1. 12. General equilibrium: An exchange economy
Varian, Chapter 30

2. The simplest market Two people, or agents Two goods
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

3. 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?

4. Endowments Endowments A has (wAx, wAy) B has (wBx, wBy) y (wAy+wBy)
Any possible
bundle of
goods for A is
in this space
Endowments
A has (wAx, wAy)
B has (wBx, wBy)
Endowment
wAy x
Person A
(wAx+wBx)
wAx

5. Preferences y (wAy+wBy) Endowment A’s indifference wAy curves x
Person A
(wAx+wBx)
wAx

6. 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)

7. The Edgeworth Box: endowments
Person B
wBx y
Endowments
A has (wAx, wAy)
B has (wBx, wBy)
wBy
Endowment
Any feasible
allocation of
goods among
the agents is
a point in this
box
wAy
Person A x
wAx

8. The Edgeworth Box: preferences
Person B
wBx y
B’s indifference
curves
wBy
Endowment
A’s indifference
curves
wAy
Person A x
wAx

9. Trade in the Edgeworth Box
Person B
wBx y
B’s indifference
curves
wBy
Points to which
A would be willing
to trade
Endowment
A’s indifference
curves
wAy
Person A x
wAx

10. Trade in the Edgeworth Box
Person B
wBx y
B’s indifference
curves
wBy
Points to which
B would be willing
to trade
Endowment
A’s indifference
curves
wAy
Person A x
wAx

11. Mutually beneficial trade
Person B
wBx y
B’s indifference
curves
If trade is voluntary
they should end up
somewhere in here
wBy
Endowment
A’s indifference
curves
wAy
Person A x
wAx

12. What does the result look like?
Person B
wBx y
At there
are no more
gains from
trade
wBy
Endowment
wAy
Person A x
wAx

13. 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

14. The Pareto Set Person B wBx y Potential bargaining outcomes
from endowment
Allocation if A has
all the bargaining
power
wBy
Pareto set
Allocation if B has
all the bargaining
power
Endowment
wAy
Person A x
wAx

15. Example: Pareto set Consumer A: Consumer B: Find the contract curve
(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

16. The Pareto Set wBx Person B 10 y 15 Endowment wAy 10 5 wBy Pareto set
Person A
wAx x

17. 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

18. What about prices? Excess supply of x Excess demand of x
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

19. Gross demand A
Person B
wBx y
wBy
Endowment
wAy
Person A x
wAx

20. Gross demand B
Person B
wBx y
wBy
Endowment
wAy
Person A x
wAx

21. Market equilibrium At these prices, excess demand and excess
Person B
wBx y
At these prices, excess
demand and excess
supply are both zero
wBy
Endowment
wAy
Person A x
wAx

22. 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
wBy
Endowment
wAy
Budget constraint
Person A x
wAx

23. Prices and excess demand
wBx
Person B y
Excess
demand for y
Amount of
y that A
wants to
buy
wBy
Endowment
Amount of y that
B wants to sell
wAy
Budget constraint
Person A x
wAx

24. A’s price offer curve Person B wBx y Agent A’s price offer curve wBy
Endowment
wAy x
Person A
wAx

25. B’s price offer curve Person B wBx y wBy Agent B’s price offer curve
Endowment
wAy
Person A x
wAx

26. 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 x
wAx

27. 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.

28. The Pareto Set wBx Person B 10 8.57 y 15 wAy 10 5 wBy 9 6 Endowment
POC A
POC B
Pareto set 5
6.43 15
Person A
wAx x

29. 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

30. 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 x
wAx

31. 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

32. 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
Trade
Endowment
wAy
Redistribute good
x from B to A
Person A x
wAx

33. Policy implications How to help the poor Give them money!
Price controls, quantity controls, etc lead to inefficient outcomes