General Equilibrium, Berardino
Cesi, MSc Tor Vergata
General Equilibrium
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Equilibrium in Consumption
GE begins (1/3)
SMS
SMS
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• 2-Individual/ 2-good Exchange economy (No production, no
transaction costs, full information..)
• Endowment (Nature): e
• Private property/ NO Social Planner
• Redistribution: Voluntary exchange (blocking some trades)
GE begins (2/3)
MRSi , j
• Not all Pareto efficient allocations are equilibria (i.e. allocation d)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Contract curve and Pareto efficiency (exchange equilibrium of the
process of voluntary exchange)
GE begins (3/3)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• What does “exchange” mean? (no Blocking)
• …Each individual face the following maximization problem:
General equilibrium (1/3) :
• 𝑖 ∈ 𝔗; 𝑤𝑤𝑤𝑤 𝔗 = 1, … , 𝐼 with preferences ≿𝑖 over allocations of
n goods with endowment 𝒆𝑖 = 𝑒1𝑖 , … , 𝑒𝑛𝑖 and 𝒆 ≡ 𝒆1 , … , 𝒆𝐼
• Allocations 𝒙 ≡ 𝒙1 , … , 𝒙𝐼 with consumer i’s bundle given by
𝒙𝑖 ≡ 𝑥1𝑖 , … , 𝑥𝑛𝑖
Conditions for allocations to be a barter equilibrium:
• a) Feasibility:
• 𝒙∈𝐹 𝒆
• with 𝐹 𝒆 ≡ 𝒙�∑𝑖∈𝔗 𝒙𝑖 = ∑𝑖∈𝔗 𝒆𝑖 : assignment of goods not
exceed the available amount
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Exchange economy
• Consumers:
General equilibrium (2/3) :
• b) Pareto efficiency:
• A feasible allocation 𝒙 ∈ 𝐹 𝒆 is Pareto efficient if there are no
other feasible allocations, y ∈ 𝐹 𝒆 , such that 𝒚𝑖 ≿𝑖 𝒙𝑖 for all
consumers, i, with at least one preference strict
• The equilibrium must be Pareto efficient, but not all Pareto
efficient allocation are equilibria
• Some Pareto efficient allocations will be blocked
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• An allocation is Pareto efficient if it is not possible to make
someone strictly better off without making someone else worse off
• c) Unblocked
• Allocation must be unblocked by any coalition (S) of consumers
• With I>2, consumers may form coalitions to block proposed
allocations
• S blocks 𝒙 ∈ 𝐹 𝒆 if there is an allocation y such that:
• ∑𝑖∈𝑆 𝒚𝑖 = ∑𝑖∈𝑆 𝒆𝑖
• 𝒚𝑖 ≿𝑖 𝒙𝑖 for all 𝑖 ∈ 𝑆, with at least one preference strict
• The set of all equilibrium allocations (satisfying a)-c)) is defined as
the Core of an Exchange Economy (𝐶 𝒆 ) with endowment e.
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
General equilibrium (3/3) :
Perfectly Competitive Market
• Don’t know each other (preferences), only prices matter
• Sufficiently “insignificant” on every market: price takers (no market
power)
• Price is known
• Decentralized markets: each agent (seller/buyer) acts in his own selfinterest while ignoring the actions of others
• Self-interest oriented
• Individual utility maximizer at that price
• No production
• No uncertainty
• Equilibrium:
• There exists a price (vector) at which markets for goods k
(simultaneously) clear (demand-supply matching)
• WE show the Existence of this price vector
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Consumers:
Existence (1/17)
• 𝑖 ∈ 𝔗; 𝑤𝑤𝑤𝑤 𝔗 = 1, … , 𝐼 with preferences ≿𝑖 on the
consumption set ℜ n+ given by the utility function 𝑢𝑖 (on ℜ n+)
• 𝑢𝑖 is:
1. Continuous
2. Strongly increasing
3. Strictly quasiconcave
At the (given) price vector 𝒑 ≡ 𝑝1 , … , 𝑝𝑛 ≫ 𝟎
• Consumer solves:
i
i
𝑖 ≤ 𝒑 · 𝒆𝑖
s.t.
