Lecture 17
General equilibrium with exchange
General Equilibrium
• So far, we have considered how buyers and sellers behave.
• Buyers decide on consumption bundles by maximize utility subject to a budget constraint
• Sellers decide on production through profit maximization
• We then derived aggregate demand and aggregate supply from those individual decisions
• Prices adjust to equate supply and demand in a market
• If demand is higher than supply, prices rise
• If supply is higher than demand, prices fall
• However, we focused on one market at the time
• In reality, changes in one market affect other markets, so we cannot really study one
market in isolation
General Equilibrium
• The coming lectures, we are going to study general equilibrium
• This is an equilibrium (supply = demand) in multiple markets
simultaneously
• The key insight is that all these markets link to one another
• We are going to build it up slowly
Example of linked markets
• In the 1970s, there was a major increase in oil prices. These had many ripple effects:
• Kenyan Flower Industry:
• Higher oil prices in the 1970s increased greenhouse heating costs.
• Result: Flower production shifted to warmer countries like Kenya.
• Candy Bars:
• Higher oil prices raise production costs for goods, including candy bars.
• Brazil diverted sugar cane from sugar to ethanol production (to lower fuel costs), raising sugar prices
globally.
• Hence, candy bars became more expensive
• Brick Driveways:
• Crude oil is refined into various products like gasoline and asphalt.
• When gasoline prices rise, refiners prioritize gasoline, reducing asphalt supply.
• Result: Asphalt prices increase, leading homeowners to switch to alternatives like concrete or brick.
Exchange
• Let’s start with an economy with no production but where
agents can exchange goods
• An exchange economy is a good approximation of basic closed
economies such as prisons or the ISS
Exchange
• Two consumers: Consumer A and Consumer B
• Two goods: Good 1 and Good 2
• Both consumers start with a particular amount of goods 1 and 2
• These are their initial endowments
• For example:
• Because there is no production in an exchange economy, the total supply is equal to
the sum of the endowments
• Supply of Good 1: 6 + 2 = 8
• Supply of Good 2: 4 + 2 = 6
• If A and B can trade with each other, how much of each good will they end up with?
Exchange: Example
• Hunger games
Representing an exchange economy
• Edgeworth and Bowley devised a diagram, now called the Edgeworth box, to
represent show all possible allocations of such exchange economy
F-Y Edgeworth Sir Arthur Bowley
(1869 - 1957)
(1845-1926)
Starting an Edgeworth Box
Height =
The dimensions of
2A 2B the box are the
4 2 quantities available
6 of the goods
A B
Width = 1 1 6 2 8
Feasible Allocations
• What allocations of the available goods in the economy are feasible?
• The Edgeworth box diagram contains all feasible allocations in the economy
• Among them, one interesting allocation is the before-trade allocation; i.e., the
endowment allocation
• In all feasible allocations, demand=supply for both goods!
The Endowment Allocation
Height = The endowment
allocation is
2A 2B
A ( 6, 4 )
4 2 and
6 B ( 2, 2).
A B
Width = 1 1 6 2 8
The Endowment Allocation
OB
6
4
OA
A ( 6 ,4 )
6
8
The Endowment Allocation
2
OB
2
6
4
OA
6
B ( 2, 2)
8
The Endowment Allocation
2
OB
2
6 The
4 endowment
allocation
OA
A ( 6 ,4 )
6
B ( 2, 2)
8
Other Feasible Allocations
• denotes a final allocation to Edgeworth. These are E’s demands
• denotes an allocation to Bowley. These are B’s demands
• In all feasible allocations, demand=supply for both goods
• Hence, an allocation is feasible if and only if:
• And
• This is true for any point in the Edgeworth box!
What will be the outcome of trade?
• We need to find out how much of good 1 and 2 are demanded by A and B
• Here, we go back to the first weeks!
• In other words, we assume that people maximize utility
• The formalization will follow next lecture
• In other words, they will only trade (parts of) their initial endowment if
doing so increases their utility.
• Like before, we can represent people’s utilities over two goods using
indifference curves.
