General vs. Partial Equilibrium
• Partial equilibrium
– Analysis of a single market
– Assumes price, quantity interactions with all other
markets small enough to be ignored
General Equilibrium/Exchange
• General equilibrium
– Considers all interactions among markets
– Simultaneously obtains prices in all markets
– Consistent relationships among all markets for all
variables (factors, incomes, etc.)
ECON 370: Microeconomic Theory
Summer 2004 – Rice University
Stanley Gilbert
Econ 370 - Exchange
General Equilibrium Analysis
Exchange: Introduction
• Start with theory of pure exchange
• Two consumers, A and B
– Assume fixed set of consumer goods (ignore how these
goods are produced)
– Assign goods to individuals: endowments
– Analyze exchange of endowments
• Endowments of goods 1 and 2 are
• ωA = (ω1A, ω2A) for Consumer A
• ωB = (ω1B, ω2B) for Consumer B
• Will consider “production” of these goods later in
general equilibrium context
– E.g., ωA = (6, 4) and ωB = (2, 2)
• Total quantities available are
• Will analyze 2-consumer, 2-good model
Econ 370 - Exchange
2
– ω1A + ω1B = 6 + 2 = 8 units of good 1
– ω2A + ω2B = 4 + 2 = 6 units of good 2
3
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1
Exchange: The Edgeworth Box
Starting an Edgeworth Box
• Theory of Exchange
– Modeled using Edgeworth box
– Shows all possible allocations of fixed quantities of
goods 1 and 2 between consumers A and B
– Start with initial allocation = endowments
– After trade, end up with final allocation
– Traded quantities must be consistent
Height =
ω 2A + ω 2B
= 4+2
=6
The dimensions of
the box are the
quantities available
of the goods
A
B
Width = ω1 + ω1 = 6 + 2 = 8
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Econ 370 - Exchange
Edgeworth Box: The Endowment
2
E
6
4
OA
ω
A
= ( 6 ,4 )
Econ 370 - Exchange
Other Feasible Allocations
OB
• (x1A, x2A) denotes an allocation to consumer A
2
• (x1B, x2B) denotes an allocation to consumer B
• Feasible allocation satisfies
– x1A + x1B ≤ ω1A + ω1B and
– x2A + x2B ≤ ω2A + ω2B
The
endowment
allocation
6
6
• So, all points in Edgeworth box (including those
on boundary) are Feasible
8
ω B = ( 2, 2 )
7
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2
Feasible Reallocations: Diagram
OB
x1B
E
xA
2
x2B
For consumer A
Mo
re
pr
ef
er
re
x2A
ω2A + ω2B
Adding Preferences - Consumer A
OA
x1A
OA
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Adding Preferences - Consumer B
ω 2B
OB
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Mo
re
E
ω 1B
pr
ef
er
re
ω 1A
x1A
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Adding Preferences – Consumer B
For consumer B
xB
2
E
ω 2A
ω1A + ω1B
d
x1B
d
For consumer B
Mo
re
pr
ef
er
re
ω1B
E
OB
ω 2B
d
x B2
xB
1
11
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3
Edgeworth Box
Pareto-Improving Reallocations
x 2A
• Pareto-improving reallocation
ω1B
x1B
ω 2A
OA
– Re-allocation of endowment improving welfare of one
consumer w/o reducing welfare of the other
OB
• Only Pareto-improving reallocations will occur
through voluntary trade
• Can be easily identified on Edgeworth box
diagram
ω 2B
ω1A
x B2
x1A
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Econ 370 - Exchange
Pareto Improving Trades
Pareto-Optimality: Diagram
x 2A
ω1B
x1B
Pareto-optimal
Allocation where convex
indifference curves are
tangent
OB
Pareto-Improving Trades
ω 2A
OA
Econ 370 - Exchange
ω 2B
ω1A
14
x B2
Only way one consumer’s
welfare can be increased is to
decrease that of the other
x1A
15
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4
Pareto-Optimality: Diagram
Contract Curve
“Contract Curve” is the set of all Pareto-optimal allocations
A is strictly better
off but B is strictly
worse off
Both A
and B are
worse off
B is strictly better
off but A is strictly
worse off
x 2A
OA
ω1A
Contract curve
17
OB
ω 2B
ω 2A
Both A
and B are
worse off
Econ 370 - Exchange
ω1B
x1B
x1A
x B2
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The Core: Definition
“Core”
• Core - set of all allocations that are:
Pareto-Improving Allocations
x 2A
– Pareto optimal allocations, and
– Welfare-improving for both consumers, given original
endowments
ω1B
x1B
OB
• Rational trade should achieve allocation in core
• Which core allocation?
