Download Intermediate Microeconomics (22014)

Part III. General
Equilibrium
Intermediate Microeconomics (22014)
Part III. General Equilibrium
Instructor: Marc Teignier-Baqué
First Semester, 2011
Part III. General
Outline Part III. General Equilibrium
Equilibrium
Exchange
Production
Welfare
1. Pure Exchange Economy (Varian, Ch 31)
1.1
1.2
1.3
1.4
1.5
Edgeworth Box
The Core
Competitive Equilibrium
Welfare Theorems
Walras' Law
2. Production (Varian, Ch 32)
3. Welfare (Varian, Ch 33)
Part III. General
Topic 6. General Equilibrium
Equilibrium
Exchange
I
Up until now, partial equilibrium analysis:
I markets for goods analyzed in isolation, ignoring eect of
other prices on the market equilibrium;
I demand and supply functions of own price alone.
I
In general, however, demand and supply in several markets
interact to determine equilibrium prices of all goods.
I Substitutes and complements.
I People's income aected by goods sold.
I
In top 6, general equilibrium analysis: all markets clear
simultaneously.
I
Considerations of Pareto eciency and also of welfare distribution
and "social preferences."
Production
Welfare
Part III. General
Equilibrium
PURE EXCHANGE ECONOMY (Varian, Ch 31)
I
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
Welfare theorems
Walras' Law
Production
Welfare
I
Since very complex problem, simplications adopted:
I Only competitive markets studied, so that consumers and
producers take prices as given.
I Situations with, at most, two goods and two consumers.
I First, pure exchange economy : xed endowments, no
description of resources conversion to consumables.
I Afterwards, production introduced into the model.
Pure exchange economy:
I Two consumers, A and B, two goods, 1 and 2.
I Endowments of goods 1 and 2:
ω A = ω A, ω A , ω B = ω B , ω B .
1
I
1
2
Given a price vector (p1 , p2 ), consumers choose their favorite
aordable allocation (as in topic 1):
p1 x1A + p2 x2A ≤ p1 ω1A + p2 ω2A
p1 x B + p2 x B ≤ p1 ω B + p2 ω B
1
I
2
2
1
2
Prices must be such that allocations chosen are feasible:
xA + xB ≤ ωA + ωB , xA + xB ≤ ωA + ωB
1
1
1
1
2
2
2
2
Part III. General
Edgeworth Box
Equilibrium
I Edgeworth box is diagram showing all possible
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
allocations of the available quantities of goods 1 and 2
between the two consumers.
Welfare theorems
Walras' Law
I
Production
Welfare
I
The dimensions of the box are the quantities available
of the goods.
The allocations depicted are the feasible allocations .
Height = 2A  2B
Width =
Width = 1A  1B
Part III. General
Endowment allocation
Equilibrium
I The endowment allocation is the before-trade
Exchange
allocation:
Edgeworth Box
The Core
Competitive
equilibrium
ω 1B
Welfare theorems
Walras' Law
Production
Welfare
OB
ω 2B
ω 2A
+
Endowment
allocation
ll
A
ω 2B ω 2
OA
ω 1A
ω 1A + ω 1B
Part III. General
Equilibrium
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
Welfare theorems
Feasible reallocations
I Which reallocation will consumers choose?
I
I
Feasible.
Pareto-improving over the endowment allocation.
I An allocation is feasible if and only if
A + xB ≤ ωA + ωB
1
1
1
A
B
A
B
x +x ≤ ω +ω
Walras' Law
x1
Production
Welfare
2
2
2
2
I All points in the box, including the boundary, represent
feasible allocations of the combined endowments:
xB
1
ω 2A
xB
2
+
ω 2B
OB
xA
2
OA
x1A
ω 1A + ω 1B
Part III. General
Pareto-improving allocations
Equilibrium
Exchange
I An allocation is Pareto-improving over the endowment
Edgeworth Box
The Core
allocation if it improves the welfare of a consumer
Competitive
equilibrium
without reducing the welfare of another.
Welfare theorems
Walras' Law
Production
I Preferences of consumers A and B:
Welfare
xA
2
Preferences of consumer B
Preferences of consumer B
Preferences consumer A
x1B
 1B
OB
 2B
 2A
OA
 1A
x1A
x2B
Part III. General
Pareto-improving allocations
Equilibrium
I An allocation is Pareto-improving over the endowment
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
allocation if it improves the welfare of a consumer
without reducing the welfare of another.
