Industrial Organization
Imperfect Information in the Product Market (Chap. 3)
Philippe Choné, Philippe Février, Laurent Linnemer and
Thibaud Vergé
CREST-LEI
2008/09
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One of the most famous result in IO
Bertrand competition with homogenous goods
Bertrand, 1883 : results (symmetric case)
Unique equilibrium and Welfare is maximal
Price equal marginal cost : pi∗ = c, πi∗ = 0
Bertrand, 1883 : assumptions
Homogenous good, constant return to scale
All buyers are informed about prices
All firms are informed about all marginal costs
Symmetric firms (N ≥ 2)
No repeat purchase ; No capacity constraint
Asymmetric case (. . . )
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Outline
1
Price competition with uninformed consumers
2
Price competition with private information on cost
Fixed demand
Elastic demand
Effect of asymmetric information on prices
Introduction of uninformed consumers
3
Advertising competition
Same marginal cost
Marginal cost is private information
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Price competition with uninformed consumers
Varian (1980)
The model
Symmetric firms i = 1 to N ≥ 2
Constant marginal cost c
Firms simultaneously choose their prices
Unit mass of consumers unitary demands, gross utility v
Unit mass of consumers, each having D(p) (alter. ass.)
A fraction I of consumers observe all prices
A fraction U = 1 − I do not
Informed consumers buy from the lowest price firms
Uninformed consumers buy randomly
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Price competition with uninformed consumers
Varian (1980) : profit functions
Notation
Let p = (p1 , . . . , pN ) denote the list of prices
Let p−i denote the list of prices but pi
Let k (p) denote the number of lowest price firms
Profit functions (pi ≤ v )
(pi − c)U/N,
Πi (pi , p−i ) =
(pi − c) (U/N + I/k ) ,
if pi > minj pj
if pi = minj pj
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Price competition with uninformed consumers
Varian (1980) : equilibrium
Equilibrium
No pure strategy Nash equilibrium
A unique symmetric Nash equilibrium (mixed strategy)
Properties of the equilibrium : p ∈ p, p
i
p=v
ii
(p − c)(U/N + I) = (v − c)U/N
(p − c) U/N + I (1 − F (p))N−1 = (v − c)U/N
iii
Extension to D(p)
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Price competition with uninformed consumers
Varian (1980) : comparative static on U (here N = 2)
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Price competition with uninformed consumers
Varian (1980) : p for U = 0.5 as a function of c
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Price competition with uninformed consumers
Varian (1980) : shape of the density (here N = 3 and U = 0.125)
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Price competition with private information on cost
À la Hansen (1988) (Spulber, 1995), see also Bagwell-Wolinsky (2002)
The model : N ≥ 2 firms
Each firm i privately informed of its marginal cost θi
θi drawn from θ, θ
Either constant marginal cost ci = θi or C(q, θi )
Distribution function F (θ), density f (θ).
Firms simultaneously choose their prices
Unit mass of consumers, each having D(p)
or Unit mass of consumers unitary demands, gross utility v
A fraction I = 1 of consumers observe prices (Spulber)
A fraction I = 1 − U observe prices (as in Varian)
Informed consumers buy to the lowest price firms
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Price competition with private information on cost
À la Hansen (1988) (Spulber, 1995). The elementary case (fixed demand).
The model with unitary demand and ci ⇔first price auction
Symmetric equilibrium : p(θ) with p(.) %
Probability that θ is the lowest : G(θ) = (1 − F (θ))N−1
Profit : (p(θ) − θ) G(θ)
Similarity with an auction : (θ − b(θ)) (F (θ))N−1
Looking for p(.)
Equilibrium
p∗ (θ) =
N−1
(1−F (θ))N−1
Rθ
θ
zf (z) (1 − F (z))N−2 dz if θ < θ
p∗ (θ) = θ
Remark : equilibrium invariant with v
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Price competition with private information on cost
À la Hansen (1988) (Spulber, 1995). The elementary case (fixed demand). Proof.
b
If
to be θ the profit would be :
θ pretends
b − θ G(θ)
b where G(θ) = (1 − F (θ))N−1
p(θ)
F. o. c. (holds for θb = θ) : p0 (θ)G(θ) + (p(θ) − θ) g(θ) = 0
That is : p0 (θ)G(θ) + p(θ)g(θ) = θg(θ)
R
That is : p(θ)G(θ) = θg(θ)
That is : (as G(θ) = 0)
Z θ
p(θ)G(θ) = −
zg(z)dz
θ
That is also :
p(θ) = θ +
1
G(θ)
Z
θ
G(z)dz
θ
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Price competition with private information on cost
À la Hansen (1988) (Spulber, 1995). General case (variable demand).
