1 general equilibrium apec 3001 summer 2006 readings: chapter 16
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General Equilibrium
APEC 3001
Summer 2006Readings: Chapter 16
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Objectives
• General Equilibrium– Exchange Economy
– With Production
• First & Second Welfare Theorems
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General Equilibrium
• Definition: – The study of how conditions in each market in a set of related markets affect
equilibrium outcomes in other markets in that set.
• Example of Exchange Economy– Two people: Mason & Spencer
– Initial Endowments:• Mason: 75 pieces of candy & 50 pieces of gum.
• Spencer: 25 pieces of candy & 100 pieces of gum.
• Total: 100 pieces of candy & 150 pieces of gum.
• Edgeworth Exchange Box:– A diagram used to analyze the general equilibrium of an exchange economy.
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
Graphical Example of Edgeworth Exchange Box
75 25
50
100
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Question: Can Mason & Spencer do better?
• To answer this question, we need to know something about Mason & Spencer’s preferences.
• Assume:– Complete
– Nonsatiable
– Transitive
– Convex
• Implication:– Mason & Spencer have utility functions that produce indifference curves that
• represent higher levels of satisfactions as we move away from the origin,
• are ubiquitous,
• are downward sloping,
• cannot cross, &
• are bowed toward the origin.
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
Edgeworth Exchange Box With Indifference Curves
75 25
50
100
I0M
I1M
I2M
I2M
> I1M > I0
M I2S
> I1S > I0
S
I0S
I1S
I2S
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How can Mason & Spencer do better?
• Pareto Superior Allocation: – An allocation that at least one individual prefers and others like at least equally
as well.
• Pareto Optimal Allocation: – An allocation where it is impossible to make one person better off without
making at least one other person worse off.
• Consider the indifferences curves for Mason & Spencer that intersect the initial endowment.
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
Gains From Trade
75 25
50
100
IEM
IES
PARETO SU
PERIOR
ALLOCATIO
NS
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
Pareto Optimal Allocations
75 25
50
100
IEM
IES
PARETO SU
PERIOR
ALLOCATIO
NS
IPIS
IPIM
a
b
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What are the Pareto Optimal allocations?
• Contract Curve: – The set of all Pareto optimal allocations.
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
The Contract Curve
75 25
50
100
IEM
IES
PARETO SU
PERIOR
ALLOCATIO
NS
IPIS
IPIM
a
b
ContractCurve
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How can Mason & Spencer get to a Pareto Optimal allocation?
• Suppose the price of candy is PC0 & the price of gum is PG
0.
• Implications:– Mason’s Income: M0
M = PC075 + PG
050
– Spencer’s Income: M0S = PC
025 + PG0100
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
Income Constraint With Prices PC0 and PG
0 for Candy and Gum
75 25
50
100
Slope = -PG0/PC
0
M0S/PG
0
M0M/PG
0
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
Mason’s and Spencer’s Optimal Consumption Given Prices PC0 and PG
0
75 25
50
100
I0S
I0M
Slope = -PG0/PC
0
M0S/PG
0
M0M
/PG0
C0M
G0M
G0S
C0S
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Is this a market equilibrium?
• No!– C0
M + C0S < 100 Excess supply of candy!
– G0S + G0
S > 150 Excess demand for gum!
• So now what can we do?– Offer a higher price for gum or lower price for candy!
– For example, PC1 < PC
0 & PG1 > PG
0.
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Mason
Spencer
Mason’sCandy
Spencer’sCandy
Mason’s Gum
Spencer’s Gum
100
150
100150
00
00
Mason’s and Spencer’s Optimal Consumption Given Equilibrium Prices PC1 and PG
1
75 25
50
100
I0S
I0M
Slope = -PG0/PC
0
M0S/PG
0
M0M
/PG0
C0M
G0M
G0S
C0S
Slope = -PG
1/PC1
M0M
/PG1
M0S/PG
1
I1M
I1S
G1S
G1M
C1S
C1M
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Is this a market equilibrium?
• Yes!– C0
M + C0S = 100 There is no excess demand or supply of candy!
– G0M + G0
S = 150 There is no excess demand or supply of gum!