Welfare theorems
Walras' Law
Production
 1B
Welfare
 2B
 2A
OA
OB
 1A
Set of Pareto‐improving allocations
I Since each consumer can refuse to trade, the only
possible outcomes from exchange are Pareto-improving
allocations.
Part III. General
Contract curve
Equilibrium
I An allocation is Pareto-optimal if the only way one
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
consumer's welfare can be increased is to decrease the
welfare of the other consumer.
Welfare theorems
Walras' Law
Production
Welfare
I The set of all Pareto-optimal allocations is called
contract curve .
Pareto‐optimal allocations are marked by .
Convex indifference curves are tangent at .  1B
OB
 2B
 2A
OA
 1A
Th
The contract curve
t t
Part III. General
The Core
Equilibrium
Exchange
Edgeworth Box
I The core is the set of all Pareto-optimal allocations
The Core
that are welfare-improving for both consumers relative
Competitive
equilibrium
to their own endowments.
Welfare theorems
Walras' Law
Production
Welfare
 1B
 2B
 2A
OA
OB
 1A
The Core: Pareto‐optimal p
trades not blocked by A or B.
I Rational trade should achieve a core allocation.
Part III. General
Equilibrium
Exchange
Trade in competitive markets
I Specic core alloation achieved depends upon the
manner in which trade is conducted.
Edgeworth Box
The Core
Competitive
equilibrium
Welfare theorems
Walras' Law
Production
I In perfectly competitive markets, each consumer is a
price-taker trying to maximize her own utility given
(p1 , p2 )
and her own endowment:
Welfare
xA
2
Consumer A optimization
A
A
p1x1A  p 2x A
2  p1 1  p 2 2
x2* A
 2A
OA
x1* A
I Similarly for consumer B.
 1A
x1A
Part III. General
Trade in competitive markets
Equilibrium
I At equilibrium prices p1 and p2 , both consumers
Exchange
Edgeworth Box
maximize their own utility and both markets clear:
The Core
A + xB = ωA + ωB
1
1
1
A
B
A
B
x +x = ω +ω
Competitive
equilibrium
x1
Welfare theorems
Walras' Law
2
Production
2
2
2
Welfare
Budget constraint for consumer B
x1*B
1B
x2* B
x2* A

 2B
A
2
OA
OB
x1* A
1A
Equilibrium allocation
Equilibrium allocation
Budget constraint for consumer A
Part III. General
First fundamental theorem of welfare economics
Equilibrium
Exchange
Edgeworth Box
The Core
Theorem
Given that consumers' preferences are well-behaved, trading
Competitive
equilibrium
in perfectly competitive markets implements a Pareto-optimal
Welfare theorems
allocation of the economy's endowment.
Walras' Law
Production
Welfare
I Note: Indierence curves are tangent, which implies that
the equilibrium allocation is Pareto optimal.
Part III. General
Equilibrium
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
Welfare theorems
Walras' Law
Second fundamental theorem of welfare economics
Theorem
Given that consumers' preferences are well-behaved, for any
Pareto-optimal allocation, there are prices and an allocation
of the total endowment that makes the Pareto-optimal
allocation implementable by trading in competitive markets.
Production
Welfare
I In other words, any Pareto-optimal allocation can be
achieved by trading in competitive markets provided
that endowments are rst appropriately rearranged.
Pareto‐optimal allocation cannot be implemented by competitive trading
implemented by competitive trading from initial endowment but it can be
implemented by competitive trading from alternative endowment .
l
d
OB
OA
Part III. General
Walras' Law
Equilibrium
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
Welfare theorems
Walras' Law
Theorem
If consumer's preferences are well-behaved, so that for any
positive prices
(p1 , p2 )
consumers spend all their budget, the
Production
summed market value of excess demands is zero. This is
Welfare
Walras' Law.
