetting modelsthe problems
associated with the nonexistence of pure strategy equilibria and the recourse to less
plausible mixedstrategy solutions.
This paper does not model costs of decision and adjustment explicitly. Rather, we
employ the solution concept of an epsilonequilibrium or "approximate equilibrium".
An ?equilibrium occurs when each agent is within e of his best payoff given the actions
47
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48 REVIEW OF ECONOMIC STUDIES
of the other agents (we shall be restricting ourselves to the case of Nash Eequilibria).
We employ the Eequilibrium concept because it gives our results a greater generality
than if we explicitly modelled costs of decision and adjustment. Whilst such costs are
the most natural interpretation of the E, we could also interpret it as relevant due to
bounded rationality in a more general sense.
The results in this paper concern the existence of approximate Bertrand equilibia,
and their approximation to the competitive equilibrium when we replicate the industry.
The starting point of our analysis is Theorem 1 (nonexistence), which shows that with
strictly convex costs, the standard assumptions that demand and cost functions posses
bounded derivatives imply that no strict equilibrium will exist.' Turning to approximate
equilibria, Theorem 2 (existence) shows that given E > 0 (no matter how small), if the
industry is large enough, then an Eequilibrium will exist. Theorem 3 (approximation)
demonstrates that if E is small enough, and the industry large enough, then any
Eequilibrium will be approximately competitive.
It needs to be made clear at the outset that the Theorem 2 (existence) holds only
for one specification of contingent demand, "CCD" defined in Assumption A3 below,
which originates in Levitan and Shubik (1972). In the literature there are two types of
contingent demand functions used, which give the demand for a higher priced firm when
demand has been partly satisfied at lower prices. Under CCD the higher priced firm
faces the industry demand less the quantity supplied by lower priced firms. Under the
alternative FCFS (see Assumption A4) contingent demand is the proportion of industry
demand left unserved by lower priced firms (this originates in Edgeworth (1897). The
difference between these specifications can be seen as arising from different rationing
mechanisms that operate when there is excess demand for a lower priced seller (see Dixon
(1986)). Under CCD all households receive a proportion of what they want: under FCFS
a proportion of households receive all they want, and the rest nothing. The two
specifications of demand have very different properties.2 Theorem 2 works for CCD
because as the industry is replicated, the firm's contingent demand becomes infinitely
elastic at the competitive price, and hence the incentive for any one firm to deviate from
the competitive price tends to zero. Under FCFS, however, replication does not affect
the contingent demand at the competitive price (Proposition 1), so that even in a very
large industry the firm will have the same incentive to deviate from the competitive price
as in a small industry.
We see three main reasons why these results are important. First, they generalise
the BertrandEdgeworth framework, indicating that it can be applied to cases where firms
have nontrivial cost functions. Secondly, the model presents a possible account of how
a competitive equilibrium might occur. If we recall Arrow's Paradox, in a competitive
market all agents are assumed "pricetakers", yet the price is assumed to be "flexible"
and attain the equilibrium value (Arrow (1959)). The model presented in this paper gives
an account of how pricesetting firms might come to behave as if they were pricetakers
(see Dixon (1982)). In a large industry with small costs of decision and price adjustment,
we can use the Theorems of existence and approximation to give an account of how the
competitive equilibrium can come about. Thirdly, price dispersion is the typical case in
an approximate Bertrand equilibrium. Even in many markets which would commonly
be accepted as "competitive", there is rarely only one price charged. The presence of
lump sum costs of decision and price adjustment thus provides a simple and intuitively
satisfying account of such "limited" price dispersion. This complements the standard
accounts of price dispersion due to consumer search costs (as in Salop and Stiglitz (1977),
Sadanand and Wilde (1982) inter alia).
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DIXON APPROXIMATE BERTRAND EQUILIBRIA
49
The closest results to this paper concern mixedstrategy equilibria in the Bertrand
Edgeworth framework. Shubik argued for the general BertrandEdgeworth case with
mixedstrategies that as the number of firms tended to infinity, the expected price each
firm sets will tend to the competitive price (1959, p. 123, 1955). This result has been
further investigated by Allen and Hellwig (1986). Our result is rather different from this,
since the Approximation Theorem can apply to small industries, although it is only in
large industries that we can be confident that an eequilibrium exists when e is small.