Adding Preferences to the Box
xA
2 For Edgeworth
2A
OA
1A x1A
Adding Preferences to the Box
xB
2 For Bowley
2B
OB
1B xB
1
Adding Preferences to the Box
xB 1B OB
1
x2B
2B
For Bowley
Edgeworth’s Box
xA
2
xB 1B OB
1
x2B
A
2 2B
OA A
1 x1A
Pareto-Improvement
• An allocation that improves the welfare of an individual without reducing the welfare of another
is a Pareto-improving allocation
• In other words, it’s a win-win situation where at least one party gains, but no one loses.
• Pareto improvements are key to understanding how resources can be allocated more efficiently.
• Many concepts in welfare economics revolve around achieving a Pareto optimal state
• Governments and institutions use the concept of Pareto improvements to design policies that can
make society better off without negatively impacting others.
• One important reason: Incomparability problem
• How can we compare the increase in ‘utility’ of one person to the decrease in ‘utility’ of another?
• Where are the Pareto-improving allocations in the Edgeworth box?
Pareto-Improvements
xA
2
xB 1B OB
1
x2B
2A 2B
OA
1A x1A
The set of Pareto-
improving allocations
Pareto-Improvements
• Since consumers can refuse to trade, the outcomes of trade ought to be
Pareto-improving allocations
• But which specific Pareto-improving allocation(s)?
Pareto-Improvements
xA
2
xB 1B OB
1
x2B
2A 2B
OA
1A x1A
The set of Pareto-
improving allocations
Let’s zoom in
Suppose
Edgeworth
trades with Bowley
some good 1 in exchange
for some of good 2
A Pareto-Improvement
This trade
improves both
consumers’ welfare
This is a Pareto-improvement
over the endowment allocation
A Pareto-Improvement
New mutual gains-to-trade
region is the set of all further
Pareto-improving allocations
Another Pareto-Improvement
Further trade
cannot improve
both consumers’
welfare!
Let’s zoom in again
Better for
Edgeworth
Better for
Bowley
Both are Let’s zoom in again
worse Edgeworth is
off strictly better off
Bowley is
strictly worse off
Bowley is strictly better off Both are
Edgeworth is strictly worse off worse
off
Pareto-Efficiency
The allocation is
Pareto-efficient
The only way one consumer’s
welfare can increase is by
decreasing the welfare of the other
Pareto-Efficiency
Where are all the Pareto-efficient feasible allocations?
Pareto-Efficiency
xA
2
xB 1B OB
1
x2B
2A 2B
OA
1A x1A
Pareto-Efficiency
• The set of all Pareto-efficient allocations is called the contract curve
• The outcome of exchange ought to be somewhere on that curve
• Otherwise Bowley and Edgeworth could find a deal leaving at least one of
them better off without hurting the other
Pareto-Efficiency
xA
2
xB 1B OB
1
x2B
2A 2B
OA
1A x1A
The contract curve
Pareto-Efficiency
• At any interior Pareto-efficient allocation, individual MRSs
should be equal
• Recall the MRS tells you the rate at which an individual is
willing to substitute one good for another
• If MRSA ≠ MRSB individuals would be better off by trading
between them
Pareto-Efficiency
• Suppose for instance Edgeworth’s MRS is 3
• He’s willing to give up up to 3 units of good 2 for 1 unit of good 1
• Suppose Bowley’s MRS is 1
• He is willing to give up 1 unit of good 2 for 1 unit of good 1 (and
viceversa)
• Hence, Edgeworth values good 1 relatively more than Bowley
• And they can trade for mutual benefit
The Shape Of The Contract Curve
• But all Pareto-efficient allocations are not necessarily
interior
• Recall that an allocation where one individual owns
everything is also PE
• The contract curve does not need to be a “curve” either
• It can be a straight line
• It can be an area!
A linear contract “curve”
xA
2
xB
1 OB
x2B
OA A
x1
The contract
x A curve is an area
2
B
x1 OB
x2B
OA
1A x1A
Pareto-Efficiency on the edge
xA
2
PE but MRSs
are different!
B
x1 OB
x2B
OA
x1A
Recap
• We have started analysing general equilibrium
• This is a situation where supply and demand are equal in
multiple markets simultaneously
• We have focused on an exchange economy, where supply is
fixed by initial endowments
• People trade their initial endowments to make themselves
better off
• Trade continues until all Pareto improvements have been
exhausted