𝒑
·
𝒙
u
x
max
i
n
x ∈ℜ +
( )
• with solution 𝒙𝑖 𝒑, 𝒑 · 𝒆𝑖 , by 1-3, unique for each 𝒑 ≫ 𝟎 and
continuous in 𝒑 on ℜ n+ +
• n-good/ n-market
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Basic assumptions (consumers)
Existence (2/17)
• Excess demand functions
• Aggregate Excess demand function for good k (real)
(
)
i∈ℑ
i∈ℑ
• 𝑧𝑘 𝒑 > 0: Aggregate demand for good k exceeds the aggregate
endowment for good k
• 𝑧𝑘 𝒑 < 0: excess supply of good k
• Aggregate excess demand function (vector)
z (p ) ≡ ( z1 (p ),..., z n (p ))
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
z k (p ) ≡ ∑ xki p, p ⋅ ei − ∑ eki
Existence (3/17)
• Given 𝑢𝑖 satisfying (1-3), then for all 𝒑 ≫ 𝟎, 𝐳 𝒑 satisfies:
• Walras’ law says that at any set of positive prices, the aggregate
excess demand is zero (follows from 𝑢𝑖 strongly increasing and
binding budget constraint)
• Excess demand in the system of markets must be matched by excess
supply of equal value, at given prices, somewhere else in the system
• 2-good case: 𝑝1 𝑧1 𝒑 = −𝑝2 𝑧2 𝒑
𝑧1 𝒑 > 0 ⇒ 𝑧2 𝒑 < 0
𝑧1 𝒑 = 0 ⇒ 𝑧2 𝒑 = 0
• By the Walras’ law if, at same prices, n-1 markets are in equilibrium
then the nth market must also be in equilibrium
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Continuity at 𝒑
• Homogeneity of degree zero in prices: 𝐳 λ𝒑 = 𝐳 𝒑 ∀ λ > 0
• Walras’ law: 𝐩 ∙ 𝐳 𝒑 = 0
Existence (4/17)
• 𝑧𝑘 𝒑 = 0 , partial equilibrium in the single market k (demand
equal to supply in the market k);
Or
• 𝐳 𝒑 = 𝟎 (demand equal to supply in every market)
• Walrasian prices 𝒑∗ : prices equalizing demand and supply in
every market
A vector p ∈ ℜ + + is called Walrasian equilibrium if z 𝒑∗ = 𝟎
*
n
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• We have a general equilibrium when
Existence (5/17)
THEOREM 1 (Existence)
• Suppose z satisfying the following conditions:
z(.) is continuous on ℜ n
++
𝐩 ∙ 𝐳 𝒑 = 0 for all 𝒑 ≫ 𝟎
n
� ≠ 0,
if 𝒑𝑚 is a sequence of price vectors in ℜ + + converging to 𝒑
and 𝑝̅𝑘 = 0 for some good k, then for some good k’, the associate
sequence of excess demands in the market k’, 𝑧𝑘𝑘 𝒑𝑚 , is
unbounded above
Then there is a price vector 𝒑∗ ≫ 𝟎 such that z 𝒑∗ = 𝟎
Condition 3 says that if the prices of some but not all goods are arbitrarily
close to zero the excess demand for at least one of those goods is
arbitrarily high (to be shown)
Proof: We need the fixed-point theorem (Brouwer).
Brouwer’s fixed-point Theorem
Let 𝑆ϲℝ𝑛 be a non-empty compact and convex set. Let f: S→S be a
continuous function. Then there exists at least one fixed point of f in S.
That is, there exists at least one 𝒙∗ ∈ 𝑆 such that 𝒙∗ = 𝑓(𝒙∗ ).
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
1)
2)
3)
Existence (6/17)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• f is a continuous mapping from 𝑎, 𝑏 into 𝑎, 𝑏 (itself). Thus, f
crosses the 45° line at least once within the square 𝑎, 𝑏 x 𝑎, 𝑏
• …nothing about uniqueness
Existence (7/17)
Proof
•
𝑧̅𝑘 𝒑 = 𝑚𝑚𝑚 𝑧𝑘 𝒑 , 1 for all 𝒑 ≫ 𝟎 (bounded above by 1) and
𝑧̅ 𝒑 = 𝑧̅1 𝒑 , . . , 𝑧̅𝑛 𝒑
• 𝑆𝜺 = 𝒑 �∑𝑛𝑘=1 𝑝𝑘 = 1 𝑎𝑎𝑎 𝑝𝑘 ≥
𝜀
1+2𝑛
∀ 𝑘 with 𝜀𝜀 0,1
Compact (closed
and bounded)
Convex
Non-empty:
i.e.
1
2+𝑛
1
2+𝑛
,
, … 𝜖𝑺𝜺
p=
1+2𝑛 1+2𝑛
because 𝜀 < 1
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
First part: We start by showing the existence of 𝒑∗ in a relatively
small set of prices, then we will extend this result by extending this
set. (hint: keep in mind the fixed point theorem, 𝑓: 𝑆 → 𝑆)
Define:
Existence (8/17)
• For each 𝒑 ∈ 𝑆𝜺 and for each k, let’s define 𝑓𝑘 𝒑 as:
𝜀+𝑝𝑘 +𝑚𝑚𝑚 0,𝑧̅𝑘 𝒑
𝑛𝜀+1+∑𝑛
𝑚=1 𝑚𝑚𝑚 0,𝑧̅𝑚 𝒑
and 𝑓 𝒑 = 𝑓1 𝒑 , … , 𝑓𝑛 𝒑
𝜀
because 𝑧̅𝑚 𝒑
𝑛𝑛+1+𝑛×1
𝜀
𝜀
<
). This implies
𝑛𝑛+1+𝑛×1
1+2𝑛
• Then ∑𝑛𝑘=1 𝑓𝑘 𝒑 = 1 and 𝑓𝑘 𝒑 ≥
𝑓𝑘 𝒑 ≥
𝜀
1+2𝑛
given 𝜀 < 1(
• f is continuous function mapping the set 𝑆𝜀 into itself
• Is 𝑓𝑘 𝒑 continuous on 𝑆𝜺 ? Check:
• 𝑧𝑘 𝒑 is continuous on 𝑆𝜺 (the same holds for 𝑧̅𝑘 𝒑 )
• denomiator and numerator of 𝑓𝑘 𝒑 are continuous on 𝑆𝜺
• denominator bounded away from zero (at least equal to 1)
≤ 1 ∀𝑚 ⇒
f: 𝑆𝜀 → 𝑆𝜀
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• 𝑓𝑘 𝒑 =
Existence (9/17)
Assume n=2, then 𝑓𝑘 𝒑 ≥
𝜀
1+2𝑛
becomes 1 > 𝑓𝑘 𝒑 ≥
𝑝∗
𝑝
𝜀
𝟓
• f maps 𝑆𝜀 into 𝑆𝜀 (it recalls the fixed point theorem f: 𝑆𝜀 → 𝑆𝜀 )
𝑓 𝑝
𝑆𝜀
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
1
𝑓 ∗ 𝑝∗
𝜀
5
𝜀
5
𝑆𝜀
1
Existence (10/17)
• We apply FPT: ∃ 𝒑𝜀 ∈ 𝑆𝜀 |𝑓 𝒑𝜀 = 𝒑𝜀 or 𝑓𝑘 𝒑𝜀 =𝑝𝑘 𝜀
n


i ) pkε nε + ∑ max 0, z m p ε  = ε + max 0, z k p ε
m =1


(
( ))
(
( ))
• But the proof is not over yet….What do we know so far?