The most important difference is that in this paper, "approximation" means that each
firm sets its price close to the competitive price. With mixedstrategies only the expected
price converges as the industry is replicated: AllenHellwig (1986) show that even in
large industries firms will set prices that are not close to the competitive price with a
positive probability. Our results also differ from the convergence of CournotNash
equilibria to the competitive equilibrium (see Novshek and Sonnenschein (1983), Roberts
(1980) inter alia). This literature considers the approximation of strict Cournot equilibria
to competitive outcomes in general equilibrium economies. In the CournotNash
framework, of course, the Law of one price is imposed via the inverse demand correspon
dence.
1. THE GENERAL BERTRANDEDGEWORTH FRAMEWORK
Without replication there is a set g of n firms producing a homogenous product. If the
industry is replicated r times, each firm in g is replicated r times, the resulting set being
denoted gr.
Each firm i e gr is free to set its own price pi E [0, oo), the rn vector of prices
being denoted p.
Assumption Al. Costs. Each firm i E g has a total cost function ci: R+ + R+, which
is strictly increasing, continuous and (weakly) convex in output xi.
From the cost function we derive the firm's supplycorrespondence si (pi), and
corresponding "supply functions" Si and oi which are derived by taking the supremum
and infimum of the correspondence:
Definition 1. (a) Supplycorrespondence si: R+ R.
si (p) = argmaxX,E[O,10] p
(b) Supply functions:
Xi
Ci (Xi)
Si(p)
oi (p) = min {s: s E si (p)}.
=sup {s:
sE Si (p)}
We do not impose an upper bound on xi because this would be inappropriate in the
context of replication, and include oo as a possible value for xi (by convention ci(oo)=
Lim ci as xi+ oo). From Definition 1(a) note that si is a closed, convex valued, upper
hemicontinuous correspondence, and si is monotonic (let q > p, then for any Sq ,si (q),
s E si (p), sq ' sP) Hence si is multivalued only at a countable number of points, as
depicted in Figure 1. Given the properties of si, both oi and Si are nondecreasing,
continuous almost everywhere, whilst Si is rightcontinuous and leftuppersemi
continuous, and oi leftcontinuous and rightlowersemicontinuous.3 Also 0(p) ' S(q)
whenever p > q. We make the assumption that firms have "lexicographic preferences"
in the sense that they prefer to produce the largest of any outputs yielding the same profit.
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50 REVIEW OF ECONOMIC STUDIES
Price
S (p)
Output
FIGURE 1
Supply correspondence
This is a reasonable assumption, and one that is implicit in the standard treatment of the
Bertrand model with constant costs,4 although it is not necessary for our results.5 Under
lexicographic preferences the firm's desired trade is therefore given by the supply function
Si (p).
The industry demand function gives the demand when all firms set the same price:
Assumption A2. Industry demand. F: R+ > [0, H].
demand is given by F(p), where:
Without replication, industry
(a) F(0) =
(b) there exists p* E [0, ao) such that: p* = sup {p E R+: F(p) > 0}
(c) F is continuous, nonincreasing, and strictly decreasing when strictly positive.
T, where E is positive and finite
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The assumption of continuity (c) can be relaxed: all that is required for the results in
this paper is that industry demand is leftcontinuous. This paper includes little explicit
analysis of demand, but it can be pictured as arising from a continuum of perfectly
informed nonstrategic consumers.
Assumptions 12 are together very general, and embrace the usual special cases dealt
with in the literature. These include the standard Bertrand case where there is constant
average costs for any positive output, and the Edgeworthian case of constant costs up to
6
capacity.
If the industry is replicated r times, then the industry demand becomes r a F(p) and
the set of firms gr. The competitive price 0 is unaffected by replication, and is defined:
0 =definf {p: F(p) Iieg Si (P) < O}
(1)
which is illustrated in Figure 2. To avoid trivialities, we assume that for all firms Si(0) > 0,
Price
P*
E~~~~~~~~~~~~~~~~
Si(p)
g
e~~~~~~~~~~~~~~~~~~~
(p)
Output
FiGURE 2
The competitive price
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52 REVIEW OF ECONOMIC STUDIES
and that p* > 0 > 0. It is important to note that from (1) and Definition 1(b):
F(O) Y ,g Si(0)'O_F(O)ic
(2)
In order to model pricesetting behaviour, we need to know the demand facing firm
j given the prices and quantities offered by other firms, the firm's contingent demand
function. In general contingent demand depends on prices set and quantities offered by
firms. However, since we assume firms have lexicographic preferences the quantity offered
becomes a function of priceif firm j has set price pj, he will offer to sell up to Si(pi).