Variable demand⇔first price auction with variable quantity
Symmetric equilibrium : p(θ) with p(.) %
Probability that θ is the lowest : G(θ) = (1 − F (θ))N−1
Profit : (p(θ) − θ) G(θ)D (p(θ))
Looks more complex but write H(θ) = G(θ)D (p(θ)) ( !)
Profit : (p(θ) − θ) H(θ)
Equilibrium
p∗∗ (θ) = θ +
1
H(θ)
p∗∗ (θ) = θ, θ ≤
Key result :
p∗∗
Rθ
θ H(z)dz
p∗∗ < pm (θ)
<
(warning : this is an equation)
(Spulber)
p∗
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Price competition with private information on cost
À la Hansen (1988) (Spulber, 1995). Effect of asymmetric information on prices.
Perfect information on θi ⇔second price auction
Under perfect information, let θ1 < θ2 < . . . < θN
Equilibrium price : p1∗ = θ2 = p2∗
Second price auction under asymmetric information
Same result
Using the equivalence theorem for fixed demand
Expected price = for 2nd and 1st price auction
Perfect information ⇒ expected price= E [p∗ ]
Asymmetric information : E [p∗∗ ] < E [p∗ ]
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Price competition with private information on cost
Introduction of uninformed consumers
The fundamental writing of the profit becomes
b − θ H(θ)
b =
p(θ)
N−1 b − θ U/N + I 1 − F (θ)
b
b
p(θ)
D(p(θ))
Same technics
Closer to real life ?
Each firm serves a demand (even if price is not the lowest)
The term U/N could be E [Ui ]
More flexible for empirical applications
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Advertising competition
Bagwell-Ramey (1994), Bagwell-Lee (2008)
General idea :
What if no price is observable before shopping ?
Still advertising expenditures are
Should consumers go to the most advertised store ?
Consumers do respond to advertising
Otherwise there would be none
Often there is no price information
Bagwell-Ramey’s idea : more consumers⇒lower price
⇒rational for consumers to respond to advertising
Bagwell-Lee’s idea : lower cost firms advertise more
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Advertising competition
Bagwell-Ramey (1994) : Firms have the same marginal cost
The model : N ≥ 2 firms
A fraction I of consumers observe advertising
A fraction U does not
Firms simultaneously set advertising (+other investments)
Consumers follow an advertising search rule
Informed consumers visit the most advertised store
Let Π∗ (E) − A be the expected profit
E expected demand ; A advertising expenditure
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Advertising competition
Bagwell-Ramey (1994) : Equilibrium (similar to Varian)
Distribution F (.) of advertising expenses, A ∈ A, A
i
A=0
ii
Π∗ U/N + F (A)N−1 I − A = Π∗ (U/N)
iii
A = Π∗ (U/N + I) − A = Π∗ (U/N)
Benham, 1972
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Advertising competition with private info. on cost
Bagwell-Lee (2008)
The model : N ≥ 2 firms
Each firm i privately informed of its marginal cost θi
θi drawn from θ, θ
Constant marginal cost ci = θi (could be C(q, θi ))
Distribution function F (θ), density f (θ).
Firms simultaneously choose their advertising levels
Unit mass of consumers, each having D(p)
or Unit mass of consumers unitary demands, gross utility v
A fraction I = 1 − U observe advertising levels
Informed consumers buy to the highest adverting firms
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Advertising competition with private info. on cost
Bagwell-Lee (2008) : equilibrium
Profit function
N−1 b
bθ
b
− A(θ)
= π m (θ) U/N + 1 − F (θ)
Π θ,
b − A(θ)
b
= π m (θ)H(θ)
Similar to an all-pay auction
Equilibrium advertising such that
b = A0 (θ)
b and A(θ) = 0
π m (θ)h(θ)
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