# General Equilibrium ```General Equilibrium
Microeconomics (Econ7790)
Benson Tsz Kin Leung1
Fall 2021
1 Hong
Kong Baptist University
1
Book Chapter
• Chapter 13 of Nicholson Snyder.
• Goal:
• To analyze equilibrium of multiple (competitive) markets.
• To derive welfare implications.
.
2
Motivation
• Ana’s demand curve for coffee depends on price of tea
3
• Suppose an illness infects tea trees, causing supply curve of
tea to shift to the left.
• equilibrium price of tea will go up
• How does this affect demand curve for coffee?
• equilibrium price of coffee?
• change in price of coffee shifts demand curve for tea
• etc. etc.
4
General Equilibrium
• Analyzes all markets at the same time
• Assume every market is perfectly competitive
• Goal: find equilibrium price and allocation for each market
• equilibrium prices: quantity demanded in each market =
quantity supplied in each market
• equilibrium allocation: how much of each good each
person/firm consumes/produces
• Is this allocation desirable?
5
Exchange Economy with no production
6
Model of an Exchange Economy
• endowment economy – no production
• I people: i = 1, 2, ..., I
• K goods: k = 1, 2, ..., K
• individual i’s utility function u i
• individual i’s endowment: how much of each good i owns
initially
prices? allocation? is the market efficient?
7
Model of an Exchange Economy
for illustration
• 2 people: Ann, Bob
• 2 goods: x an y
• endowment: exA , eyA for Ann, exB , eyB for Bob
8
Edgeworth Box
9
What is a good outcome?
10
Social Allocation
• how much of each good each individual consumes:
x A, y A , x B , y B
• feasible:
xA + xB
≤ exA + exB = ex
yA + yB
≤ eyA + eyB = ey
11
Comparing Two Allocations
• Allocation 1: x1A , y1A , x1B , y1B and allocation 2:
x2A , y2A , x2B , y2B
• Which allocation does Ann prefer? which does Bob prefer?
12
Pareto Criterion
Allocation 1 is better than allocation 2 if everyone agrees that
allocation 1 is better.
Definition
Allocation 1, x1A , y1A , x1B , x1B , is Pareto superior to
allocation 2, x2A , y2A , x2B , x2B if no one is worse off under
allocation 1 than under allocation 2:
u A x1A , y1A ≥ u A x2A , y2A
and
u B x1B , y1B ≥ u B x2B , y2B
and either Ann or Bob is strictly better off under allocation 1
than under allocation 2 (one of the inequality is strict).
13
Pareto Criterion
Definition
Allocation 1 is a Pareto improvement upon allocation 2 if
allocation 1 is Pareto superior to allocation 2.
14
Pareto Criterion
Only Ann and Bob.
15
Pareto Criterion
Definition
Allocation x∗A , y∗A , x∗B , x∗B is Pareto efficient if it is feasible
and if there exists no other feasible allocation that is Pareto
superior to x∗A , y∗A , x∗B , x∗B .
16
Pareto Criterion
Definition
Allocation star x∗A , y∗A , x∗B , y∗B is Pareto efficient if it is
feasible and if there exists no other feasible allocation that is
Pareto superior to x∗A , y∗A , x∗B , y∗B .
17
Social Allocation
• how much of each good each individual consumes:
x A, y A , x B , y B
• feasible:
xA + xB
≤ exA + exB = ex
yA + yB
≤ eyA + eyB = ey
• preference monotone: Pareto efficient allocation wastes no
endowment
=⇒
xB
= ex − x A
yB
= ey − y A
18
Social Allocation
xB
= ex − x A
yB
= ey − y A
19
Pareto Criterion
Is allocation Q Pareto superior to allocation E ?
20
Pareto Criterion
Is allocation E Pareto efficient?