• Simply: ∃ a price vector 𝒑𝜀 satisfying (i) for every 𝜀𝜀 0,1
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Then by applying the above result to the definition of 𝑓 𝑝 (and
rearranging), we have:
Existence (11/17)
• Check out: sequence 𝒑𝜀 is bounded because 𝒑𝜀 ∈ 𝑆𝜀 (every
price is between 0 and 1). Thus 𝒑𝜀 must converge.
• Assume to pick up the sequence 𝒑𝜀 converging to a vector
defined (by chance) as 𝒑∗
• What we do know about this vector: since all 𝑝𝑘 sum up to 1,
then we must have 𝒑∗ ≥ 𝟎 with 𝒑∗ ≠ 𝟎.
• We need to show that 𝒑∗ ≫ 𝟎
• For this proof we use condition (3).
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Second part: let 𝜀 approach to zero, then it gives a sequence of
price vector 𝒑𝜀 satisfying (i).
Existence (12/17)
*
• Then…for some 𝑘� we must have pk = 0
• But by condition (3) there must exist some 𝑘 ′ with pk* ' = 0 such
that 𝑧𝑘𝑘 𝒑𝜀 is unbounded above (remind: we have 𝜀 → 0)
• Since 𝒑𝜀 → 𝒑∗ , pk* ' = 0 implies pkε ' → 0, therefore for some k=k’…
n


i ) pkε nε + ∑ max 0, z m p ε  = ε + max 0, z k p ε
m =1


(
zero
( ))
bounded above
(
≠0
( ))
By unboundedness of 𝑧𝑘𝑘 𝒑𝜀 (from condition (3)).
CONTRADICTION: 0∙(positive value)=𝜀 + 1
Cannot exist a NON strictly positive price vector 𝒑∗ for market k’
(then neither for k)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• We show that 𝒑∗ ≫ 𝟎 by contradiction: assume 𝒑∗ ≫ 𝟎 does
not hold
Existence (13/17)
• We have shown that 𝒑𝜀 → 𝒑∗ ≫ 𝟎 as 𝜀 → 0.
Last step: showing the second part of the statement, z 𝒑∗ = 𝟎
• We use the continuity of 𝒛� . on ℜ n obtained from z(.)
++
n
 ε
ε 
ε 
limε →0  pk nε + ∑ max 0, z m p  = ε + max 0, z k p 
m =1

 

n
p
*
k
(
( ))
( ))
(
(
( ))
( )) for all k.
*
*
max
0
,
p
max
0
,
p
z
=
z
∑
m
k
m =1
• multiplying by 𝑧𝑘 (𝒑∗ ) and summing over k
n
 n
* 
p • z p ∑ max 0, z m p  = ∑ z k p* max 0, z k p*
 m =1
 k =1
*
( )
*
(
( ))
( )
(
( ))
• By the Walras’ law the LHS is zero, then together with 𝒑∗ ≫ 𝟎 implies
𝑧𝑘 𝒑∗ = 0.∎
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
(
Let’s check when aggregate excess demand 𝒛 𝒑 satisfies (3)
• Theorem When 𝑢𝑖 is Continuous, Strongly increasing and Strictly
quasiconcave and ∑𝐼𝑖=1 𝒆𝑖 ≫ 𝟎 then 𝒛 𝒑 satisfies (1)-(3).
Proof
• 1) Let’s focus on (3). Consider the sequence of strictly positive price
� ≠ 𝟎 such that 𝒑
�𝑘 =0 for some good k.
vectors 𝒑𝑚 → 𝒑
� ∙ ∑𝐼𝑖=1 𝒆𝑖 > 0 , then 𝒑
� ∙ ∑𝐼𝑖=1 𝒆𝑖 =
• From ∑𝐼𝑖=1 𝒆𝑖 ≫ 𝟎 we have 𝒑
∑𝐼𝑖=1 𝒑
� ∙ 𝒆𝑖 > 0. Thus there must be at least one consumer i with
� ∙ 𝒆𝑖 >0.
𝒑
• 2) Part. By contradiction we assume the demand of consumer i
along the sequence 𝒑𝑚 , 𝒙𝑖 𝒑𝑚 , 𝒑𝑚 ∙ 𝒆𝑖 , is bounded :
• Then it must converge…and assume 𝒙𝑖 𝒑𝑚 , 𝒑𝑚 ∙ 𝒆𝑖 ≡ 𝑥 𝑚 → 𝑥 ∗
• Since individual is utility (strongly increasing) maximizer (under
constraint) we have: 𝒑𝑚 ∙ 𝒙𝑚 = 𝒑𝑚 ∙ 𝒆𝑖 for every m, with:
� ∙ 𝒙∗ = 𝒑
� ∙ 𝒆𝑖 >0
Budget constraint: 𝑙𝑙𝑙𝑚→∞ (𝒑𝑚 ∙ 𝒙𝑚 = 𝒑𝑚 ∙ 𝒆𝑖 ) ⇒ 𝒑
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Existence (14/17)
Existence (15/17)
� > 𝑢𝑖 𝒙∗
• 1) 𝑢𝑖 𝒙
�∙𝒙
�=𝒑
� ∙ 𝒆𝑖 >0
• Recalling that pk = 0, that implies: 2) 𝒑
• By continuity of 𝑢𝑖 we have that (1) e (2) imply, ∀𝑡 ∈ 0,1
� > 𝑢𝑖 𝒙∗
• 𝑢𝑖 𝑡𝒙
� ∙ 𝑡𝒙
� <𝒑
� ∙ 𝒆𝑖
• 𝒑
�, 𝒙𝑚 → 𝒙∗ and continuity of 𝑢𝑖 , for large m
• Because 𝒑𝑚 → 𝒑
� > 𝑢𝑖 𝒙𝑚
• 𝑢𝑖 𝑡𝒙
� < 𝒑𝑚 ∙ 𝒆𝑖
• 𝒑𝑚 ∙ 𝑡𝒙
• ….contradicting that 𝒙𝑚 solve the maximization problem at 𝒑𝑚
• Thus sequence of demand 𝒙𝑚 𝒑𝑚 , 𝒑𝑚 ∙ 𝒆𝑖 is unbounded
{ } is unbounded above.