Contingent demand for j in an rreplica industry can then be written as a function of
prices only:
Definition 2. Contingent Demand for Firm j, dj: Rm > R,.
drj = drj(p).
We have analysed the possible specifications of contingent demand in some detail
elsewhere (Dixon (1986)). We shall employ the two standard specifications found in the
literature. They both assume perfectly informed consumers who buy from the lowest
priced firm that will sell to them. They differ over how the demand for a higher priced
firm is determined when households have already purchased from a lower priced firm.
The first, Firstcomefirstserved (FCFS), originates in Edgeworth (1897), and has
been used more recently in Allen and Hellwig (1983, 1984), Bekman (1965), and Dasgupta
and Maskin (1986). The most natural interpretation of this specification is that if there
is excess demand for the output of a lower priced firm(s), then a subset of consumers
are sold as much as they want, and hence have no residual demand for higher priced
firms (we ignore any marginal customer who is only partly satisfied). The demand for
higherpriced firms is given by the sum of the demand functions of the customers
unsatisfied at lower prices. If consumers are heterogeneous, then the contingent demand
under FCFS will be random, depending on who has been served first. We assume that
demand is nonrandom, which can be justified by the assumption that household preferen
ces are identical and homothetic (Dixon 1986).
The second specification, Compensated Contingent Demand (CCD), originates in
Levitan and Shubik (1972), and has been used by Brock and Scheinkman (1985), Kreps
and Scheinkman (1983), Osborne and Pitchik (1985). Under CCD the demand for the
higher priced firms is the industry demand minus the total quantity sold by lower priced
firms. This is what the contingent demand would be if there was only one household in
the market, and any "income effects" from lower priced purchases where compensated.
With many identical households it can be seen as arising from an "equal shares" rationing
rule, which means that if there is excess demand for a firm(s), then each household
receives an equal share of the available output. This can be generalized to heterogeneous
households if there is a proportionalrationing scheme (as employed by Dubey (1982))
households receive a certain proportion of their demand, this being determined so as to
equate purchases with supply at that price. Of course, proportional rationing is manipu
lable, since any household will receive more if it asks for more.7 Alternatively, households
with unit demand functions and "reservation price rationing" (households with higher
reservation prices are served first) give rise to CCD (Gelman and Salop (1983)).
With excess supply the exact method by which demand is allocated amongst firms
setting the same price is not important for our results. The existing literature tends to
make very specific assumptions about this. However, since households are indifferent
between firms setting the same price, there is no obvious general reason for a specific
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rule (Dixon (1986)). We simply require the contingent demand for any firm j to lie
between an upper bound V,j (based on the optimistic assumption that households will
shop at j first), and a lower bound Wrj (based on the pessimistic assumption that
households shop at j last). If no other firms are setting the same price as j, then Vr, = W,j.
The only conditions that we impose on contingent demand functions drj are that (a)
contingent demand for j lies between Wrj and Vtj; (b) the contingent demands of firms
setting the same price "add up" so that only one side of the market is rationed and there
is voluntary trading;9 (c) demand for one firm is positive iff it is positive for all firms
setting the same price.
Assumption A3. Compensated Contingent Demand CCD. Let i, j E g,.
Define
W,j (p) =max [0,r *F( pj)Y, p,cp Si( Pi)]
V,j (p) = max [0, r * F(pj P>
(a) da (p) e [ Wj (p), Vj (p)]
(b) Adding up: E j min [Si(pi), dri(p)] = min [Vj(p), SPi=pJ Si(pi)]
(c) If Pi = pj, dri>O iff drj>O.
In order to specify contingent demand under FCFS, first define:
pi <P, Si (pi)
kri (pi) = Si (pi)/ r * F(pi)
Krj (p) = max [0, 1 Ep< pi kri (pi)]
kri represents the proportion of the total industry demand at pi that firm i can satisfy.
K,, gives the proportion of consumers left over once they have tried to purchase from
lower priced firms.
Assumption A4. First Come First Serve. Let i j e g,, i 0 j.