21
Pareto Criterion
Allocation E is not Pareto efficient because allocation D is feasible
and Pareto superior to E
22
Pareto Criterion
We can find a Pareto improvement upon allocation E because
Ann’s MRS at exA , eyA 6= Bob’s MRS at exB , eyB
slope of Ann’s indifference curve at E 6=
slope of Bob’s indifference curve at E
23
Pareto Efficient Allocations
At a Pareto efficient allocation, Ann and Bob’s indifference curves
have the same slope:
A A
B
B
Ann’s MRS at x , y
= Bob’s MRS at x , y
24
Pareto Efficient Allocations
25
Pareto Efficient Allocation Can Be Unfair
26
The Set of All Pareto Efficient Allocations
27
Pareto Criterion
• Often many Pareto efficient allocations exist
• Does not say which Pareto efficient allocation is better
• a Pareto efficient allocation may not be desirable — may be
extremely unfair
• The society may deem an extremely unfair Pareto efficient
allocation worse than a fairer but Pareto inefficient allocation
28
Which Allocation will Happen?
29
Which Allocation will Happen if Bob is a Slave?
30
Which Allocations Might Happen if Trade Has to be Voluntary?
31
Which Allocation will Happen if Ann Has All the Bargaining
Power?
if Ann can choose any offer as long as Bob will agree to trade
32
Institution Determines Allocation
• institution determines who can do what and payoffs given
each person’s action, it affects allocation
• example
• slavery
• farm communes
• private property right and competitive market
33
Walrasian Equilibrium (Competitive Equilibrium)
• Suppose there is a perfectly competitive market for each good
(institutional assumption)
• market for good k is perfectly competitive if each individual
can buy and sell any amount of good k at the going price pk
• reasonable assumption if there are many small buyers and
many small sellers in each market, each taking price as given
• Rationality assumption for individual behavior:
• each consumer chooses the most preferred bundle in the
budget set
• each firm (if there is any) chooses input combinations (and
output) to maximize profits
• equilibrium price of the market for good k “clears the
market”: at this price,
quantity demanded = quantity supplied
34
Exchange economy (no production)
35
Walrasian Equilibrium (Competitive Equilibrium)
A Walrasian equilibrium consists of price for each market, px∗ , py∗ ,
and allocation, x ∗A , y ∗A , x ∗B , y ∗B , such that
• each market clears
x ∗A + x ∗B
= exA + exB
y ∗A + y ∗B
= eyA + eyB ;
• each invidual chooses optimally
x ∗A , y ∗A = Ann’s Marshallion demand at px∗ , py∗ , px∗ exA + py∗ eyA
x ∗B , y ∗B = Bob’s Marshallion demand at px∗ , py∗ , px∗ exB + py∗ eyB
36
Walrasian Equilibrium (Competitive Equilibrium)
• each market clears
x ∗A + y ∗A = exA + exB
y ∗A + y ∗B
= eyA + eyB ;
• there exists no x̃ A , ỹ A such that
px∗ x̃ A + py∗ ỹ A ≤ px∗ exA + py∗ eyA
u A x̃ A , ỹ A &gt; u A x ∗A , y ∗A
• There exists no x̃ B , ỹ B such that
px∗ x̃ B + py∗ ỹ B ≤ px∗ exB + py∗ eyB
u A x̃ B , ỹ B &gt; u A x ∗B , y ∗B
37
Walrasian Equilibrium (Competitive Equilibrium)
38
First Welfare Theorem
Theorem
If x ∗A , y ∗A , x ∗B , y ∗B is a Walrasian equilibrium allocation,
then it is Pareto efficient.
39
First Welfare Theorem
Theorem
If x ∗A , y ∗A , x ∗B , y ∗B is a Walrasian equilibrium allocation,
then it is Pareto efficient.
Proof
• Let px∗ , py∗ be the equilibrium price that generates this
allocation.
• Suppose to the contrary that x ∗A , y ∗A , x ∗B , y ∗B is not
Pareto efficient.
• Then we can find another allocation x̃ A , ỹ A , x̃ B , ỹ B
that is Pareto superior to x ∗A , y ∗A , x ∗B , y ∗B .
• At least one person is better off under x̃ A , ỹ A , x̃ B , ỹ B
than under x ∗A , y ∗A , x ∗B , y ∗B , and no one is worse off.
• Either Ann is better off, or Bob is better off.
