Cournot Duopoly and Bertrand Duopoly

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Cournot Duopoly(and Bertrand Duopoly)

• Guilherme Carmona
• Winter 2007

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History

• Antoine Augustin Cournot
• 1838 Recherches sur les principes mathématiques
  de la théorie des richesses
• First formal model of competition between few
  firms
• Assumption firms choose quantity
• Cournot equilibrium is NE of his game!

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A bit more history

• Bertrand (1883) criticized model
• Firms set prices!
• Controversy not yet finished
• Both models used in practice!
• Cournot model used more because better behaved

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Cournot Duopoly

• Two firms, 1 and 2, produce quantities q1 and q2
  of a homogenous good
• Production is simultaneous, and price is
  determined later when both firms throw their
  production on the market
• Cost functions C1 and C2
• Demand function Q D(p)Inverse demand function
  p P(Q) D-1(Q)

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As a game

• Description in normal form
• Players N 1,2
• Strategies qi ? Si 0,K , K gt 0
• Payoffs pi(qi,qj) P(qiqj)qi Ci(qi)
• Dutta pi(qi,qj) (a b(qiqj) c)qi
• Linear demand and cost

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Best Responses or Reaction Functions

• Observation pi(qi,qj) pi(qi,qj) ?qi ? Si
  if and only if qi ? argmaxqi pi(qi,qj)
• Define best response by ri(qj) argmaxqi
  pi(qi,qj) ?qj ? Sj(assuming a unique maximizer
  to simplify things)

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BR(2)

• Assume that p is twice continuously
  differentiable
• Necessary first-order condition?pi(ri(qj),qj)/?q
  i 0 ?qj ? Sj
• Second-order condition ?2pi(ri(qj),qj)/?qi2 lt 0
  ?qj ? Sj
• Best response in linear modelri(qj) (a c)/2b
  qj/2

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General method

• Nash equilibrium q1 r1(q2)
  q2 r2(q1)gt intersection of best responses
• With q (q1, q2) and r (r1, r2), this is
  fixed point q r(q)
• Allows the use of strong mathematical tools

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Nash equilibrium

• Linear model solve q1 (a-c)/2b q2/2
  q2 (a-c)/2b q1/2
• Or make use of symmetry q (a-c)/2b
  q/2
• In both cases q (a-c)/3b
• Between monopoly and competitive outcome

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More firms

• Assume n symmetric firms
• Let Q q1 qn, Q-i Q - qi
• Profits are pi(qi,Q-i) P(qiQ-i)qi
  C(qi)
• FOC Pqi P C 0
• Symmetric NE (P-C)/P 1/ne
• Lerner index decreases in n with more firms
  prices are lower!

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More firms (2)

• Symmetric equilibrium in linear model Q-i
  (n-1)qiand q (a-c)/2b (n-1)q/2
• Equilibrium valuesq (a-c)/b(n1), Q
  n(a-c)/b(n1)P (anc)/(n1), p
  (a-c)2/b(n1)2
• For n -gt 8, approaches competitive
  outcome

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Bertrand Duopoly

• Same model as Cournot, but strategies are prices
  pi ?0,8)
• Homogeneous goods consumers buy at firm with
  lowest price
• Caution many models of price competition used in
  practice assume differentiated goods in order to
  avoid discontinuous payoffs (as we will see in a
  moment). Are often also called Bertrand
  competition, where price competition with
  differentiated goods would be more precise.

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Bertrand Sharing rule and profits

• Problem need to decide what happens at equal
  price sharing rule (an additional assumption)
• Standard each gets half of the customers
• Profits, assuming linear cost (pi c)D(pi)
  if pi lt pj p (pi c)D(pi)/2 if pi
  pj 0 if pi gt pj

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BertrandNash equilibrium

• Problem profits are discontinuous! (and exactly
  where it counts)
• No best response exists!
• Can NE exist?
• Yes, because definition has nothing to do with
  BRs

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BertrandNash equilibrium (2)

• Show
• (c,c) is a NE by checking that nobody will
  deviate
• No other price pair is NE
• Strategy p c is weakly dominated!
• Famous Bertrand paradox zero profits
• Limit of discrete case with ever smaller
  smallest coin

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Readings and exercises

• Readings ch. 8, 28
• Exercises
• PS 1 6, 7, 12
• Dutta 7.6-7.8, 7.12, 7.13, 7.15