• There exist some good 𝑘 ′ such that xkm'
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
� = 𝒙∗ + 0, … , 0, 1𝑘 , 0, … , 0 , since 𝑢𝑖 is
• Let’s introduce 𝒙
strongly increasing on ℜ n+ we have:
Existence (16/17)
• The condition for the demand of k’ to be unbounded above and
m
affordable is pk ' → 0
m
• Then: pk ' = lim m pk ' = 0
• Thus:
• Aggregate supply for k’ is fixed (endowment)
• All consumers demand a non-negative amount of k’
• then because consumer i’s demand for k’ is unbounded above….
• …..aggregate excess demand for k’ is unbounded above along the
sequence 𝒑𝑚 .∎
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
� ∙ 𝒆𝑖 then the sequence of i’s
• Because i’s income converge to 𝒑
income 𝒑𝑚 ∙ 𝒆𝑖 is bounded.
Existence (17/17)
• When 𝑢𝑖 is i) continuous, ii) strong increasing and iii) strictly
quasiconcave on ℜ n+ and ∑𝐼𝑖=1 𝒆𝑖 ≫ 𝟎 then there exists at least one
price vector, 𝒑∗ ≫ 𝟎, such that z 𝒑∗ = 𝟎
Intuition:
• We know that i)-iii) make z(.) continuous, homogeneous of
degree zero and respecting Walras’s law
• ∑𝐼𝑖=1 𝒆𝑖 ≫ 𝟎 makes z(.) satisfy (3) in theorem 1.
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Theorem for the Existence of GE (revised conclusion)
WEA: Pareto efficient
allocation
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
WE and Pareto efficiency (1/6)
WE and Pareto efficiency (2/6)
• Is the feasible bundle received in the WEA the
most preferred in the consumer’s budget set
at the Walrasian prices?
+
• Is the set of WEA on the Contract Curve?
=
Pareto Efficiency
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Is a WEA Pareto efficient?
Definition of WEA
• Let 𝒑∗ be a Walrasian equilibrium for some economy with initial
endowment e, and let:
𝒙 𝒑∗ ≡ 𝒙1 𝒑∗ , 𝒑∗ ∙ 𝒆1 , … , 𝑥 𝐼 𝒑∗ , 𝒑∗ ∙ 𝒆𝐼 ,
Then 𝒙 𝒑∗ is called a Walrasian equilibrium allocation (WEA).
a) A WEA must be (clearly) feasible
Formally: Let 𝒑∗ be a Walrasian equilibrium with initial endowment e.
Let 𝒙 𝒑∗ be the associate WEA. Then 𝒙 𝒑∗ ∈ 𝐹 𝒆
b) A WEA must be the most preferred (any other WEA feasible and
preferred must be too expensive)
n
Formally-Lemma 1: Assume 𝑢𝑖 is strictly increasing on ℜ +, that
n
�𝑖 , and that 𝒙𝑖 ∈ ℜ +
consumer’s demand at p≥ 𝟎 is 𝒙
i.
ii.
�𝑖 , then 𝐩 ∙ 𝒙𝑖 > 𝐩 ∙ 𝒙
�𝑖
If 𝑢𝑖 𝒙𝑖 > 𝑢𝑖 𝒙
�𝑖 , then 𝐩 ∙ 𝒙𝑖 ≥ 𝐩 ∙ 𝒙
�𝑖
If 𝑢𝑖 𝒙𝑖 ≥ 𝑢𝑖 𝒙
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
WE and Pareto efficiency (3/6)
WE and Pareto efficiency (4/6)
• Proof: Part i, assume (ii) holds. Proof by contradiction on
(i). Assume (i) does not hold:
• It is possible a small increase on 𝒙𝑖 toward
�𝑖 still lower than 𝒙
�𝑖
𝒙
�𝑖 > 𝑢𝑖 𝒙𝑖 > 𝑢𝑖 𝒙
�𝑖
• But this would imply 𝑢𝑖 𝒙
�𝑖 < 𝐩 ∙ 𝒙
�𝑖
and 𝐩 ∙ 𝒙
�𝑖 contradicts (2)
• Replacing 𝒙𝑖 with 𝒙
• Part 2, see the book.
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
�𝑖 , then 𝐩 ∙ 𝒙𝑖 < 𝐩 ∙ 𝒙
�𝑖
Assume 𝑢𝑖 𝒙𝑖 > 𝑢𝑖 𝒙
WE and Pareto efficiency (5/6)
• 𝑊 𝒆 : Set of WEA
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• No uniqueness
WE and Pareto efficiency (6/6)
…and about the Pareto efficiency (remind the CC and the Core) ?
𝑖
.