40
First Welfare Theorem — Proof continued
Proof
Suppose Ann better off. Then
•
px∗ x̃ A + py∗ ỹ A &gt; px∗ exA + py∗ eyA
because x ∗A , y ∗A maximizes Ann’s utility given prices
px∗ , py∗ and her resulting income of px∗ exA + py∗ eyA
• Bob is not worse off. So
px∗ x̃ B + py∗ ỹ B ≥ px∗ exB + py∗ eyB .
If x̃ B , ỹ B costs less than his income, then a bundle exists
that costs no more than Bob’s income and is preferred to
x ∗B , y ∗B .
41
First Welfare Theorem — Proof continued
Proof
•
px∗ x̃ A + py∗ ỹ A &gt; px∗ exA + py∗ eyA
px∗ x̃ B + py∗ ỹ B ≥ px∗ exB + py∗ eyB .
• So
px∗ x̃ A + py∗ ỹ A + px∗ x̃ B + py∗ ỹ B &gt; px∗ exA + py∗ eyA + px∗ exB + py∗ eyB
42
First Welfare Theorem — Proof Continued
Proof
•
x̃ A , ỹ A , x̃ B , ỹ B must be feasible by definition of Pareto
superiority. x ∗A , y ∗A , x ∗B , y ∗B is also feasible by the
definition of equilibrium. So
x̃ A + x̃ B
= exA + exB = x ∗A + x ∗B
ỹ A + ỹ B
= eyA + eyB = y ∗A + y ∗B
So
px∗ x̃ A + x̃ B + py∗ ỹ A + ỹ B = px∗ exA + exB + py∗ eyA + eyB
px∗ x̃ A + py∗ ỹ A + px∗ x̃ B + py∗ ỹ B = px∗ exA + py∗ eyA + px∗ exB + py∗ eyB .
43
Competitive Equilibrium and First Welfare Theorem
equilibrium allocation affected by initial endowment E
44
Competitive Equilibrium and First Welfare Theorem
equilibrium allocation affected by initial endowment E
45
Competitive Equilibrium and First Welfare Theorem
equilibrium allocation affected by initial endowment E
46
Competitive Equilibrium and First Welfare Theorem
First Welfare Theorem does not say equilibrium allocation is
necessarily “better” than any other allocation.
47
Second Welfare Theorem
Theorem
Every Pareto efficient allocation can be the equilibrium allocation
under some endowment exA , eyA , exB , eyB .
48
Second Welfare Theorem
Theorem
Every Pareto efficient allocation can be the equilibrium allocation
under some endowment exA , eyA , exB , eyB .
49
Second Welfare Theorem
Theorem
Every Pareto efficient allocation can be the equilibrium allocation
under some endowment exA , eyA , exB , eyB .
50
Second Welfare Theorem
Theorem
Every Pareto efficient allocation can be the equilibrium allocation
under some endowment exA , eyA , exB , eyB .
51
First and second welfare theorem
• Competitive equilibrium yields “good” outcome
• All “good” outcomes can be implemented by a competitive
equilibrium by changing the distribution of endowments
52
Example
• Suppose x is amount of rice and y is the amount of vegetables
• exA , eyA = (1, 0), exB , eyB = (0, 1)
• Ann and Bob’s preference can be both represented by
√
√
u (x, y ) = α x + y
with α &gt; 1
53
Example
√
√
u (x, y ) = α x + y
• feasibility and no wasted resources
• allocation
xB
= exA + exB − x A = 1 − x A
yB
= eyA + eyB − y A = 1 − y A
x A , y A , 1 − x A , 1 − y A on the edgeworth box:
Ann’s MRS
Bob’s MRS
=
∂u
∂x
∂u
∂y
=
∂u
∂x
∂u
∂y
r
yA
xA
r
yB
|(x A ,y A ) = α
|(x B ,y B ) = α
xB
s
=α
1 − yA
1 − xA
54
Example
Pareto efficient allocation:
Ann’s MRS at x A , y A = Bob’s MRS at x B , y B
r
α
so
yA
=α
xA
s
1 − yA
1 − xA
yA
1 − yA
=
xA
1 − xA
y A − x Ay A = x A − x Ay A
So
xA = yA
xB = 1 − xA = 1 − yA = yB
55
Example
So every allocation x A , y A , x B , y B = ((t, t) , (1 − t, 1 − t))
where t ∈ [0, 1] is Pareto efficient
56
Example —- Competitive Equilibrium
• Need to find px∗ , py∗ such that market clears when both Ann
and Bob optimizes.