• What is true at this point?– MRSM = MRSS
– We are on the contract curve, so we are at a Pareto Optimal allocation!
• First Welfare Theorem: – Equilibrium in competitive markets is Pareto Optimal.
• Second Welfare Theorem: – Any Pareto optimal allocation can be sustained as a competitive equilibrium.
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General Equilibrium with Production
• Production Possibility Frontier: – The set of all possible output combinations that can be produced with a given
endowment of factor inputs.
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Firm C(Candy)
Firm G(Gum)
Firm C’sCapital
Firm G’sCapital
Firm C’s Labor
Firm G’s Labor
KE
LE
KE
LE
00
00
Edgeworth Box for Candy and Gum Production
G0
C2
C0
C1
C2 > C1 > C0
G1
G2
G2 > G1 > G0
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Firm C(Candy)
Firm G(Gum)
Firm C’sCapital
Firm G’sCapital
Firm C’s Labor
Firm G’s Labor
KE
LE
KE
LE
00
00
Efficient Production of Candy and Gum Production
C1
G1G2
C2
More gumwith same amount of
candy!
More candywith same amount of
gum!
PARETO SU
PERIOR
ALLOCATIO
NS
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Firm C(Candy)
Firm G(Gum)
Firm C’sCapital
Firm G’sCapital
Firm C’s Labor
Firm G’s Labor
KE
LE
KE
LE
00
00
Contract Curve for Candy and Gum Production
C1
G0
G1
C2
G2
C0
MRTSC = MRTSG
C2 > C1 > C0
G2 > G1 > G0
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Competitive Cost Minimizing Production
• MRTSC = MPLC/MPK
C = w/r
• MRTSG = MPLG/MPK
G = w/r
• So, MRTSG = w/r = MRTSG
• Competitive production will result in Pareto Efficient production!
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Graphical Example of Production Possibility Frontier
Candy
GumG0 G1 G2
C0
C1
C2Slope = C/G
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Production Possibility Frontier
• Marginal Rate of Transformation: – The rate at which one output can be exchanged for another at a point along the
production possibility frontier: |C/G|.
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Note that TCG = wLG + rKG and TCC = wLC
+ rKC
TCG = wLG + rKG and TCC = wLC + rKC
Also, LG = LE – LC and KG = KE – KC
LG = – LC and KG = –KC
Therefore, TCG = -wLC - rKC = -TCC
TCG/ (GC) = -TCC/ (CG)
MCG/C = -MCC/G
|C/G| = MCG/MCC
The Marginal Rate of Transformation is the ratio of Marginal Cost!
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Profit Maximization in Competitive Industry
• MCC = PC
• MCG = PG
• Implications:– MRT = MCG/MCC = PG/PC
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Utility Maximization with Competitive Markets
• MRSM = PG/PC
• MRSS = PG/PC
• Implications:– MRT = MRSM
= MRSS
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Competitive Equilibrium with Production
Candy
Gum
|Slope| = PG/PC
Mason
Spencer
IM
IS
GM
CM CS
GS
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Summary
• For a general equilibrium with production to be Pareto Efficient, three types of conditions must hold:– Firms must equate their marginal rates of technical substitution.
– Consumers must equate the marginal rates of substitution.
– Consumers’ marginal rates of substitution must equal the marginal rate of transformation.
Competitive Markets Yield This Outcome!
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• Competitive markets result in the Pareto efficient production and distribution of goods and services!
• Any Pareto efficient production and distribution of goods can be supported by a competitive market.
Adding Production Does Not Change The Implications of The First and Second Welfare
Theorems!
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So, is there anything that can mess up these welfare theorems?
• Yes!
• Government Intervention– Taxes
– Subsidies
• Market Failure– Externality: Either a benefit or a cost of an action that accrues to someone
other than the people directly involved in the action.
– Public Goods: (1) nondiminishability and (2) nonexcludability of consumption.
• Noncompetitive Behavior– Monopoly
– Oligopoly
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What You Should Know
• General Equilibrium Conditions– Exchange Economy
– With Production
• Pareto Optimal Allocations
• First & Second Welfare Theorems & Caveats