If
𝑢
is strictly
𝑖∈𝔗
• Proof. By contradiction. Assume 𝑊 𝒆 ∉ 𝐶 𝒆 . Since 𝒙 𝒑∗ is WEA
then 𝒙 𝒑∗ ∈ 𝐹 𝒆 (is feasible). However:
• 𝒙 𝒑∗ ∉ 𝐶 𝒆 ⇒∃ a (blocking) coalition S and allocation y such that:
• 1) ∑𝑖∈𝑆 𝒚𝑖 = ∑𝑖∈𝑆 𝒆𝑖 and 2) 𝑢𝑖 𝒚𝑖 ≥ 𝑢𝑖 𝒙𝑖 𝒑∗ , 𝒑∗ ∙ 𝒆𝑖
𝑖 ∈ 𝑆, with at least one inequality strict. Also we have:
• (i) 𝒑∗ ∑𝑖∈𝑆 𝒚𝑖 = 𝒑∗ ∑𝑖∈𝑆 𝒆𝑖
for all
• By 2) and Lemma 1, for each 𝑖 ∈ 𝑆, we must have
• 𝒑∗ ∙ 𝒚𝑖 ≥ 𝒑∗ ∙ 𝒙𝑖 𝒑∗ , 𝒑∗ ∙ 𝒆𝑖 =𝒑∗ ∙ 𝒆𝑖 (with at least one strict)
Summing over all i in S:
• 𝒑∗ ∑𝑖∈𝑆 𝒚𝑖 > 𝒑∗ ∑𝑖∈𝑆 𝒆𝑖 (that contradicts (i)) (at least one must get
more than the endowment to be a member of the coalition)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Consider an exchange economy 𝑢𝑖 , 𝑒 𝑖
n
increasing on ℜ + , then 𝑊 𝒆 ⊂ 𝐶 𝒆
Welfare (1/4)
• From the previous results: Core is a non-empty set; all Core
allocations are Pareto efficient. Thus…
• 1° Welfare Theorem: every WEA is Pareto Efficient
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Trade off: efficiency (Pareto) vs. Equity
• 2° Welfare Theorem: Consider an economy 𝑢𝑖 , 𝑒 𝑖 𝑖∈𝔗 with
aggregate endowment ∑𝐼𝑖=1 𝒆𝑖 ≫ 𝟎, and 𝑢𝑖 continuous, strong
� is
increasing and strict quasiconcave in ℜ n . Suppose that 𝒙
+
Pareto efficient for 𝑢𝑖 , 𝒆𝑖 𝑖∈𝔗 and the endowment is
�. Then 𝒙
� is a WEA
redistributed so that the new endowment is 𝒙
�𝑖 𝑖∈𝔗
of the resulting exchange economy 𝑢𝑖 , 𝒙
� was not
• Graphical proof (intuition): assume by contradiction that 𝒙
a WE, but it would contradict Pareto efficiency
•
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Welfare (2/4)
Welfare (3/4)
• What we have: if �
𝒙 is Pareto efficient, then it must be feasible in the
�𝑖 𝑖∈𝔗 , that is ∑𝐼𝑖=1 𝒙
�𝑖 = ∑ 𝑖=1 𝒆𝑖 ≫ 0, then this
economy 𝑢𝑖 , 𝒙
economy has a WEA, let’s say �
𝒙.
• What is remained to show: �
𝒙=�
𝒙
� is WEA, then:
• Since 𝒙
•
�𝑖 is the consumer’ i demand function with endowment 𝒙
�𝑖 , then this implies:
𝒙
�𝑖 ≥ 𝑢 𝑖 𝒙
�𝑖 for all 𝑖 ∈ 𝔗
a) 𝑢𝑖 𝒙
� must be feasible for the economy 𝑢𝑖 , 𝒙
�𝑖
• 𝒙
∑𝐼𝑖=1 𝒙
�𝑖
=
∑ 𝐼𝑖=1 𝒆𝑖
=
𝑖∈𝔗
𝐼
∑𝑖=1 𝒙
�𝑖
� is feasible for the original economy 𝑢𝑖 , 𝒆𝑖 𝑖∈𝔗
• Then 𝒙
• But then every inequality in (a) (for every i) must be binding, otherwise
� cannot be Pareto efficient.
𝒙
�𝑖 = 𝑢 𝑖 𝒙
�𝑖 must implies 𝒙
�𝑖 = 𝒙
�𝑖 because otherwise by strict
• ………𝑢𝑖 𝒙
�𝑖 + 1 − 𝛼 𝒙
�𝑖 with
quasiconcavity of 𝑢𝑖 , consumer would choose 𝒙𝛼 = 𝛼𝒙
�𝑖 = 𝑢 𝑖 𝒙
�𝑖 , contradicting that 𝒙
�𝑖 is utility-maximizing in
𝑢 𝑖 𝒙𝛼 > 𝑢 𝑖 𝒙
the WE.
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Formal Proof
� is Pareto efficient then 𝒙
� is a WEA for
• Corollary of 2° WT: if 𝒙
� after redistribution of initial
some Walrasian equilibrium 𝒑
endowment to any allocation 𝒆∗ ∈ 𝐹 𝒆 , such that,
� ∙ 𝒆∗𝒊 = 𝒑
�∙𝒙
�𝑖 for every 𝑖 ∈ 𝐼
𝒑
• Redistribution + free market=Pareto efficiency
• Less equity concerns
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Welfare (4/4)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Equilibrium in Production
Producers (1/3)
• Production possibility set 𝑌𝑗 , 𝑗 ∈ ℐ
• 𝟎 ∈ 𝑌𝑗 ⊆ ℝ𝑛
• 𝑌𝑗 is closed and bounded
• 𝑌𝑗 is strongly convex
• No constant and increasing return to scale (profitmaximizing production plant is unique)
General Equilibrium, Berardino
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• Fixed number of firm, ℐ ∈ 1, . . , 𝐽
n
• Production plan of each firm: 𝑦 𝑗 ∈ ℜ +
j
• Input yk < 0
j
y
• output k > 0
Producers (2/3)
• Firm j’s maximization problem:
j
max
p
⋅
y
j
j
• Since objective function is continuous and the constraint set is
bounded and closed, a maximum for the profit function will
exist, given by:
j
Π j (p ) = max
⋅
p
y
j
j
y ∈Y
• The profit maximizing production plan, 𝒚𝑗 𝒑 , is:
•
• Unique by strong convexity, whenever 𝒑 ≫ 𝟎
n
• Continuous on ℜ + +
Π j (p ) is continuous on ℜ n+
Firm’s supply
function (Remind
Production
Theory)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
y ∈Y
Producers (3/3)
• Aggregate production possibilities
𝑌 ≡ 𝒚 �𝒚 = � 𝒚𝑗 , 𝑤𝑤𝑤𝑤𝑤 𝒚𝑗 ∈ 𝑌𝑗
• Y respects all the assumptions on 𝒀𝑗 .
