; Cournot Duopoly Cournot Duopoly and Bertrand Duopoly Guilherme Carmona Winter 2007
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# Cournot Duopoly Cournot Duopoly and Bertrand Duopoly Guilherme Carmona Winter 2007

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```									   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

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