• can normalize py∗ = 1 because general inflation does not
change demand functions
57
Example —- Competitive Equilibrium
• Given px∗ , py∗ = (px∗ , 1), Ann will choose x ∗A , y ∗A that
maximizes her utility given her budget constraint
px∗ x A + y A ≤ px∗ exA + eyA
• FOC
MRS =
r
α
y ∗A
= px∗
x ∗A
So
y
∗A
=
px∗
= px∗
py∗
px∗
α
2
x ∗A
58
Example —- Competitive Equilibrium
• Substitute into budget line:
px∗ exA + eyA = px∗ x ∗A + y ∗A = px∗ x ∗A +
So
x ∗A =
px∗
α
2
x ∗A
px∗ exA + eyA
∗ 2
px∗ + pαx
59
Example —- Competitive Equilibrium
• Given px∗ , py∗ = (px∗ , 1), Ann’s optimal consumption bundle is
px∗ exA + eyA
∗ 2
px∗ + pαx
∗ 2
px
=
x ∗A
α
x ∗A =
y ∗A
• Do the same for Bob:
px∗ exB + eyB
∗ 2
px∗ + pαx
∗ 2
px
=
x ∗B
α
x ∗B =
y ∗B
60
Example —- Competitive Equilibrium
• Market for x has to clear: x ∗A + x ∗B = exA + exB
exA
+
exB
=x
∗A
+x
∗B
px∗ exA + exB + eyA + eyB
=
∗ 2
px∗ + pαx
• exA + exB = 1; eyA + eyB = 1 then
px∗
α
2
=1
So
px∗ = α
• Plug it back into Ann and Bob’s demand function to find
equilibrium allocation.
61
Example —- Competitive Equilibrium
• The more people like rice, the more expensive it is
• Write exA + exB = ex ; eyA + eyB = ey
px∗
α
2
ex = ey
r
ey
px∗ = α
ex
• More rare rice is, the more expensive it is
62
With production - one firm one consumer
63
Competitive equilibrium
• One firm that produces y with production function y = f (x).
• Consumer with utility function u(x, y ).
• Consumer with endowment ex , ey and receives firm’s profit.
64
Competitive equilibrium
• Firm maximize profit π = py∗ f (x) − px∗ x. (Solution x f )
• Consumer maximizes utility function u(x, y ) with price px∗ , py∗
and I . (Solution x c , y c )
• Income I = px∗ ex + py∗ ey + π.
• Market clears: x c = ex − x f and y c = ex + f (x f )
65
Example —- Competitive Equilibrium
√
• f (x) = β x
√
• u (x, y ) = xy
• ex = 1, ey = 0
66
Example —- Competitive Equilibrium
• Normalize py∗ = 1
• Firm:
• FOC: py∗ f 0 (x f ) − px∗ = 0
• √β f = px∗
2 x
• xf =
β2
4(px∗ )2 ,
yf =
β2
2px∗ ,
π=
β2
4px∗
67
Example —- Competitive Equilibrium
• Consumer:
2
β
px∗ + 4p
∗
I
∗
x
x = ∗ =
∗
2px
2px
∗
I
p
β2
y∗ = ∗ = x + ∗
2py
2
8px
68
Example —- Competitive Equilibrium
• Market clears:
y f = y ∗ + ey
β
px∗
β
=
+ ∗
∗
2px
2
8px
3β
= px∗
4px∗
√
3
∗
px =
β
2
• price of x increases in β.
69
Example —- Competitive Equilibrium
• Check:
xf =
∗
β2
1
=
4(px∗ )2
3
ex − x = 1 −
β2
4px∗
2px∗
px∗ +
=
1
3
70
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