A + p xA = p ωA + p ωA
2 2
1 1
2 2
B
B
B
B
x +p x = p ω +p ω
p1 x1
p1
1
2 2
1
1
2
2
⇒
p1
A + xB − ωA − ωB + p xA + xB − ωA − ωB = 0
2
1
1
1
2
2
2
2
x1
Part III. General
Implications of Walras' Law
Equilibrium
I One implication of Walras' Law for a two-commodity
Exchange
Edgeworth Box
The Core
Competitive
equilibrium
exchange economy is that if one market is in equilibrium
then the other market must also be in equilibrium.
Welfare theorems
Walras' Law
Production
Welfare
p1
A + xB − ωA − ωB +p xA + xB − ωA − ωB = 0
2
2
1
1
1
2
2
2
x1
⇒ If x1A + x1B = ω1A + ω1B ,
A + xB = ωA + ωB.
2
2
2
then x2
I Another implication of Walras' Law for a two-commodity
exchange economy is that an excess supply in one
market implies an excess demand in the other market.
p1
A + xB − ωA − ωB +p xA + xB − ωA − ωB = 0
2
1
1
1
2
2
2
2
x1
⇒ If x1A + x1B < ω1A + ω1B ,
A + xB > ωA + ωB.
2
2
2
then x2
Part III. General
Outline Part III. General Equilibrium
Equilibrium
Exchange
Production
Robinson Crusoe
Economy
Competitive
Equilibrium
Welfare Theorems
1. Pure Exchange Economy (Varian, Ch 31)
Welfare
2. Production (Varian, Ch 32)
2.1 Robinson Crusoe economy
2.2 Competitive equilibrium
2.3 Welfare theorems
3. Welfare (Varian, Ch 33)
Part III. General
Equilibrium
Exchange
Production
Robinson Crusoe
Economy
Competitive
Equilibrium
Welfare Theorems
Welfare
PRODUCTION (Varian, Ch 32)
I Add input and output markets, rms' technologies.
I Robinson Crusoe's Economy:
I
I
I
One agent: Robinson Crusoe.
Endowment: a xed quantity of time.
Decision: use time for labor (production of coconuts) or
leisure.
I Technology: coconuts are obtained from labor according
to the production function C
= f (L).
Coconuts
Production function
Feasible production
l
plans
0
24
Labor (hours)
Part III. General
Equilibrium
Exchange
Production
Robinson Crusoe
Economy
Robinson Crusoe's preferences
I Indierence curves in the leisure-coconut diagram:
coconut is a good, leisure is a good:
Coconuts
More preferred
Competitive
Equilibrium
Welfare Theorems
Welfare
24
0
Leisure (hours)
I Indierence curves in the labor-coconuts diagram:
coconut is a good, labor is a bad.
Coconuts
More preferred
0
24
Labor (hours)
Part III. General
Robinson Crusoe's choice
Equilibrium
Exchange
Production
I Robinson chooses time allocation and, as a result, his
consumption of coconuts:
Robinson Crusoe
Economy
Competitive
Equilibrium
Welfare Theorems
Coconuts
Welfare
MRS = MPL
Production function
C*
Outputt
Labor
0
24
Leisure
L*
24
0
Labor (hours)
Leisure (hours)
Part III. General
Competitive equilibrium in the Robinson economy
Equilibrium
Exchange
Production
Robinson Crusoe
Economy
Competitive
Equilibrium
I Robinson esquizofrenia:
I
Welfare Theorems
Welfare
I
We rst consider Robinson as a prot-maximizing
rm , who takes prices as given and decides how much
hours to hire and how much to produce.
Then, we consider Robinson as a utility-maximizing
consumer who gets the rm prots and decides his
hours of work and his consumption of coconuts.
I Let p be the coconuts price and w the wage rate.
I
Use coconuts as the numeraire good; i.e. price of a
coconut = 1.
Part III. General
Equilibrium
Exchange
Robinson as a rm
I Optimization problem of the rm: given w , choose labor
demand and coconut supply to maximize prots:
Production
max π
Robinson Crusoe
Economy
L
Competitive
Equilibrium
Welfare Theorems
Welfare
I
= C − wL = f (L) − wL ⇒
MP (L
∗
)=w
Labor demanded: L∗ , output supplied: C ∗ = f (L∗ ).