• Aggregate profit 𝒑 ∙ 𝒚 has a unique maximum over 𝑌 when 𝒑 ≫ 𝟎
Define 𝒚 𝒑 : aggregate profit-maximizing plan (continuous in p)
� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐴𝐴 )
• Theorem: Aggregate profit maximization (𝒚
For any price 𝒑 ≥ 𝟎, we have:
� ≥ 𝒑 ∙ 𝒚 for all 𝒚 ∈ 𝑌
• 𝒑∙𝒚
�𝒋 ∈ 𝑌𝑗 , j∈ ℐ, we may write
If and only if for some 𝒚
�𝒋 , and
• i) 𝒚 = ∑𝑗∈ ℐ 𝒚
�𝒋 ≥ 𝒑 ∙ 𝒚𝒋 for all 𝒚𝒋 ∈ 𝑌𝑗 , j∈ ℐ
• ii) 𝒑 ∙ 𝒚
� maximizes aggregate profit iff it can be decomposed into
Intuition: 𝒚
individual firm profit-maximizing production plans
General Equilibrium, Berardino
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𝑗∈ ℐ
• Consumers may supply goods and services by labour-commodity
• Consumer i’s share in firm j (profit) 0 ≤ 𝜃 𝑖𝑖 ≤ 1, with ∑𝑖∈ 𝔗 𝜃 𝑖𝑖 = 1
for all 𝑗 ∈ ℐ
• Consumer i’s maximization problem:
( )
i
i
x
u
max
i
n
x ∈R+
s.t. : p ⋅ x ≤ p ⋅ e i + ∑ θ ij Π j (p ) = m i (p )
j∈J
• Solution 𝒙𝒊 𝒑, 𝑚𝒊 𝒑
(unique) exists whenever 𝒑 ≫ 𝟎:
• Continuous in 𝒑 (because 𝑚𝒊 𝒑 is continuous in 𝒑) on ℜ n
++
General Equilibrium, Berardino
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Consumers
Equilibrium
Let’s define the “refined” economy 𝑢𝑖 , 𝑒 𝑖 , 𝜃 𝑖𝑖 , 𝑌𝑗
• Aggregate excess demand for commodity k:
(
𝑖∈𝔗,𝑗∈ ℐ
)
zk (p ) ≡ ∑ xki p, mi (p ) − ∑ ykj (p ) − ∑ eki
j∈J
i∈ℑ
• And the aggregate excess demand
z (p ) ≡ ( z1 (p ),..., z n (p ))
• Theorem. If 𝑢𝑖 is continuous, strongly increasing and
quasiconcave , 𝟎 ∈ 𝑌𝑗 ⊆ ℝ𝑛 , 𝑌𝑗 is closed, bounded and strong
convex, 𝒚 + ∑𝑖∈𝔗 𝒆𝑖 ≫ 𝟎 for some 𝒚 ∈ ∑𝑗∈ℐ 𝒀𝑗 , then there
exist at least a price 𝒑∗ ≫ 𝟎 such that 𝒛 𝒑∗ = 𝟎.
General Equilibrium, Berardino
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i∈ℑ
•
•
•
•
•
1-consumer, 1-firm economy
𝑌 = −ℎ, 𝑦 |0 ≤ ℎ ≤ 𝑏, 𝑎𝑎𝑎 0 ≤ 𝑦 ≤ ℎ𝛼
Where b>0 and 𝛼𝛼 0,1
Y is closed, bounded, strongly convex and includes 0
2
Consumption, 2-good case ℜ +
• 𝑢 ℎ, 𝑦 = ℎ1−𝛽 𝑦 𝛽 ; 𝛽𝜖 0,1
• Endowment (hours): 𝒆 𝑇, 0 , with b>T (h<b)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Example (1/5)
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Example (2/5)
Example (3/5)
• Solution (find 𝒑∗ ≫ 𝟎 or (𝑤 ∗ , 𝑝∗ ))
• Firm:
max py
⇒ max phα − wh
h
y
• Where 𝑦 = ℎ𝛼 and ℎ > 0.
• Foc: 𝛼𝑝ℎ𝛼−1 − 𝑤 = 0
• ℎ𝑓 =
𝛼𝑝
𝑤
1
1−𝛼
; 𝑦𝑓 =
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Firm’s supply function
• Consumer’s demand function
• Market clearing condition
𝛼𝛼
𝑤
𝛼
1−𝛼
; 𝜋 𝑤, 𝑝 =
1−𝛼
𝛼𝛼
𝑤
𝛼
𝑤
1
1−𝛼
Example (4/5)
• Consumer
max u (h, y )
• s.t.