Define
Wrj(p) = max [0, Krj(p) * r * F(pj) p4
Vrj(p) = Krj(p) * r F(pj).
pj Si(pi)]
(a) drj (p) E I Wri (p) I Vr (P)]
(b) Adding up: 4P= mi [S1, dri] = mi [Vq(p), Z,=
(c) As in Assumption A3(c)
Under both specifications of contingent demand, drj will be left lower semicontinuous
in own price pj. With either specification, if the output of firms setting some price, p say,
is positive, then the outputs of all firms setting prices below p are given by Si (pi).
Having defined the firm's demand and outputs as a function of prices set, we can
now define the payoffs as functions of prices set. Profits are of course pj * xj cj (xj). The
profit function j: R+ > [O,ao] we define as:
Si (pi)]
Cj (Pj) def Pj sj (pj) cj (sj (pj))
(3)
The payoff function for the model is then r,r: R+n >R+
{ p4 (pi)
~pj d.j(p)  cj(drj (p))
if
otherwise
dr
(p)o .
oj (p)
(4)
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Thus we have a game where firms' payoffs depend only on prices, which inverts the logic
of the CournotNash model, where the inverse industry demand curve makes the market
price a function of quantities, and hence profits depend only on quantities. We now define:
Definition 3. eequilibrium.7'
no firm j E gr and no price q e [0, ao) such that:
Let e
O. If p e R" is an eequilibrium, there exists
Xrj (q, pj)  rj ( p) >?
or equivalently:
sup Trj (q, p)
Trj (p)
In an eequilibrium, should one exist, we have an rn vector of prices set by firms.
For large E any priceconfiguration may be an equilibrium. The smaller ? becomes, the
more priceconfigurations are ruled out. We are not assuming that firms are required to
maintain fixed prices over the trading period, as in Benassy (1976) and Iwai (1982, pp.
1415). Firms are free to change prices whenever they wish to do so. Given the lump
sum costs of decision and adjustment, however, it is optimal expost for firms to keep
their prices fixed in equilibrium. Thus in the model presented here both prices and
quantities are flexible. There is of course an asymmetry between prices and quantities
in this model, since the costs of varying output are continuous and convex by Assumption
1, whilst the cost of changing price are discontinuous and nonconvex.
2. EPSILONEQUILIBRIA IN A REPLICATED INDUSTRY
Before considering the general existence of ?equilibria, we shall first consider the problem
of nonexistence of strict equilibria.
Theorem 1 (Nonexistence).
A4. Let r E z+, and rn > 1. If F is differentiable, F' is boundedfrom below and cj are strictly
convex and differentiable, then no strict equilibrium exists.
Assume Assumption A12 and either Assumption A3 or
Proof Note that if c; are strictly convex then sj are single valued and bounded
functions for pj on [0, p*]. As is well known, the only strict purestrategy equilibrium
possible is 0 where all firms set the competitive price (Shubik (1959, p. 100 Theorem 2)).
To see that 0 is not an equilibrium, consider the contingent demand for a firm raising
his price to q,. Under CCD we have:
Xrj (q,, a<j) = qJ
r
[ F(qJ) ? Y,j si (0)] cj (r F(q,)  E i,jsj S(0))
Hence:
ar,11aq,=(q,
 cj)  r* Ft+ Si(0).
Evaluated at q, = 0, given that F' is bounded, a rj/Oq, = sj(O)> 0. This is, of course,
exactly analogous to Hotelling's Lemma, and holds for much the same reason. Since the
output sj (0) is optimal for q4 = 0, and 0 = c', a small increase in q, from 0 has no firstorder
effect on profits. Since ci is strictly convex, c! is also bounded over some suitably chosen
closed interval, [0, si (P*)] say. The argument also holds for FCFS.
Theorem 1 suggests that existence will be due either to some kinkiness or weak
convexity in the cost function, or a demand curve which is horizontal around O."
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We shall now consider the general properties of equilibria under replication when
The first result is that for any ? >0, no matter how small, under CCD an
eequilibrium will exist if the economy is sufficiently large.
?>0.
Theorem 2 ((Existence).
Under Assumption A1A3 (CCD), let
?r
definf {?: an sequilibrium exists in an rreplica industry}
For large r ?r is well defined, and as r tends to 00, ?r tends to 0.
