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Conjectural Variations and General Oligopoly Equilibria
Ludovic JULIEN
Introduction
- Motivations:
Conjectural Variations and General Oligopoly Equilibria Ludovic J - - PowerPoint PPT Presentation
Conjectural Variations and General Oligopoly Equilibria Ludovic J - - PowerPoint PPT Presentation
Conjectural Variations and General Oligopoly Equilibria Ludovic J ULIEN Introduction Motivations : (i) - Modelling expectations in GE under strategic interactions (ii) - Asymmetries in GE under SI (iii) - Foundations of perfect competition
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- 1. Introduction (suite)
- 1. Atomic approach (Aumann (1964)),
- 2. Asymptotic approach (Codognato-Gabszewicz (1991), (1993) and
Gabzewicz-Vial (1972)),
- 3. Game-Theoretic approach (Postlewaite-Roberts (1976), Gale (2000)).
- Casting conjectural variations into GE
- Idea of CV: to take into account the perceptions of each individual about his
rivals’ responses to a change in his own individual decision (Bowley (1924), Figuières et al. (2004), Julien (2006)).
- + Locally constant conjectures (Perry (1982), Julien (2008)),
- Idea today: intregation of CV in the asymptotic approach.
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- 2. The model
- Pure exchange economy:
- L divisible goods
- H agents, indexed h,
- Preferences:
- Endowments:
∏ =
= = L h h
x x U
l l l
l
1
) (
α
H h ,..., 1 = L ,..., 1 = l ) 1 , ( ∈ α ) ,..., 1 ,..., ( =
h
ω
l l
n n h ,..., 1
1 +
=
−
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The economy (suite)
- Strategy sets:
- Beliefs:
DEFINITION 1. An economy is a collection .
{ }
1 : ≤ ≤ =
l l l h h h
s s S
l l l
ν = ∂ ∂ ∑
≠ − − h h h h
s s
l l
n n h ,..., 1
1 +
=
−
{ }
L H h h h h
v X
,..., 1 ,..., 1
, ), ( ,
= =
=
l l l
f ω ξ
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The economy (suite)
- DEFINITION 2. A CVGE for is given by a
vector of strategies an allocation and a vector of conjectural variations such that: (i) (ii) (iii)
( ) { }
l l l
f ν ω ξ , , , ,
h h h h
S X =
) ~ ,..., ~ ,..., ~ ( 11
HL h
s s s
l
HL
IR x
+
∈ ~
) ,..., ,..., ( 1
L
ν ν ν ν
l
=
h s s x x
h h h h
∀ =
−
)), ( ~ ), ( ~ (( ~
l l l l
ν ν
l
l l l l l
∀ = ∑
∑
, )) ~ ( ~ ), ~ ( ~ ( ~
h h h h h
s s s p x ω h s s p x U s s s p x U
h h h h h h h h h
∀ ≥
− −
))), ( ~ ), ( , ( ( ))) ( ~ ), ( ~ )), ( ~ ( ~ ( ~ (
l l l l l l l l l l
ν ν ν ν ν
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- 3. Market prices and strategic plans
- Market clearing prices:
- Strategic plans:
∏
= ≠ = − − − −
⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −
L k k k h h h h k k
s n n s s s s Arg
k
l l l l l l l l
l l l
1 1 1 1
] ) 1 ( [ ) ( ) 1 ( 1 max
α α α α
α α
∑ ∑
− − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛
h h h hk k k k
s s p p
l l l l
) 1 ( ) 1 ( α α α α
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- 4. Equilibrium strategies & allocations
) 1 )( 1 ( )] 1 ( )[ 1 ( ~
1 1 l l l l l l l l l
ν α ν α + − − − + − − − =
− −
n n n n sh ) 1 )( 1 ( ) ( ~
1 1 l l l l l l l l
ν α α + − − − − =
− −
n n n n xh )] 1 )( 1 ( )[ )[ ( )] 1 ( )[ )( 1 ( ~
1 1 1 1 1 k k k k k k k k k k k k hk
n n n n n n n n n n x ν α α ν α α + − − − − − + − − − − =
− − − − − l l l l
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- 5. Main results
PROPOSITION 1. When
the conjectural general equilibrium coincides with the competitive equilibrium. No replication procedure or asymptotic identification.
l
l
∀ − = , 1 ν
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Main results (suite)
DEFINITION 3. A locally consistent CGE for is a CGE for the vector of conjectures such that if is solution to then: PROPOSITION 2. The Competitive equilibrium is a locally consistent conjectural general equilibrium. ξ
) ,..., ,..., ( 1
L
ν ν ν ν
l
= ) ~ ,..., ~ ,..., ~ ( 11
HL h
s s s
l
h h
V Arg s max ∈
l
. , ) , ~ ( l
l l l l l
∀ = ∂ ∂∑
≠ − −
ν ν
h h h h h
s s s
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