I Graphically, rm demands L such that production
function tangent to isoprot line:
Coconuts
w = MPL
Isoprofit line:
 *  C *  wL *
Production function
C*
*
0
L*
24
Labor (hours)
Part III. General
Equilibrium
Robinson as consumer
I Optimization problem of the consumer: choose labor
Exchange
supply and coconut demand to maximize utility subject
Production
to the budget constraint:
Robinson Crusoe
Economy
Competitive
Equilibrium
Welfare Theorems
max U (C , L) s.t. C
C ,L
Welfare
I
= π ∗ + wL ⇒
∂ U (C , L) /∂ L
=w
∂ U (C , L) /∂ C
Labor supplies: L∗ , coconuts demanded: C ∗ .
I Graphically, consumer chooses C and L such that the
indierence curve is tangent to the budget constraint:
Coconuts
MRS = w
Budget constraint:
C   *  wL.
C*
*
0
L*
24
Labor (hours)
Part III. General
Market equilibrium
Equilibrium
Exchange
I In equilibrium, wage rate must be such that
Production
Robinson Crusoe
Economy
Competitive
Equilibrium
Welfare Theorems
quantity labor demanded = quantity labor supplied
(quantity output supplied = quantity output demanded)
Welfare
Coconuts
MRS = w = MPL
C*
*
0
L*
24
Labor (hours)
Part III. General
Equilibrium
First Fundamental Theorem of Welfare Economics
Theorem
Exchange
If consumers' preferences are convex and there are no
Production
externalities in consumption or production, a competitive
Robinson Crusoe
Economy
Competitive
Equilibrium
Welfare Theorems
Welfare
market equilibrium is Pareto ecient.
= MP :
Competitive equilibrium achieves Pareto eciency: w is
the common slope of the isprot line and the budget
constraint.
I Pareto eciency: MRS
I
Coconuts
0
MRS = MP
24
Labor (hours)
Part III. General
Equilibrium
Second Fundamental Theorem of Welfare
Economics
Exchange
Production
Robinson Crusoe
Economy
Competitive
Equilibrium
Welfare Theorems
Welfare
Theorem
If consumers' preferences are convex, rms' technologies are
convex, and there are no externalities in consumption or
production any Pareto ecient economic state can be
achieved as a competitive market equilibrium.
Part III. General
Non-convex technologies
Equilibrium
Exchange
I The First Welfare Theorems still holds if rms have
Production
non-convex technologies since it does not rely upon
Robinson Crusoe
Economy
rms' technologies being convex.
Competitive
Equilibrium
Welfare Theorems
Welfare
I The Second Welfare Theorem does not hold if rms
have non-convex technologies.
Coconuts
Coconuts
MRS = MPL
0
Iff competitive
i i equilibrium
ilib i
exists, the common slope
is the relative wage rate w
th t implements
that
i l
t the
th Pareto
P t
efficient plan by
decentralized pricing.
24
Labor (hours)
MRS = MPL.
0
This Pareto optimal
allocation cannot be
implemented by a
competitive equilibrium.
24
Labor (hours)
Part III. General
Outline Part III. General Equilibrium
Equilibrium
Exchange
Production
Welfare
1. Pure Exchange Economy (Varian, Ch 31
2. Production (Varian, Ch 32)
3. Welfare (Varian, Ch 33)
Part III. General
WELFARE (Varian, Ch 33)
Equilibrium
Exchange
Production
Welfare
I Social choice: Dierent economic states will be
preferred by dierent individuals. How can individual
preferences be aggregated into a social preference
over all possible economic states?
I Fairness: Some Pareto ecient allocations are unfair
(for example, one consumer eats everything). Under
what conditions, competitive markets guarantee that a
fair allocation is achieved?
Part III. General
Social welfare functions
Equilibrium
Exchange
Let ui (x ) be individual i's utility from overall allocation x.
Production
Welfare
I Utilitarian social welfare function:
n
W
=
∑ ui (x )
i =1
I Weighted-sum social welfare function:
n
W
=
∑ ai ui (x ) ,
i =1
ai
>0
I Minimax welfare function:
W
= min {u1 (x ) , u2 (x ) , ...., un (x )}
Part III. General
Fair allocations
Equilibrium
Exchange
Production
Welfare
I An allocation is fair if it is
I
I
Pareto ecient
envy free (no agent prefers the allocation of other
agents to their own).
I If every agent's endowment is identical, then trading in
competitive markets results in a fair allocation (may not
be true for non-competitive markets).