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 (Arrow (1959), Debreu (1959)) Today (iii): Can we deduce PC from an IC framework?

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 .

2. The model • Pure exchange economy : = l 1 ,..., L - L divisible goods = h 1 ,..., H - H agents, indexed h, = l L α α ∈ = ∏ ( 0 , 1 ) U ( x ) x - Preferences: l l h h = l 1 ω = = 1 + ( 0 ,..., 1 ,..., 0 ) h n 1 ,..., n - Endowments: − l l h

The economy (suite) - Strategy sets: { } = ≤ ≤ : 0 1 S s s l l l h h h ∂ ∑ s − l h = ν − ≠ = 1 + - Beliefs: h h h n 1 ,..., n − ∂ l l l s l h { } = ξ = ω l 1 ,..., L f D EFINITION 1. An economy is a collection . X , ( ), , v = l l h h h h 1 ,..., H

The economy (suite) { ( ) } ξ = ω ν - D EFINITION 2. A CVGE for is given by a f X , , , S , l l l h h h h ~ ~ ~ ~ ∈ HL ( 11 s ,..., s ,..., s ) vector of strategies an allocation and a x IR + l h HL ν = ν ν ν ( 1 ,..., ,..., ) vector of conjectural variations such that : l L ~ ~ ~ = ν ν ∀ x x (( s ( ), s ( )), h (i) − l l l l h h h h = ∑ ∑ ~ ~ ~ ~ ~ ω ∀ l ( ( ), ( )) , x p s s s (ii) l l l l l h h h h h ~ ~ ~ ν ~ ν ~ ν ≥ (iii) U ( x ( p ( s ( )), s ( ), s ( ))) − l l l l l l h h h h h ν ~ ν ∀ U ( x ( p , s ( ), s ( ))), h − l l l l h h h h

3. Market prices and strategic plans ∑ s ⎛ ⎞ α − α hk p ( 1 ) ⎜ ⎟ = l l l h • Market clearing prices: ⎜ ⎟ ∑ α − α ⎝ ⎠ p ( 1 ) s l k k k h h • Strategic plans: α ⎛ ⎞ ⎧ α − α ⎫ α − = 1 k ( 1 s ) ( s ) k L l l ∏ ⎜ ⎟ l ⎨ l l ⎬ h h Arg max s ⎜ ⎟ − α − α + − − k 1 ⎝ ⎠ ⎩ ⎭ 1 [ s ( n n 1 ) s ] l = k 1 − − l l l l k 1 h h ≠ l k

4. Equilibrium strategies & allocations − α − − + ν ( 1 )[ n n ( 1 )] ~ = − l l l l 1 s h − − − α + ν l n n ( 1 )( 1 ) − l l l l 1 α − ( n n ) ~ = − l l l 1 x h l − − − α + ν n n ( 1 )( 1 ) − l l l l 1 α − α − − − + ν ( 1 )( n n )[ n n ( 1 )] ~ = − − l l k k 1 k k 1 k x α − − − − − α + ν hk ( )[ )[ ( 1 )( 1 )] n n n n n n − − − l l k 1 k k 1 k k 1 k k

5. Main results ν = − ∀ l PROPOSITION 1. When 1 , the conjectural l general equilibrium coincides with the competitive equilibrium . No replication procedure or asymptotic identification.

Main results (suite) ξ D EFINITION 3. A locally consistent CGE for is a CGE ν = ν ν ν for the vector of conjectures such that if ( 1 ,..., ,..., ) l L ~ ~ ~ ∈ s Arg max V ( 11 s ,..., s ,..., s ) is solution to then: l h h l h HL ∂ ∑ ~ ν ( , ) s s − = ν ∀ − ≠ l l l h h l h h , . ∂ l s l h P ROPOSITION 2. The Competitive equilibrium is a locally consistent conjectural general equilibrium.

Perspectives Asymmetries in GE under strategic interactions Generalization with functional forms Learning Economic Policy