𝑝𝑝 + 𝑤𝑤 = 𝑤𝑤 + 𝜋 𝑤, 𝑝
• (net of profit) In fact from the endowment: 𝑤, 𝑝 ∙ 𝑇, 0 = 𝑤𝑤
• Solutions: ℎ𝑐 =
1−𝛽 𝑤𝑤+𝜋 𝑤,𝑝
𝑤
; 𝑦𝑐 =
𝛽 𝑤𝑤+𝜋 𝑤,𝑝
𝑝
• Since WE prices are positive and Excess demand is homogeneous of
degree zero we assume 𝑝∗ = 1
• By Walras’ law we only need to find 𝑤 ∗ 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 ℎ𝑐 + ℎ 𝑓 = 𝑇:
•
𝑤∗
=𝛼
1−𝛽 1−𝛼
𝛼𝛼𝑇
1−𝛼
>0
General Equilibrium, Berardino
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h, y
B: Technologically
possible but infeasible
(more than T)
C: Feasible but not
utility maximizer for
the consumer
−
w
p
Pareto Efficient
WEA
•C
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Example (5/5)
Welfare with production (1/7)
• WEA is feasible:
• 𝒙, 𝒚 =
𝒙1 , … , 𝒙𝐼 , = 𝒚1 , … , 𝒚𝑗
𝑖∈𝔗,𝑗∈ ℐ
n
is feasible if 𝒙𝑖 ∈ ℜ +
for all 𝑖, 𝒚𝑗 ∈ 𝑌𝑗 for all 𝑗, and ∑𝑖∈𝔗 𝒙𝑖 = ∑𝑖∈𝔗 𝒆𝑖 + ∑𝑗∈ℐ 𝒚𝑖
• 1° Welfare Theorem with Production: If 𝑢𝑖 is strictly increasing
on ℜ n+ then every WEA is Pareto Efficient.
General Equilibrium, Berardino
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• WEA Pareto efficient in the economy 𝑢𝑖 , 𝑒 𝑖 , 𝜃 𝑖𝑖 , 𝑌𝑗
Welfare with production (2/7)
• The WEA is also feasible: ∑𝑖∈𝔗 𝒙𝑖 = ∑𝑖∈𝔗 𝒆𝑖 + ∑𝑗∈ℐ 𝒚𝑗
�, 𝒚
�), feasible, such that: 𝑢𝑖 𝒙
�𝑖 ≥
• Pareto inefficiency implies ∃ (𝒙
�𝑖 ≥ 𝒑∗ ∙ 𝒙𝑖
𝑢𝑖 𝒙𝑖 with at least one with strict inequality, and 𝒑∗ ∙ 𝒙
�𝑖 > 𝒑∗ ∙ ∑𝑖∈𝔗 𝒙𝑖 (at least one gets more)
• Summing over i: 𝒑∗ ∙ ∑𝑖∈𝔗 𝒙
� 𝑗 > 𝒑∗ ∙ ∑𝑖∈𝔗 𝒆𝑖 + ∑𝑗∈ℐ 𝒚𝑗
• By feasibility: 𝒑∗ ∙ ∑𝑖∈𝔗 𝒆𝑖 + ∑𝑗∈ℐ 𝒚
� 𝑗 > 𝒑∗ ∙ ∑𝑗∈ℐ 𝒚𝑗 , for some j this would
After rearranging , 𝒑∗ ∙ ∑𝑗∈ℐ 𝒚
� 𝑗 > 𝒑∗ ∙ 𝒚𝑗 , but this contradicts that in the WE 𝒚𝑗
imply: 𝒑∗ 𝒚
maximizes firm j’s profit at 𝒑∗ ∎
General Equilibrium, Berardino
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Proof. By contradiction.
Assume 𝒙, 𝒚 is a WEA at 𝒑∗ but not Pareto efficient.
Welfare with production (3/7)
• 2° Welfare Theorem with production: Suppose an economy
with:
a) 𝑢𝑖 continuous, strong increasing and strict quasiconcave
b) 𝟎 ∈ 𝑌𝑗 ⊆ ℝ𝑛 , 𝑌𝑗 closed, bounded and strongly convex
c) 𝒚 + ∑𝑖∈𝔗 𝒆𝑖 ≫ 𝟎 for some aggregate production vector 𝒚
�, 𝒚
�)
d) A Pareto efficient allocation (𝒙
�
Then there exist income transfers 𝑇1 , … , 𝑇𝐼 |∑𝑖∈𝔗 𝑇𝑖 = 0 and 𝒑
such that:
�𝑖 maximizes 𝑢𝑖 𝒙𝑖 s.t. 𝒑
� ∙ 𝒙𝑖 ≤ 𝑚 𝑖 𝒑
� + 𝑇𝑖 , 𝑖 ∈ 𝔗
1. 𝒙
�𝑖 maximizes 𝒑
� ∙ 𝒚𝑖 s.t. 𝒚𝑗 ∈ 𝑌𝑗 , 𝑗 ∈ ℐ
2. 𝒚
General Equilibrium, Berardino
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• Is a Pareto efficient allocation also WEA ?
Welfare with production (4/7)
1.
2.
a new production set for each firm j still satisfying (b): 𝑌�𝑗 ≡ 𝑌𝑗 −
�𝑖
𝒚
�𝑖 , 𝜃 𝑖𝑖 , 𝑌�𝑗
A new economy ℰ̅ = 𝑢𝑖 , 𝒙
𝑖∈𝔗,𝑗∈ ℐ
Lemma: since the original economy ℰ = 𝑢𝑖 , 𝒆𝑖 , 𝜃 𝑖𝑖 , 𝑌𝑗 , has a WE, then
also ℰ̅ satisfies all the necessary conditions for the existence of the WE
with production (see revised theorem for the existence). Thus economy
�𝒚
� at 𝒑
� ≫ 𝟎.