Proof Let ? > 0. Define qrj as the price above which firm j's contingent demand
is zero, given that the other firms (i $j, i E gr) set the competitive price:
qrj = inf {q: drj(q, Oj) = O}
We now show that under CCD, qrj  0 as r e 00. For r 2, either:
(a) r F(0)
ioj Si(0)O
or
(b) r F(O),6j
If (a) holds (as in the Bertrand case), then qrj = 6. If (b) holds then q,j> 0. Let
A > 0. There exists ro such that for r > ro, qrj < 0 + A. To see why we can expand the sum
of other firms desired outputs at 0:
Si(0) _ O.
r F(q,j)= r Zirg Si(0)Sj(0)
(i=jifapplicable)
so that
F (qrj)  , Eg Si (0)
Si (o).
(5)
r
But from (1), S,ig Si(0)_ F(G), so that from (5):
F(qrj)  F(0)'1
 Sj(0)/r.
(6)
Since F is continuous and strictly decreasing when positive, we can choose r large enough
so that the R.H.S. of (6) is small enough to ensure that qrj < 0 + A.
Given that qrj tends to 0, the firms incentive to defect from 0 will also tend to 0.
Either Sj (0) is bounded from above by some A> 0, or it is not. If Sj (0) is unbounded,
then qrj  0 for r 2. Hence:
SUpq?o JTrj(q, Oj) Ir,,(P)
= 0.
(7)
If Sj(0) is bounded, then:
SUpq? oirj(q,
0j)
Irj()=(qrj
)
A.
(8)
Hence as r *0 and qrj > 0, from (7) and (8):
SUp [i7r,,( q, 0 <)ir,,(O8) ]
? II
The intuition behind Theorem 2 is that as the industry is replicated, whilst the industry
elasticity of demand remains constant, the firm's elasticity tends to infinity around 0.
Under CCD the slope of the contingent demand function for pj> 0 (when positive)
becomes r times the slope of F, whilst the size of the firm remains constant. This is
depicted in Figure 3 for the case where Sj (0) is bounded. If we interpret CCD as arising
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Price
qrj
d.
qr
q. j
\
' j
\
~~~r'
> r
s (0)
Output
FIGURE 3
Contingent demnand around 0 as the industry is replicated
from an equalshares or proportionalrationing scheme, the result is also intuitive. In a
large industry starting from 0, when firm j raises his price the reduction in total supply
at 0 is relatively small. Under equal shares or Proportional rationing, each individual
household will only be affected slightly by this reduction, since it is spread over all
households. By continuity, the additional amount any householder will be willing to pay
to satisfy his frustrated demand will be very small. Hence the contingent demand facing
firm j will be almost perfectly elastic.
Theorem 2 is very specific in the sense that the proof only deals with the competitive
price. Theorem 3 embraces the set of all approximate equilibria: if ? is small enough,
and the industry large enough, then in any ?equilibrium in which all firms have positive
output, all prices set will be close to the competitive price. To see why we need to impose
the restriction that all firms have positive outputs, consider the standard Betrand model
where all firms have the same constant costs with no capacity limit. If there are more
than two firms, then any price vector is a strict equilibrium so long as two firms set the
competitive price. Since all firms earn zero profits whatever price is charged, even for
? = 0 there exist equilibria with firms setting prices arbitrarily far from 0 with zero output.
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Theorem 3 (Approximation). Assumption A12, A3 or A4. Let A >0. There exists
E > 0 rO > 0, such that if p E R+n is an Eequilibrium, and x >> 0, then
Ipj0I<A
foralljegr,
r>ro.
Proof See appendix. II
Taken together, the existence and approximation Theorems 2 and 3 are very general.
They imply that in a "large" industry with small costs of decision and price adjustment,
the trading process represented by Assumptions A14 will give rise to an outcome that
is almost competitive. The approximation will consist both in the fact that the prices set
are close to 0, and that industry output is close to its competitive level. Approximation
Theorem 3 also implies that any price dispersion will be small. How large is "large" in
this context is not theoretically determinable, depending on such factors as the elasticity
of industry demand and the firm's cost functions. In the Betrand case of identical firms
with constant returns to scale a strict equilibrium exists with two firms, and both firms
set 0.