– Depends on allocation mechanism: Competition,
Monopoly, negotiation, etc.
ω 2B
ω 2A
OA
Core
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Econ 370 - Exchange
ω1A
x1A
x B2
20
5
Trade in Competitive Markets
Trade in Competitive Markets
Budget Line
– Consumers are price takers
– Endowment determines location of budget line
(analogous to income constraint)
– Endowment may not be optimal Consumption bundle
– Consumers trade to reach optimum
p1 x1A + p2 x2A = p1ω1A + p2ω2A
Optimum
x *2 A
• Note: not realistic in two person case
ω 2A
– Two groups
– True for many individuals
Econ 370 - Exchange
For consumer A
x 2A
• Suppose final allocation is determined by a
competitive price mechanism
Endowment
OA
21
x1* A
x1A
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Trade in Competitive Markets
22
Trade in Competitive Markets
For consumer B
• So, given p1, p2, consumer A wants to
x B2
– Reduce consumption of good 1
– Increase consumption of good 2
– (Relative to endowments of 1 and 2)
Budget Constraint
p1 x1B + p2 x 2B = p1ω1B + p 2ω2B
• Thus, net demands for commodities 1, 2 are
Endowment
ω 2B
– x1*A – ω1A,
– x2*A – ω2A
Optimum
x *2B
OB
Econ 370 - Exchange
ω1A
23
Econ 370 - Exchange
ω1B
x1*B
x1B
24
6
Trade in Competitive Markets
General equilibrium
• So, given p1, p2, consumer B wants to
• General Equilibrium occurs for (p1, p2) such that
both commodity markets clear:
– Increase consumption of good 1
– Decrease consumption of good 2
– (Relative to endowments of 1 and 2)
– x1*A + x1*B = ω1A + ω1B
– x2*A + x2*B = ω2A + ω2B
• Requires (p1, p2) tangent to both IC’s at same
point, so Supply = Demand for both goods
• Thus, net demands for commodities 1, 2 are
– x1 – ω1
– x2*B – ω2B
*B
B,
• Recall price line is budget line for both A, B, and
endowment point is same for A, B
Econ 370 - Exchange
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Econ 370 - Exchange
Trade in Competitive Markets
x
B
1
x
A
2
Econ 370 - Exchange
Trade in Competitive Markets
• At prices (p1, p2) both markets clear
ω1B
A
1
ω
• There is a general equilibrium
OB
ωB2
ω2A
OA
26
x B2
• Trading in competitive markets achieves a specific
Pareto optimal reallocation of endowments
• Pareto optimality in exchange also referred to as
efficiency in consumption
x1A
27
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7
Efficiency in Consumption
Efficiency in Consumption
• Efficiency in consumption requires
MRSA =
• Efficiency in consumption achieved with
competitive markets
dx 2A dx 2B
=
= MRSB
dx1A dx1B
• All consumers face the same p1 / p2
• So, all have the same MRS
• Or mutually beneficial trades exist
• So, efficiency in consumption achieved
• (MRS’s reflect relative valuations of x1)
Econ 370 - Exchange
• Or, Pareto optimal exchange
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Econ 370 - Exchange
Walras’ Law: Introduction
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Walras’ Law: Derivation
• Walras’ Law is an identity
• Assume every consumer’s preferences are wellbehaved so, for any (p1, p2), spends all of budget
• Identity
– a statement that is true for any positive prices (p1, p2)