ℰ̅ has a WEA 𝒙,
�, 𝒚
� can also be a WE of the original
Next step: check whether 𝒙
𝑖 𝑖 𝑖𝑖 𝑗
�𝑖 . Let’s use
economy,ℰ = 𝑢 , 𝒆 , 𝜃 , 𝑌 , first we need to show �
𝒙𝑖 =𝒙
feasibility.
�, 𝒚
� is feasible for the original economy by definition
• Remind: 𝒙
� ? also feasible for the original economy?
• Is 𝒙,
• Let’s check. Each consumer can afford his endowment because 𝟎 ∈ 𝑌�𝑗
implies that equilibrium profits of every firm j are non-negative. Thus:
�𝑖 ≥ 𝑢 𝑖 𝒙
�𝑖
Consumption side: 𝑢𝑖 𝒙
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Proof
• Let’ s construct:
Welfare with production (5/7)
Production side
�𝒚
� is feasible also in the original economy
• We now show that a 𝒙,
a)
b)
� 𝑗 ∈ 𝑌�𝑗 is such that 𝒚
� 𝑗 =𝒚
� 𝑗 -𝒚
�𝑖 for some 𝒚
� 𝑗 ∈ 𝑌𝑗 (see the definition of
𝒚
𝑌�𝑗 )
�𝒚
� is feasible in ℰ̅ because it is a WEA in ℰ̅
𝒙,
• a) + feasibility (a+b) imply:
�𝑗
• ∑𝑖∈𝔗 �
𝒙𝑖 = ∑𝑖∈𝔗 �
𝒙𝑖 + ∑𝑗∈ℐ 𝒚
�𝑖 + ∑𝑗∈ℐ 𝒚
� 𝑗 −𝒚
�𝑗
= ∑𝑖∈𝔗 𝒙
�𝑖 − ∑𝑗∈ℐ 𝒚
� 𝑗 +∑𝑗∈ℐ 𝒚
�𝑗
= ∑𝑖∈𝔗 𝒙
�𝑗
= ∑𝑖∈𝔗 𝒆𝑖 +∑𝑗∈ℐ 𝒚
�𝒚
� is feasible in the original economy ℰ. (still not enough…).
• Then 𝒙,
�, 𝒚
� is Pareto efficient, then we must have:
Since 𝒙
�𝑖 = 𝑢 𝑖 �
𝑢𝑖 𝒙
𝒙𝑖
�𝑖 =𝒙
�𝑖
• By quasiconcavity it implies: 𝒙
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Note that for now we have:
Welfare with production (6/7)
• 𝑢𝑖 𝒙𝑖 is strongly increasing then budget constraint equalizes at
�
�𝑖 . This implies:
𝒙𝑖 =𝒙
�∙𝒚
� 𝒋 =0 (zero profit condition for each firm j).
• ∑𝑗∈ℐ 𝜃 𝑖𝑖 𝒑
� 𝒋 =0 maximizes firm j’s profits at 𝒑
� under 𝑌�𝑗 in ℰ̅
• 𝒚
� 𝑗 then…
• Since 𝑌�𝑗 ≡ 𝑌𝑗 − 𝒚
� 𝑗 maximizes 𝒑
� ∙ 𝒚𝑗 s.t. 𝒚𝑗 ∈ 𝑌𝑗 , 𝑗 ∈ ℐ
• 𝒚
•
� 𝑗 characterize a WE also in
production side OK: the pareto efficient vector 𝒚
the original economy.
Transfer side (T): Back to the consumer side. We have shown that:
�𝑖 maximizes 𝑢𝑖 𝒙𝑖 s.t. 𝒑
� ∙ 𝒙𝑖 ≤ 𝒑
�∙𝒙
�𝑖
• (*) 𝒙
• Note that the transfer at the original economy to get (*) is such that
� ≡𝒑
�∙�
�∙�
� ,
𝑇𝑖 +𝑚𝑖 𝒑
𝒙𝑖 or 𝑇𝑖 ≡ 𝒑
𝒙𝑖 -𝑚𝑖 𝒑
• with consumer’s income at the original economy:
� =𝒑
� ∙ 𝒆𝑖 +∑𝑗∈ℐ 𝜃 𝑖𝑖 𝒑
�∙𝒚
�𝑗
𝑚𝑖 𝒑
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
• Thus we have:
�𝑖 maximizes 𝑢𝑖 𝒙𝑖 s.t. 𝒑
� ∙ 𝒙𝑖 ≤ 𝒑
�∙𝒙
�𝑖 +∑𝑗∈ℐ 𝜃 𝑖𝑖 𝒑
�∙𝒚
�𝒋
𝒙
Welfare with production (7/7)
�∙𝒙
�𝑖 − ∑𝑖∈𝔗 𝐩
� ∙ 𝐞i − ∑𝑖∈𝔗 ∑j∈ℐ θij 𝒑
�∙𝒚
�𝑗
• ∑𝑖∈𝔗 𝑇𝑖 = ∑𝑖∈𝔗 𝒑
From ∑𝑖∈ 𝔗 𝜃 𝑖𝑖 = 1
�∙𝒙
�𝑖 − ∑𝑖∈𝔗 𝐩
� ∙ 𝐞i − ∑𝑗∈ℐ 𝒑
�∙𝒚
�𝑗
• ∑𝑖∈𝔗 𝑇𝑖 = ∑𝑖∈𝔗 𝒑
�, 𝒚
� implies:
• See that feasibility of 𝒙
�∙𝒙
�𝑖 = ∑𝑖∈𝔗 𝐩
� ∙ 𝐞i + ∑𝑗∈ℐ 𝒑
�∙𝒚
�𝑗
• ∑𝑖∈𝔗 𝒑
• ⇒∑𝑖∈𝔗 𝑇𝑖 =0∎
General Equilibrium, Berardino
Cesi, MSc Tor Vergata
Last part to show: transfers must sum up to zero.