Econ 414, Daniel R. Vincent
Lecture Six
Market Games in
Quantities and
Prices
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Outline of Lecture: Prices
] Price competition is very different from quantity
competition.
] Description of duopoly competition in prices (Bertrand)
] Finding equilibrium in homogeneous products.
] Price competition when products are differentiated.
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Econ 414, Daniel R. Vincent
Price Competition Between
Two Firms
] Price competition is different from quantity
competition
] Price competition leads to marginal cost
pricing with as few as two firms
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Price competition between two firms
Market Quantity, Q = 130 - P
Market, Q = x1 + x2 + … + xn = ∑xi
Price vector, p = (p1, p2)
here p1 and p2 are firm 1’s and 2’s prices respectively
x1(p) is the demand facing firm 1
Firm 1’s profit, P1(p) = (p1 - c) x1(p)
Similarly, Firm 2’s profit, P 2(p) = (p2 - c) x2(p)
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Demand functions for the two firms
The demand curve for firm 1:
x1(p) = 130 - p1
= (130 - p1)/2
=0
when p1 < p2
when p1 = p2
when p1 > p2
The demand curve for firm 2:
x2(p) = 130 - p2
= (130 - p2)/2
=0
when p2 < p1
when p2 = p1
when p2 > p1
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Bertrand demand, firm 1
P1
x1 = 0
x1 = 65 - P1/2
P2
0
x1 = 130 - P1
0
130
x1
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Econ 414, Daniel R. Vincent
Bertrand Equilibrium
Suppose that prices can be set in units of
pennies.
If Firm 1 believes that Firm 2 is setting price
p2>c+.01, what is Firm 1’s Best Response?
Answer: Set p*1= p2-.01. Why?
If Firm 2 thinks Firm 1 is setting price
p1>c+.01, what is Firm 2’s Best Response?
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Bertrand Equilibrium
For every price above c+.01, each firm has an
incentive to undercut the other.
What are the only pair of prices where this
incentive is not present?
At p*1= p*2=c+.01.
The Bertrand Equilibrium price (with
homogeneous goods) is virtually the same as
the perfectly competitive price!
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Econ 414, Daniel R. Vincent
Bertrand Variations
] Bertrand equilibrium with a cost
advantage
] Bertrand equilibrium with many firms
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Bertrand Limit Theorem
When n is greater than or equal
to 2, all products are perfect
substitutes, and no firm has a
cost advantage, then the
Bertrand game equilibrium
implies that price equals
marginal cost
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Econ 414, Daniel R. Vincent
Market Games with
Differentiated Products
] Price and quantity competition when
products are differentiated
] Cournot and Bertrand equilibrium still
different, but the difference is muted
] Monopolistic competition as the limit of
market game equilibrium
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Differentiated Products
All differentiated products
have one thing in common:
if the price is slightly above
the average price in the
market, a firm doesn’t lose all
its sales
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Econ 414, Daniel R. Vincent
Two firms in a Bertrand competition
] The demand function faced by firm 1:
x1(p) = 180 - p1 - (p1 - average price)
x1(p) = 180 -1.5 p1 +.5p2
] The demand function faced by firm 2:
x2(p) = 180 -1.5 p2 +.5p1
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Two firms in a Bertrand competition
] Firm 1 has profits
P1(p1,p2) = (p1 - 20) x1
= (p1 - 20) (180 -1.5 p1 +.5p2 )
Firm 2’s profit function
P 2(p1,p2)= (p2 - 20) (180 - 1.5p2 +0.5p1)
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Econ 414, Daniel R. Vincent
Maximizing profits
Firm 1 maximizes its profits when its
marginal profit is zero:
0 = ∂P1/∂p1
= (p1 - 20) (-1.5) + (180 - 1.5p1 + 0.5p2)
⇒ 0 = 210 - 3p1 + 0.5p2
Firm 1’s best response function:
p1 = f1(p2) = 70 + p2/6
Similarly, Firm 2’s best response function:
p2 = f2(p1) = 70 + p1/6
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Bertrand best responses, two firms,
differentiated products
p2
p1 = f1(p2) = 70 + p2/6
p2 = f2(p1) = 70 + p1/6
p* = (84, 84)
p1
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Bertrand Equilibrium
] The Bertrand equilibrium of the market game
is located at (84, 84)
] The market price is $84, significantly higher
than the marginal cost which is given at $20
] Each firm sales (180 - 84) units = 116 units
] Each firms profit = (84 - 20) × 116
= $7424
] Therefore, each firm could spend over $7000
in differentiating its products and can still
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come out ahead
Bertrand competition with n firms
] Firm 1’s market demand
x1= 180 - p1 - (n/2) ( p1 - average price)
] Firm 1’s profit function
P1(p) = (p1 - 20) x1
] When the first order condition is satisfied
0 = ∂ P 1/∂p1 = (p1 - 20) (-K) + x1
⇒ K (p1 - 20) = 180 - p1 (+0 with symmetry)
where K = (n + 1)/2
∴ p1* = 180 /(K+1) + 20K/(K+1)
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Bertrand competition with infinite
number of firms
] n → ∞ ⇒ K → ∞ and 1/K → 0
] Taking limit of p1* as n goes to infinity
lim p1* = lim 20/(1 + 1/K) = 20
] In this limit, price is equal to marginal
cost and profits vanish.
] This limit is monopolistic competition
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Appendix. Uniqueness of
Equilibrium
] The contraction mapping theorem
] Best response mappings as contraction
mappings
] Best response mapping fixed points are
game equilibria
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