12. General equilibrium:
An exchange economy
Varian, Chapter 30
The simplest market
• Two people, or agents
– Agent A and Agent B
• Two goods
– Good x and Good y
• Agents interact by exchanging or trading goods
• There is no production of either good
Questions we’re interested in
• Will unregulated exchange lead to “good”
outcomes?
– Under what conditions?
• How does trade occur?
– Bargaining?
– Using prices?
• Can an outsider (e.g., government)
intervene to improve things?
Endowments
(wAy+wBy) y
• Endowments
• Any possible
bundle of
goods for A is
in this space
A has (wAx, wAy)
B has (wBx, wBy)
Endowment
wAy
x
Person A
wA
x
(wAx+wBx)
Preferences
(wAy+wBy) y
wAy
Endowment
A’s indifference
curves
x
Person A
wA
x
(wAx+wBx)
The simplest market
• Allocation: A pair of consumption bundles
– (xA, yA), (xB, yB)
• Feasible allocation: pair of consumption
bundles that add up to total endowment
– (xA, yA)+(xB, yB)=(wAx, wAy)+(wBx, wBy)
The Edgeworth Box: endowments
wBx
Person B
y
• Endowments
A has (wAx, wAy)
wBy
B has (wBx, wBy)
Endowment
wAy
Person A
wAx
x
• Any feasible
allocation of
goods among
the agents is
a point in this
box
The Edgeworth Box: preferences
wBx
y
Person B
B’s indifference
curves
wBy
wAy
Person A
A’s indifference
curves
wAx
Endowment
x
Trade in the Edgeworth Box
wBx
y
Person B
B’s indifference
curves
wBy
Points to which
A would be willing
to trade
wAy
Person A
A’s indifference
curves
wAx
Endowment
x
Trade in the Edgeworth Box
wBx
y
Person B
B’s indifference
curves
wBy
Points to which
B would be willing
to trade
wAy
Person A
A’s indifference
curves
wAx
Endowment
x
Mutually beneficial trade
wBx
y
Person B
B’s indifference
curves
wBy
If trade is voluntary
they should end up
somewhere in here
wAy
Person A
A’s indifference
curves
wAx
Endowment
x
What does the result look like?
wBx
Person B
y
At
there
are no more
gains from
trade
wBy
Endowment
wAy
Person A
wAx
x
Describing potential outcomes
• Pareto set, or contract curve: The set of all
points that could be the outcome of a
bargain
– Depends on w
– Can only make one individual better off by
making the other worse off
The Pareto Set
wBx
y
Potential bargaining outcomes
from endowment
Person B
Allocation if A has
all the bargaining
power
wBy
Pareto set
wAy
Person A
Allocation if B has
all the bargaining
power
wAx
Endowment
x
Example: Pareto set
• Consumer A:
– (wAx, wAy)=(5,10)
– u(xA, yA)=1/3 ln(xA) + 2/3 ln(yA)
• Consumer B:
– (wBx, wBy)=(10,5)
– u(xB, yB)=1/2 ln(xB) + 1/2 ln(yB)
• Find the contract curve
The Pareto Set
y 15
wBx
10
Person B
Endowment
wAy 10
5 wBy
Pareto set
Person A
5
wAx
15
x
Can we narrow down outcomes?
• So far we’ve said nothing about the mechanism
or process by which people trade
• We’ve found that agents should get to the
contract curve…..
• …..but we’re not sure what point they’ll reach on
that curve
• If trading is via prices, this indeterminacy can be
resolved
What about prices?
• A price-based process
–
–
–
–
–
Set py=1
Try a value of px=p
Gives slope of budget line for both consumers
Find gross demands of both consumers
Vary p until demand is a feasible allocation
• Excess supply of x
– Lower p
• Excess demand of x
– Raise p
Gross demand A
wBx
Person B
y
wBy
Endowment
wAy
Person A
wAx
x
Gross demand B
wBx
Person B
y
wBy
Endowment
wAy
Person A
wAx
x
Market equilibrium
wBx
y
Person B
At these prices, excess
demand and excess
supply are both zero
wBy
Endowment
wAy
Person A
wAx
x
Prices and excess supply
Amount of x that
A wants to sell
wBx
Person B
y
Amount of x that
B wants to buy
Excess supply of x
Endowment
wAy
Budget constraint
Person A
wAx
x
wBy
Prices and excess demand
wBx
y
Person B
Excess
demand for y
Amount of
y that A
wants to
buy
wAy
Person A
wBy
Amount of y that
B wants to sell
wAx
Endowment
Budget constraint
x
A’s price offer curve
wBx
Person B
y
Agent A’s price
offer curve
Endowment
wAy
Person A
wBy
x
wAx
B’s price offer curve
wBx
Person B
y
wBy
Agent B’s price
offer curve
Endowment
wAy
Person A
wAx
x
Using price offer curves to find the
market equilibrium
Person B
wBx
y
Agent A’s price
offer curve
wBy
Agent B’s price
offer curve
Endowment
wAy
Person A
wAx
x
Example: Price offer curves
• Consumer A:
– (wAx, wAy)=(5,10)
– u(xA, yA)=1/3 ln(xA) + 2/3 ln(yA)
• Consumer B:
– (wBx, wBy)=(10,5)
– u(xB, yB)=1/2 ln(xB) + 1/2 ln(yB)
• Find the IOCs and the equilibrium
allocation and price.
The Pareto Set
wBx
10 8.57
y 15
wAy 10
9
5 wBy
6
Endowment
POC A
Pareto set
Person A
Person B
5 6.43
wAx
POC B
15
x
The First Fundamental Theorem
of Welfare Economics
• The First Fundamental Theorem of
Welfare Economics
– All equilibria resulting from a competitive
market are Pareto efficient
– There are no gains from trade available from
the result of a price-based exchange
• There may be other undesirable properties
of a market outcome
The First Fundamental Theorem of
Welfare Economics
Person B
wBx
y
The market equilibrium
allocation is on the contract
curve, so is Pareto efficient
wBy
Contract curve
Endowment
wAy
Person A
wAx
x
The Second Fundamental
Theorem of Welfare Economics
• The Second Fundamental Theorem of
Welfare Economics
– If preferences are convex, any Pareto efficient
allocation can be reached by a competitive
market, with the correct redistribution of the
endowments
– Using redistribution then allowing price-based
trade, a planner can choose any PE allocation
The Second Fundamental Theorem
of Welfare Economics Person B
wBx
y
Pareto efficient
allocations
Can the market get us
to any Pareto efficient
allocation we want?
wBy
Answer: Yes, with
convex preferences
and redistribution
Endowment
wAy
Person A
Trade
wAx
Redistribute good
x from B to A
x
Policy implications
• How to help the poor
– Give them money!
– Price controls, quantity controls, etc lead to
inefficient outcomes