The Existence Theorem 2 holds only for CCD, not for FCFS. For FCFS, the elasticity
of demand may be unaffected by replication:
Proposition 1. Assumptions A12, A4. If allfirms have strictly convex costs, there
exist constants Mj _0 such that for all r, j E gr,
sup Jrj (q, 0.j) j (O) = Mj
Proof Given that other firms set 0, consider j's contingent demand if it sets price
q e (0, p*]. If all firms have strictly convex costs, then sj is single valued, and from
Definition 3, F(0) =gsi
(0).
drj(q, O&)=max [O, 1(F07l
. r. F(q)
=S s(t0)  F(q)l F(O) > O
(q < p*).
This is unaffected by replication.

The reason behind this result is very simple. Consider the initial position where all
firms are setting the competitive price, 0. If costs are strictly convex then all firms are
supply constrained, so that irj (0) = (j (0). Consider a Nash deviant raising his price
above 0. Under FCFS his contingent demand is given by consumers left unserved by the
other firms. The number of households left over will be exactly sufficient to demand all
the Nash deviant's supply sj(O) were he to set 0. But this number is totally unaffected
by replication, since sj (0) is unaffected. So no matter how large the industry, given O<,
the Nash deviant will face the same sized captive market of consumers unable to buy
from the lower priced firms.
APPENDIX: PROOF OF THEOREM 3
We first need some more definitions and notation. Consider the subset B of types of firms whose supplies oa
are bounded for prices in the interval [, p*]: B ={i E g: oi (p) is bounded for p E [O, p*]}. Then define the
output r, for each firm in B:
ri = max [oi(p*), E]
(F(O) =E by Assumption A2)
and let Y = maxi,B ri if B nonempty, E otherwise.
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In our proof we shall employ truncated supply functions to avoid algebraic difficulties when oj is
unbounded. Define cr*: R+ + [0, Y] where oa(p) =def min [Y, or (p)]. Given the properties of oi, cr* is bounded,
nondecreasing, leftcontinuous, and rightlowersemicontinuous. Corresponding to this truncated supply
function we define the truncated profit function 6*(p) = p  cr(p)  cj (cr*(p)).
Four Lemmas will be employed in the Theorem. The first Lemma concerns the truncated excessdemand
function F(p)  gr*(p)
which is strictly decreasing when F(p)>0,
semicontinuous. All results hold for CCD and FCFS.
and is leftcontinuous and rightupper
Lemma 1. Let A > 0. There exists 8 > 0 s.t. if F(p) ,g o4 (p) > 8 then:
p < 0 + A.
Proof. It follows immediately from leftcontinuity, strong monotonicity and the fact that F(O) 
Ego(0)_0.
11
Lemma 2 (Undercutting Lemma). Let p E R'.
e gr, Pi ' p may include j) then
If for some p E R+ and some j E gr Yp,p
xi'
o1(p)(i
sup rrrj(q, p1j)
j 7(p)(q e R+)q.
Proof
For q < p, drj(q, p j) > p 2p Xi > oj*(p) _ oj*(q) so that 1Trj(q, pj)
ej(q).
Hence
SuPq.R+
1Trj(q,Pj)SUPq<p
rrj(q,pj)~?(p)
I
Lemma 3. Let 8>0,
some p and for some h c gr
p, x e R+. There exists s>O such that if p is an sequilibrium and x >> O, and for
supq1p lTrj(q, pj) _ 6*(p)
foralije
h
then: o7j*(pj)>o7j*(p)3/2n
whenever pj<pj
eh
Proof. Since p is an sequilibrium:
Trj(Pj, Pj) _j* (p)s
for all j e gr
A necessary condition for this to be satisfied for firms j e h, pj < p is that:
0 40Pj) > 0*( p)  8(al)
(to see why, note that for pj ? p, 7rj = ?j j*)
cr4 are leftcontinuous and nondecreasing, so that given 8 there exists y > O such that for pj < p.
oj*(p)j*(pj)
< 8/2n
whenever pp1 <y
(a2)
j is continuous and strictly increasing. Continuity of its inverse around j(p) ensures that given y > 0, there
exists s > 0 such that
IP pjj<
y
whenever l1(p) 6:Npj)
< ?.
(a3)
Hence by choosing 8 small enough so that (a3) is satisfied, (al) implies (a2) for firms j e h, pj 'p. 11
Lemma 4. Let r E R+, p e R.
Then for all j E gr:
suPq<e rrj(q, Pj) =j(
)
Proof. We shall prove this for CCD. Recall the lower bound Wrj in Assumption A3:
drj(q, pj)i'r*
F(q)Ep,*j
S(pi).
