; Cournot Duopoly Cournot Duopoly and Bertrand Duopoly Guilherme Carmona Winter 2007
Documents
Resources
Learning Center
Upload
Plans & pricing Sign in
Sign Out
Your Federal Quarterly Tax Payments are due April 15th Get Help Now >>

Cournot Duopoly Cournot Duopoly and Bertrand Duopoly Guilherme Carmona Winter 2007

VIEWS: 78 PAGES: 16

  • pg 1
									   Cournot Duopoly
(and Bertrand Duopoly)
     Guilherme Carmona
        Winter 2007
               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!
       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”
       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)
              As a game
•   Description in normal form
•   Players: N = {1,2}
•   Strategies: qi  Si = [0,K] , K > 0
•   Payoffs: πi(qi,qj) = P(qi+qj)qi – Ci(qi)
•   Dutta: πi(qi,qj) = (a – b(qi+qj) – c)qi
    – Linear demand and cost
     Best Responses or
     Reaction Functions
• Observation:
    πi(qi*,qj*) ≥ πi(qi,qj*) qi  Si
  if and only if
       qi*  argmaxqi πi(qi,qj*)
• Define best response by
    ri(qj) = argmaxqi πi(qi,qj) qj  Sj
  (assuming a unique maximizer to
          simplify things)
               BR(2)
• Assume that π is twice continuously
  differentiable
• Necessary first-order condition:
  ∂πi(ri(qj),qj)/∂qi = 0 qj  Sj
• Second-order condition:
   ∂2πi(ri(qj),qj)/∂qi2 < 0 qj  Sj
• Best response in linear model:
  ri(qj) = (a –c)/2b – qj/2
       General method
Nash equilibrium:     q1* = r1(q2*)
                      q2* = r2(q1*)
  => 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
     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”
           More firms
• Assume n symmetric firms
• Let Q = q1 +…+ qn, Q-i = Q - qi
• Profits are
      πi(qi,Q-i) = P(qi+Q-i)qi – C(qi)
• FOC: P’qi + P – C’ = 0
• Symmetric NE: (P-C’)/P = 1/nε
  – Lerner index decreases in n: with more
      firms prices are lower!
       More firms (2)
• Symmetric equilibrium in linear
  model: Q-i = (n-1)qi
  and q = (a-c)/2b – (n-1)q/2
• Equilibrium values:
  q = (a-c)/b(n+1), Q = n(a-c)/b(n+1)
  P = (a+nc)/(n+1), π = (a-c)2/b(n+1)2
• For n -> ∞, approaches “competitive
       outcome”
      Bertrand Duopoly
• Same model as Cournot, but strategies are
  prices pi [0,∞)
• 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.
  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 < pj
       π = (pi – c)D(pi)/2 if pi = pj
          0                 if pi > pj
         Bertrand:
      Nash 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 BR’s
        Bertrand
   Nash 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”
 Readings and exercises
• Readings: ch. 8, 28
• Exercises:
  – PS 1: 6, 7, 12
  – Dutta 7.6-7.8, 7.12, 7.13, 7.15

								
To top