How to Curb the PriceCompetition Paradox

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How to Curb the Price-Competition Paradox Hans
Carlsson Department of Economics Lund
University February 6, 2008
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How to Curb the Price-Competition Paradox Hans
Carlsson Department of Economics, Lund
University February 6, 2008, Abstract A
well-known result says that a static duopoly
where both firms have constant unit costs and
compete in prices is characterized by a unique
Nash-equilibrium outcome in which the low-cost
firm serves the whole market at a price equal to
the other firms unit cost. In the symmetric-cost
case the market is split and both firms earn zero
profits. We argue that the result is based on an
inappropriate use of the Nash concept because the
equilibria involve the use of weak best replies.
To obtain a more satisfactory solution we look
for strategy sets which are closed under
undominated best replies (so-called curb sets,
see Basu and Weibull, 1991). We show there is a
unique minimal curb set in the asymmetric-cost
case. The solution is compatible with market
prices ranging from the high-cost firms unit
cost to the low-cost firms monopoly price. This
result remains valid in settings with increasing
or slightly decreasing returns to scale.
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• A well-known text-book story
• Model
• Two firms 1 and 2
• Constant unit costs c1 and c2 c2 ? c1 gt 0
• One-shot interaction firms choose prices
  simultaneously
• The entire demand goes to the firm charging the
  lowest price in case of a tie market demand is
  split equally.
• Predicted outcome
• If c1 lt c2, firm 1 gets the entire demand at a
  price equal to c2.
• In the special case c1 c2 c price will be
  equal to c and both firms make zero profits.

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Paradox in a market with only two firms none of
them will be able to make a profit. Difficult to
explain Why is the only rational action to
choose a price which for sure leads to zero
profits when a higher price may give a
profit? So Maybe we got it wrong.
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A digression Why not Bertrand model,
Bertrand paradox? Joseph Bertrands
contribution Bertrand, J (1883). "Revue de la
Theorie Mathematique de la Richesse Sociale et
des Recherches sur les Principles Mathematiques
de la Theorie des Richesses," Journal des
Savants 1883, pp. 499-508. The article covers
10 pages but only 10 lines with four sentences is
devoted to what is now referred to as the
Bertrand model.
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Such is the study, made in chapter VII, of
the struggle between two proprietors who own
mineral springs of the same quality and have no
fear of other competition. It would be in their
interest to join together in partnershipor at
least to fix a common price so as to get from the
buyers the greatest possible profit, but this
solution is rejected. Cournot con-jectures that
one of the competitors will lower his price to
attract buyers, and that the other, in order to
bring them back, will lower his more. They will
continue until each of them would no longer gain
anything more by lowering his price. A peremptory
objection arises With this hypothesis a solution
is impossible the price reduction would have no
limit. In fact, whatever jointly determined price
were adopted, if only one of the com-petitors
lowers his, he gains, disregarding all
unimportant exceptions, all the sales, and he
will double his returns if his competitor allows
him to do so. If Cournots formula masks this
result, it is because, through a peculiar
oversight, he introduces under the names D and D
the quanti-ties sold by the two competitors, and
treating them as independent vari-ables, he
assumes that the one quantity happening to change
through the will of one owner, the other would
remain constant. The contrary is ob-viously true.
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So what is Bertrands contribution? Idea of
price competition though he didnt seem to like
it (he thought collusion was more likely).
For discussions about Bertrands contribution,
see for instance Dimand, R. W. and M. H. I.
Dore (1999), "Cournot, Bertrand, and game
theory A further note", Atlantic Economic
Journal 27, pp. 325-333. What usage is
acceptable? Bertrand model, Bertrand
paradox OK Bertrand competition Why?
Bertrands model No! Bertrand theory or
even worse Bertrands theory Outrageous!
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• Attitudes to the paradox
• Usual reaction
• Paradoxical outcome is the solution to the
  model.
• Look for more sensible outcomes in variants
  (repeated interaction, differentiated goods,
  capacity constraints)
• I will argue
• It is legitimate to question the correctness of
  the solution to the original model.
• A Nash equilibrium even if unique is not
  necessarily a satisfactory solution.
• In the price competition model Nash equilibrium
  is weak meaning at least one player has best
  replies other than the equilibrium strategy.