Since q < 0,
drj (q, pj) > r* F(O) , P<0 a(O) > oj (0)> oj(q).
Hence
supq<0Xrj(q,pj)=supq<Oej(q)j(0)=j().

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We now prove Theorem 3. First, we show that we can choose s and r so that all prices are below 0 + A.
Secondly, at the end of the proof we show that we can choose s so that all prices are above 0 A. The first
stage, of the proof deals with two exhaustive cases. In case 1 the output of firms setting the highest price Pm
is greater than Y, and then in case 2 the contrary.
Case 1
EP PM xiY.
(a4)
From (a4) and the definition of Y, we are able to employ Lemmas 13. From Lemma 2, if any firm j undercuts
Pm, then it can earn at least up to 67(Pm). If p is an sequilibrium, then:
frj (Pj, Pj)  j (Pm)s
Given Lemma 1 choose 8>0 such that F(p)Y,
cr*(p)> 8 implies p <0+ A. Given (a5), from Lemma 3,
there exists s > 0 such that for any j such that pj < pm,
forallj Egr.
c4(pj)> LT1(Pm)
8/2n,
p1<pm
(a6)
Turning to firms setting pj = pm, (a4) implies that xi must satisfy:
Pm xj cj(xj) _j (Pm)>.
(a7)
Under Assumption Al the L.H.S. of (a7) is continuous and increasing up to o7(pm), so by choosing s small
enough we can ensure that given 8 > 0,
xi > 0*(pm)83/2n,
Pi = Pm.
(a8)
Recalling the definition of CCD,
r F(pm)Epl=Pm xi+Epl<pm Si (pi)
P1=Pmxi+YPi<Pm a (Pi)(aO
(a)
From (a6) and (a8), by choosing E small enough, (alO) becomes:
r*F(pm)>.gr
[o*( Pmn)p8/2n]
(al 1)
which implies
F(pm)Yg
j*(Pm) >8/2
(a12)
so that from Lemma 1
Pm < 0+A.
This proves that when (a4) holds, we can choose E and r so that all prices are less than 0+A. Turning
to its contrary:
Case 2
z xi<Y
(a13)
Pi =Pm
In order to show that the Theorem holds in this case, we assume the contrary to derive a contradiction.
The relevant contrary is that no matter how small E > 0, and how large r, there exists an Eequilibrium such
that x >> 0 and pm ? 0 + A. We are then able to derive a contradiction, which happens to be that pm < 0+ A.
From (a9), since x >> 0, and by assumption pm ? 0 + A,
r* F(O+A)Y_Pi
<PM Si (pi)>
(a14)
(al4) enables us to derive a lower bound X > 0 on the output of firms setting Pm, and hence an upper bound
nm on the number of these firms.
Consider the contingent demand of any firm j if it sets its price at pj = 0+ (A/2). From (a14):
drj(0+A/2, pj)> r [F(0+A/2)F(0+A)]>0.
Define ro=Y/[F(0+A/2)F(0+A)].
For r > ro,
drj(0 + A/2, pj) > Y _ aj*(0 + A/2).
Thus, if p is an Eequilibrium, then
irj(pj, P_j) ej* (0 +A/2)  E.
(a15)
(a16)
(a17)
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60
REVIEW OF ECONOMIC STUDIES
(a17) provides a lower bound on the profits of all firms for r> ro, and hence a lower bound on their
outputs. Turning first to firms setting pj = pm, (a17) means that for small E we can define Xj > 0:
p* * Xj  cj(Xj) = Xj(0 +A/2)  E.
Define X = min Xj. From (a18) for r> rO and E small:
(a18)
O<X_X~,j_p=
xi<Y
foralljEgr.
(a19)
So that the number of firms setting Pm is less than nm:
nm = Y/X.
