General Equilibrium

1
General Equilibrium

• APEC 3001
• Summer 2006
• Readings Chapter 16

2
Objectives

• General Equilibrium
• Exchange Economy
• With Production
• First Second Welfare Theorems

3
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.

4
Graphical Example of Edgeworth Exchange Box
Spencers Gum
Spencer
150
100
0
100
0
75
25
Spencers Candy
Masons Candy
0
100
150
0
50
Mason
Masons Gum
5
Question Can Mason Spencer do better?

• To answer this question, we need to know
  something about Mason Spencers 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.

6
Edgeworth Exchange Box With Indifference Curves
I2M gt I1M gt I0M
I2S gt I1S gt I0S
Spencers Gum
Spencer
150
100
0
100
0
I0S
I1S
75
25
I2S
Spencers Candy
Masons Candy
I2M
I1M
I0M
0
100
150
0
50
Mason
Masons Gum
7
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.

8
Gains From Trade
Spencers Gum
Spencer
150
100
0
100
0
IES
75
25
Spencers Candy
PARETO SUPERIOR ALLOCATIONS
Masons Candy
IEM
0
100
150
0
50
Mason
Masons Gum
9
Pareto Optimal Allocations
Spencers Gum
Spencer
150
100
0
100
0
IES
75
25
IPIS
b
Spencers Candy
PARETO SUPERIOR ALLOCATIONS
Masons Candy
a
IPIM
IEM
0
100
150
0
50
Mason
Masons Gum
10
What are the Pareto Optimal allocations?

• Contract Curve
• The set of all Pareto optimal allocations.

11
The Contract Curve
Spencers Gum
Spencer
150
100
0
100
0
Contract Curve
IES
75
25
IPIS
b
Spencers Candy
PARETO SUPERIOR ALLOCATIONS
Masons Candy
a
IPIM
IEM
0
100
150
0
50
Mason
Masons Gum
12
How can Mason Spencer get to a Pareto Optimal
allocation?

• Suppose the price of candy is PC0 the price of
  gum is PG0.
• Implications
• Masons Income M0M PC075 PG050
• Spencers Income M0S PC025 PG0100

13
Income Constraint With Prices PC0 and PG0 for
Candy and Gum
Spencers Gum
Spencer
150
100
M0S/PG0
0
100
0
75
25
Spencers Candy
Slope -PG0/PC0
Masons Candy
0
100
150
0
50
Mason
M0M/PG0
Masons Gum
14
Masons and Spencers Optimal Consumption Given
Prices PC0 and PG0
Spencers Gum
Spencer
150
100
M0S/PG0
0
G0S
100
0
I0S
75
25
C0S
Spencers Candy
Masons Candy
C0M
I0M
Slope -PG0/PC0
0
100
150
0
G0M
50
Mason
M0M /PG0
Masons Gum
15
Is this a market equilibrium?

• No!
• C0M C0S lt 100 ? Excess supply of candy!
• G0S G0S gt 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 lt PC0 PG1 gt PG0.

16
Masons and Spencers Optimal Consumption Given
Equilibrium Prices PC1 and PG1
Spencers Gum
Spencer
150
100
M0S/PG0
M0S/PG1
0
G0S
G1S
100
0
I0S
75
25
I1S
C0S
Spencers Candy
Masons Candy
C1S
C1M
C0M
I0M
I1M
Slope -PG1/PC1
Slope -PG0/PC0
0
100
150
0
G0M
M0M /PG1
G1M
50
Mason
M0M /PG0
Masons Gum
17
Is this a market equilibrium?

• Yes!
• C0M C0S 100 ? There is no excess demand or
  supply of candy!
• G0M G0S 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.

18
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.

19
Edgeworth Box for Candy and Gum Production
G2 gt G1 gt G0
Firm Gs Labor
Firm G (Gum)
C2
LE
0
KE
0
C1
C0
Firm Gs Capital
Firm Cs Capital
G0
G1
G2
0
KE
LE
0
Firm C (Candy)
C2 gt C1 gt C0
Firm Cs Labor
20
Efficient Production of Candy and Gum Production
Firm Gs Labor
Firm G (Gum)
LE
0
KE
C2
0
C1
More candy with same amount of gum!
Firm Gs Capital
PARETO SUPERIOR ALLOCATIONS
Firm Cs Capital
More gum with same amount of candy!
G1
G2
0
KE
LE
0
Firm C (Candy)
Firm Cs Labor
21
Contract Curve for Candy and Gum Production
G2 gt G1 gt G0
Firm Gs Labor
Firm G (Gum)
LE
0
C2
KE
0
C1
MRTSC MRTSG
C0
Firm Gs Capital
Firm Cs Capital
G0
G1
0
KE
LE
0
Firm C (Candy)
G2
C2 gt C1 gt C0
Firm Cs Labor
22
Competitive Cost Minimizing Production

• MRTSC MPLC/MPKC w/r
• MRTSG MPLG/MPKG w/r
• So, MRTSG w/r MRTSG
• Competitive production will result in Pareto
  Efficient production!

23
Graphical Example of Production Possibility
Frontier
Candy
Slope ?C/?G
C2
C1
C0
G0
G1
G2
Gum
24
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.

25
Note that TCG wLG rKG and TCC wLC rKC
? ?TCG w?LG r?KG and ?TCC w?LC r?KC
Also, LG LE LC and KG KE KC
? ?LG ?LC and ?KG ?KC
Therefore, ?TCG -w?LC - r?KC -?TCC
? ?TCG/ (?G?C) -?TCC/ (?C?G)
? MCG/?C -MCC/?G
? ?C/?G MCG/MCC
The Marginal Rate of Transformation is the ratio
of Marginal Cost!
26
Profit Maximization in Competitive Industry

• MCC PC
• MCG PG
• Implications
• MRT MCG/MCC PG/PC

27
Utility Maximization with Competitive Markets

• MRSM PG/PC
• MRSS PG/PC
• Implications
• MRT MRSM MRSS

28
Competitive Equilibrium with Production
Candy
Slope PG/PC
Spencer
GS
IS
CM
CS
IM
Mason
Gum
GM
29
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!
30
Adding Production Does Not Change The
Implications of The First and Second Welfare
Theorems!

• 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.

31
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

32
What You Should Know

• General Equilibrium Conditions
• Exchange Economy
• With Production
• Pareto Optimal Allocations
• First Second Welfare Theorems Caveats