– these need not be equilibrium prices
• For consumer A
• p1x2*A + p2x2*A = p1ω1A + p2ω2A
• Walras’ Law important for constructing general
equilibrium models
• For consumer B
• p1x2*B + p2x2*B = p1ω1B + p2ω2B
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8
Walras’ Law: Statement
Walras’ Law: Derivation
p1 ( x1* A + x1* B − ω1A − ω1B ) + p 2 ( x 2* A + x 2* B − ω 2A − ω 2B ) = 0
p1 x1* A + p2 x2* A = p1ω1A + p2ω2A
p1 x1*B + p2 x2*B = p1ω1B + p2ω2B
• Walras’ Law:
– For any positive prices p1 and p2,
– summed market value of excess demands = zero
Summing:
p1 ( x1* A + x1* B ) + p 2 ( x 2* A + x 2* B ) = p1 (ω1A + ω1B ) + p 2 (ω 2B + ω 2B )
Rearranging:
p1 ( x1* A + x1* B − ω1A − ω1B ) + p 2 ( x 2* A + x 2* B − ω 2A − ω 2B ) = 0
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First Implication of Walras’ Law
+
x1* B
− ω1A
− ω1B
34
Second Implication of Walras’ Law
Suppose market for commodity A is in equilibrium:
x1* A
Econ 370 - Exchange
If there is excess supply of commodity 1:
x1* A + x1* B − ω1A − ω1B < 0
=0
Then, Walras’ Law implies the other market (for
commodity B) must be in equilibrium
Then, there must be excess demand for 2:
p1 ( x1* A + x1* B − ω1A − ω1B ) + p 2 ( x 2* A + x 2* B − ω 2A − ω 2B ) = 0
p1 ( x1* A + x1* B − ω1A − ω1B ) + p 2 ( x 2* A + x 2* B − ω 2A − ω 2B ) = 0
⇒ x*2A + x*2B − ω2A − ω2B = 0
⇒ x*2A + x *2B − ω2A − ω2B > 0
General: n – 1 mkts in eq’m, n-th mkt is in equilibrium.
So, reducing p1/p2 moves both mkts to equilibrium
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9
Efficiency In Consumption
Theorems of Welfare Economics
• First part of Adam Smith’s famous Invisible Hand
Theorem
• 1st Fundamental Theorem:
– If consumers’ preferences are well-behaved (convex,
continuous),
– Then trading in perfectly competitive markets will
achieve a Pareto optimal reallocation of the economy’s
endowment
– Efficiency in consumption allocation is an important
social goal
– Private consumers act to maximize own utility
– But, in competitive markets, private actions achieve
social goal, as if guided by an Invisible Hand
• 2nd Fundamental Theorem
– If consumers’ preferences are well-behaved (convex,
continuous), and
– Given appropriate rearrangement of endowments
among consumers,
– Any Pareto optimal allocation can be achieved by
trading in competitive markets
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2nd Fundamental Theorem: Diagram
38
2nd Fundamental Theorem: Diagram
Achieved w/ competitive trading from endowment
x1B
x
A
2
B
1
ω
OA
The contract curve
Econ 370 - Exchange
OB
ωB2
ω2A
A
1
ω
x1B
x B2
x1A
39
x 2A
ω1B
ωB2
ω2A
OA
Econ 370 - Exchange
OB
A
1
ω
x B2
x1A
40
10
2nd Fundamental Theorem: Diagram
2nd Fundamental Theorem: Diagram
Can this allocation be achieved w/ trading from ω?
No
x1B
x 2A
B
1
ω
ωB2
ω2A
OA
x1B
OB
A
1
ω
x1A
x B2
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x 2A
ω1B
ωB2
ω2A
OA
Econ 370 - Exchange
OB
A
1
ω
x B2
x1A
42
2nd Fundamental Theorem: Diagram
But can be achieved from alterative endowment θ
x1B
x 2A
ω1B
OB
θ
ωB2
ω2A
OA
Econ 370 - Exchange
A
1
ω
x B2
x1A
43
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