(a20)
Recalling (alO), we obtain:
r  F(pm)i Egr ort(Pi)+E P=Pm
r*(pi) < nm * Y < Y2/X and recalling (a19), (a21) becomes:
Xi EP=pm vi (Pi)
(a2l)
From (a20)
o
p
r*F(pm)>Z Egr vi (

)
2X
(a22)
By assumption cr*(pm) _ c(O +(A/3).
is continuous and strictly increasing, for E small enough (a14) implies pi _? 0+ A/3. Hence:
If pi < Pm, then xi = Si (pi), and iri (p) = {(pi) = *(pi) Since 6:*
xi =Si (pi) _ Si (0 +A/3) _ r*( + A/3)
Pi < Pm
(a23)
From (a22)(a23)
r*F(PM)>Eg ar*(0+A/3)
X
>r F(0)
X
x
)
Hence:
F( pm) F(O) >

(a24)
Since F is continuous and strictly decreasing, we can choose r large enough so that Pm  0< A, the desired
contradiction. Hence in both cases 1 and 2 we were able to choose E an r so that all prices set in an Eequilibrium
with positive output were less than A above the competitive price.
A lower bound on prices set is easily found using Lemma 4. Let A > Oconsider any firmj. Either ej() = O,
or 6j(0) > 0. If ej(0) = 0, then for all prices pj < 0, xj = oj (pj) = 0, which violates the positive output requirement.
If ej(0)>0, since p is an Eequilibrium:
supq,R+ 1rj(q, pj) _ supq<
Ij(q) j()
E
Since the inverse of ej is continuous, there exists E such that
pj> 0A
which completes the proof.
11
Acknowledgement. This is part of my Thesis (1984). I owe an unbounded debt of gratitude to my supervisor
Jim Mirrlees for his advice and guidance. The comments of the editors and referees were invaluable. The
financial assistance of the ESRC is acknowledged. Faults remain my own.
NOTES
1. Mixed strategies will exist in the special case of identical firms (see Dixon 1984a). However, we find
mixed strategies implausible in BertrandEdgeworth models. Since pricesetting is not irreversible, the no regret
property of purestrategy equilibria is particularly attractive.
2. The relative merits of CCD and FCFS are discussed in Dixon (1986) and Davidson and Deneker
(1982). When excess demand is anticipated (as in this model), we observe both types of rationing. CCD avoids
the costs of offending consumers by turning them away empty handed, whilst FCFS involves serving fewer people.
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DIXON
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61
3. Definitions of semicontinuity. Consider the function f: R' 3 R, and any sequence {xJ} such that
x,, )x, x,, and x in R'. f is upper semicontinuous if as n oo:
Limsup f(xn )
f(x).
f is lower semicontinuous if as n o oo:
Liminff(xn ) _?f(x).
4. The only reason that the competitive price is an equilibrium in the standard Bertrand model is that as
either firm considers raising its price above the competitive price, its demand becomes zero as the other firm
meets all demand, despite the fact that its profits remain zero.
5. We could augment the firms' strategies to include the amount that they are willing to trade up to at a
given price.
6. It may be objected that the Edgeworthian cost function is discontinuous at capacity (when it becomes
infinite). This does not matter here because by Assumption A2 firms will never set prices above p*. We can
always construct an artificial cost function with a kink at capacity that is sufficiently large to ensure that the
firm's supply does not exceed capacity for prices below p*.
7. Manipulability is only an insuperable problem with perfect informationwhen the household knows
firms' supply functions and industry demand. Otherwise the household knows only that it will receive some
unknown proportion of its expressed demand. Thus the risk of being given what it asks for may be sufficient
to place an upper bound on households expressed demands.
8. These include the usual "equal shares" assumption (Dasgupta and Maskin (1982), Dixon (1984a));
Allen and Hellwig (1983) assume that demands for firms are proportional to supplies; and Gelman and Salop's
(1983) lexicographic preference assumption makes consumers prefer an incumbent's output to the entrant's at
the same price.
9. "Adding up" here does not mean that the contingent demands for firms setting the same price must
sum to the contingent demand there would be if only one firm set the price (i.e. Vj). As noted in Dixon (1986),
contingent demands need not satisfy Walras' Law. To see this, consider a duopoly where both firms set the
same price and supply nothing: each firm's contingent demand will equal the industry demand, since that is
how much it could sell given that the other firm has zero supply. What do need to "add up" are the actual
trades, which depend on demands and supplies: this is what condition A3(b) captures (the L.H.S. are actual
trades, the R.H.S. the total trades determined by the min condition).
10. There are two meanings to the term "E" or "approximate" equilibrium. The alternative to definition
3 is that each players action is within E of his best response (the approximation is within the strategy space,
rather than payoffs).
11. For a rather ingenious example of existence due to demand being perfectly elastic at 0 see Shubik
(1955, p. 425425).
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