Tâtonnement and the Core
Tâtonnement
Walrasian auctioneer calls prices in period t, everyone reports demand, auctioneer calculates
k
k k k d p (t ) k
z ( p )=Σi x i ( p )−ω , adjust prices at t+1 s.t. =z ( pt )
dt
Announces a new price vector iff previous price vector is not an equilibrium
Auctioneer permits trading only at equilibrium prices
Typically doesn’t converge to comp. eqbm as multiple eqba or dynamics get stuck
o Clear problems in that if all goods are in eqbm apart from one, changing the price of
that one good will affect the excess demands for all other goods in some way ⇒
becomes almost impossible to reach eqbm
Gen eq places little structure on z ( p ) so we must restrict it
Gross Substitutes
∂ z j( p)
Goods i and j gross substitutes if >0 i ≠ j
∂ pi
Note excess demand: both supply and demand respond to price
changes, rather than just demand
WARP
Consumer’s demand satisfies WARP if for any ( p , m ) and
( p' , m' ) s .t . x=x ( p ,m ) ≠ x ( p' , m' ), p x' ≤ m⇒ p' x >m ' ie if
prices and income change, consumers won't select a bundle they
could have previously afforded
Equilibrium
If all goods gross substitutes ∀ p or AD satisfies WARP, then
¿
i) If p is comp. eqbm price vector, it is unique (up to a transformation) eqbm price
vector
¿
ii) Tâtonnement process converges to p from any starting point
Core
Coalition, K, of agents blocks allocation if its members can improve on it by trading among
themselves – core is set of allocations that can be blocked
o An allocation is blocked if a coalition can improve upon
it by trading only amongst themselves
Core consists of allocations on CC giving both agents higher
utilities than reservation utility – all P-efficient – the set of
allocations that is not blocked
They improve on give allocation x until they reach some
' ' '
allocation x s . t . Σ x i=Σ wi ; x ≻ x ∀ i∈ K
Offer curve: locus of optimal choices when prices vary –
intersect in comp. eqbm, and they are above reservation utility so are in core
A line drawn from the endowment to the offer curve represents a price line to which the
consumer’s indifference curve is tangent at that point
Hence comp. eqbm of Walrasian economy is in core – can justify as outcome of cooperative
trading – this result holds regardless of the number of agents. In fact, the core shrinks to only
the competitive market equilibrium allocation(s) as the number of agents rises
The fact the Walrasian equilibrium is in the core shows that no group would ever wish to
withdraw and trade amongst themselves
Caveats
Requires all agents to bargain costlessly
Large informational demands on agents to make improvements
More Agents
Tâtonnement
Walrasian auctioneer calls prices in period t, everyone reports demand, auctioneer calculates
k
k k k d p (t ) k
z ( p )=Σi x i ( p )−ω , adjust prices at t+1 s.t. =z ( pt )
dt
Announces a new price vector iff previous price vector is not an equilibrium
Auctioneer permits trading only at equilibrium prices
Typically doesn’t converge to comp. eqbm as multiple eqba or dynamics get stuck
o Clear problems in that if all goods are in eqbm apart from one, changing the price of
that one good will affect the excess demands for all other goods in some way ⇒
becomes almost impossible to reach eqbm
Gen eq places little structure on z ( p ) so we must restrict it
Gross Substitutes
∂ z j( p)
Goods i and j gross substitutes if >0 i ≠ j
∂ pi
Note excess demand: both supply and demand respond to price
changes, rather than just demand
WARP
Consumer’s demand satisfies WARP if for any ( p , m ) and
( p' , m' ) s .t . x=x ( p ,m ) ≠ x ( p' , m' ), p x' ≤ m⇒ p' x >m ' ie if
prices and income change, consumers won't select a bundle they
could have previously afforded
Equilibrium
If all goods gross substitutes ∀ p or AD satisfies WARP, then
¿
i) If p is comp. eqbm price vector, it is unique (up to a transformation) eqbm price
vector
¿
ii) Tâtonnement process converges to p from any starting point
Core
Coalition, K, of agents blocks allocation if its members can improve on it by trading among
themselves – core is set of allocations that can be blocked
o An allocation is blocked if a coalition can improve upon
it by trading only amongst themselves
Core consists of allocations on CC giving both agents higher
utilities than reservation utility – all P-efficient – the set of
allocations that is not blocked
They improve on give allocation x until they reach some
' ' '
allocation x s . t . Σ x i=Σ wi ; x ≻ x ∀ i∈ K
Offer curve: locus of optimal choices when prices vary –
intersect in comp. eqbm, and they are above reservation utility so are in core
A line drawn from the endowment to the offer curve represents a price line to which the
consumer’s indifference curve is tangent at that point
Hence comp. eqbm of Walrasian economy is in core – can justify as outcome of cooperative
trading – this result holds regardless of the number of agents. In fact, the core shrinks to only
the competitive market equilibrium allocation(s) as the number of agents rises
The fact the Walrasian equilibrium is in the core shows that no group would ever wish to
withdraw and trade amongst themselves
Caveats
Requires all agents to bargain costlessly
Large informational demands on agents to make improvements
More Agents