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Digression 2 I O people who apply game
theory often have too much faith in the notion
of Nash equilibrium. As supporting evidence,
consider this quote from Bresnahans (1981)
seminal paper Oligopoly models are examples
of what game theorists call Nash equilibrium.
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Requiring strict incentives The problem with
weak NE suggests this condition () A
solution should contain all best replies to all
strategy combinations allowed by the
solution. Clearly A solution concept
satisfying () and existence in all games has to
be set-valued. The following claim is
immediate If a solution is a product set of
strategies and satisfies (), then it is a curb
set.
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Curb and curb sets (Basu and Weibull,
1991) Consider an n-player game with pure
strategy sets Si, i 1, , n. For a mixed
strategy combination ?-i BRi(?-i) is set
of best replies to ?-i Let T T1 ?? Tn, Ti
? Si, be a product set of strategies. BRi(T-i)
is set of best replies to all ?-i ? ?j?i
?(Tj) Definition (curb) A product set of
strategies, T T1 ?? Tn, is closed under best
replies (curb) if, for all i, BRi(T-i) ?
Ti. A curb set is minimal if it does not contain
a smaller curb set.
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To get sharper predictions we exclude play of
dominated strategies. ?i(T-i) is
undominated best replies to T-i. Definition
(curb) A product set of strategies, T , is
curb if, for all i, ?i(T-i) ? Ti. A curb
set is minimal if it does not contain a smaller
curb set.
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• A simple linear model
• Two firms 1 and 2
• Constant unit costs c1 and c2 c2 ? c1 gt 0
• Demand D(p) a bp for some a gt c2 and b gt 0
• Profits to firm i
• (pi ci)D(pi) if pi lt pj
• ½(pi ci)D(pi) if pi pj
• 0 if pi gt pj .
  i, j 1, 2, i ? j.
• Notation pim is monopoly price ( (a
  ci)/2)
• (pim maximizes (p ci)D(p).)

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A complication best replies do not exist for
relevant ranges when prices can be chosen from a
real interval. Use a discrete model Assume
prices have to be integer multiples of some
monetary unit k gt 0 and study the model's
properties for small k. Notation G(k)
game with monetary unit k P(k) nkn?N
set of feasible prices ( pure strategies) in
G(k) pim(k) firm i's (lowest) monopoly price
in G(k) (notice that ?pim(k) pim? lt k)
ci(k) lowest feasible price strictly above ci
(assume k is small enough so that ci(k) lt
pim(k) Pi(k) set of strategies in P(k) that
are undominated for firm i Clearly Pi(k)
p ? P(k) ci(k) ? p ? pim(k)
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Results (Nash equilibria) Notation For a
mixed strategy si ? ?(P(k)) with support supp si,
si- min supp si si max supp
si. Proposition 1 (a) If c1 lt c2 and k is
sufficiently small, then in any Nash equilibrium
in undominated strategies player 1 plays the pure
strategy p1 c2(k) k. (b) If c1 c2 and s is
a Nash equilibrium in undominated strategies,
then si- ? c(k) and si ? c k, i 1,
2. Corollary 1 The Nash equilibrium analysis
implies that as k goes to zero market price
converges to c2 in the asymmetric case and to c
in the symmetric case.
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Results (minimal curb sets) Proposition 2
(a) If c1 lt c2 and k is sufficiently small, then
G(k) has a unique minimal curb set V(k). The
components of V(k) are given by V1(k) p1
? P1(k) c2(k) k ? p1 ? p1m(k) V2(k)
P2(k) ( p2 ? P2(k) c2(k) ? p2 ?
p2m(k)). (b) If c1 c2 and V is a minimal curb
set of G(k), then V c(k)?c(k) or V c(k)
k?c(k) k. Corollary 2 If k is
sufficiently small, the unique minimal curb set
V(k) in the asymmetric case is compatible with
any feasible market price in the interval c2(k)
k, p1m(k). In the symmetric case as k goes to
zero any market price compatible with a minimal
curb set converges to c in the symmetric case.
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Proof Conclusion Paradox seems highly
non-generic from curb perspective. Before
assessing the significance of this result I will
address a possible objection CRS can be
considered non-generic I will show results do
not critically depend on that assumption.
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• Constant and increasing returns to scale
• Two firms 1 and 2
• Increasing, concave cost functions C1 and C2
  for all q, C1(q) ? C2(q)
• Demand decreasing function D ? ? ?.
• We assume each firm has a unique monopoly price
  pim.
• pim(k) i's (lowest) monopoly price in G(k).
• We let pi- denote the highest lower bound on the
  prices at which firm i may make a profit
• pi- inf p ? ? pD(p) gt Ci(D(p))
• pi-(k) the lowest feasible price in G(k)
  strictly above pi-.

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Results (Nash equilibria) Proposition 1 (a)
For any p ? p2-, there exists k(p) gt 0 such that
if k lt k(p) and (s1, s2) is a Nash
equilibrium, then min si lt p. (b) If
p1- lt p2- and k is sufficiently small, then in
any Nash equilibrium (s1, s2), si ? p2-.
Corollary 1 The Nash equilibrium analysis
implies that for small k market price lies in the
interval c1, c2 in the asymmetric case and
close to c in the symmetric case.
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Results (minimal curb sets) Proposition 2
If p1- lt p2- and k is sufficiently small, then
G(k) has a unique minimal curb set V(k). The
components of V(k) have the following
properties min V1(k) max p ? P1(k) p ?
p2-, max V1(k) p1m(k), and V2(k)
P2(k). Recall Pi(k) is the set of
undominated pure strategies for i. Corollary 2
If k is sufficiently small, the unique minimal
curb set V(k) in the asymmetric case allows
market prices ranging from c2 or below to p1m(k).
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• Slightly decreasing returns to scale
• Results
• Limit Nash equilibrium properties unknown
• Minimal curb as above.

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