Market | Welfare 1
The First Welfare Theorem
In the last lecture two concepts were introduced,Pareto e–ciency and general equilibrium,How do they relate?
Theorem,The flrst welfare theorem states that every general equilibrium involves a Pareto e–cient allocation.
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.,,,,,,,,,,,
............
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
...
...........................................................................................................................................................................................................
..........................................................................................................................................................................................................
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
...........
.
.......
..
......
......
......
......
......
......
......
......
......
......
......
......
...........
.
.......
..
......
.......
.......
.....
...
.......
..........
...........
...........
...............
................
...
................
...............
...........
............
..........
.......
.
.....
.......
.......
......
.A
B
xA
xBx2A
x2B
The proof works by contradiction,Suppose there was a non-Pareto e–cient allocation which was also a general
equilibrium,This point cannot lie on the contract curve (the dashed line) since it is not e–cient.
Suppose it was a point like xA,If it is a general equilibrium as well,consumer A must be maximising given prices
(illustrated by the budget line),But consumer B must be maximising also | say at point xB.
xA cannot be equal to xB as it is not on the contract curve,Therefore xB is a difierent point.
But now the market for good 2 does not clear,x2A + x2B > !2A + !2B | a contradiction (this is not an equilibrium).
Market | Welfare 2
Market Failure
Assumptions need to be made for this theorem to work,There are three crucial ones.
1,No Externalities,Each agent’s consumption decision does not afiect the utility of any other agent.
2,Price Taking Behaviour,Each agent in the economy behaves as a price taker.
3,Prices are Known,All the prices for each of the goods must be known to each of the agents,Importantly,the
consumers do not have difierent (asymmetric) information concerning the goods.
The flrst assumption is critical,The next lecture deals with the case of externalities in more depth.
The last is the minimal information requirement,Agents need only know price,They need not know the demand or
output decisions of others,or how much of a good is available,They behave \selflshly" given the known prices.
Market failure arises when any of these assumptions is not met,Externalities,market power and asymmetric
information are all examples of market failure.
Market | Welfare 3
The Invisible Hand
What are the implications of this theorem?
In a general equilibrium everyone maximises utility \selflshly" given prices,Firms \selflshly" maximise proflts.
However,as a result of this selflsh behaviour (given the assumptions discussed earlier) a socially desirable outcome
arises,An allocation is achieved where no-one can be made better ofi without making someone else worse ofi.
This is a formalisation of the invisible hand argument of Adam Smith.
What role is there for government intervention in such an e–cient world? There are two possibilities:
1,Market failure,If one of the assumptions fail,the allocation may no longer be e–cient.
2,Distributive Goals,Pareto e–ciency says nothing about distributional fairness.
Market | Welfare 4
The Second Welfare Theorem
An equilibrium is e–cient,are e–cient allocations always part of an equilibrium?
Theorem,The second welfare theorem states that every Pareto e–cient allocation can be supported by a general
equilibrium set of prices given a suitable reallocation of the endowment.
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
.........................................................................................................................................................................................................
..........................................................................................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.........................................,
........
.......
.......
.......
......
....
........
.....
......
......
......
......
.......
.......
.....
....
.......
..........
..............
.................
...........
.................
...............
.............
........
.....
.....
.....
.......
.......
......
......
......
......
...
A
B
x
Equilibrium Prices
!
0
!?
The demonstration proceeds constructively,Which set of prices can support the above Pareto e–cient allocation
(x) as a general equilibrium? The answer is the given by the budget line that separates the two indifierence curves.
In order to support x the endowment would need to reallocated from ! to !0.
Market | Welfare 5
Convexity and the Theorem
The crucial assumption here is that of convexity,Preferences need to be well-behaved for the theorem to work.
What would happen if they were not? Consider the following case,A has well-behaved preferences,B does not.
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
...........................................................................................................................................................................................................
..........................................................................................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.....
......
......
......
......
.......
......
.....
.....
.......
..........
..............
.................
...........
.............
..............
........
.........
..........
..........
...
.............
...........
.......
.................
..................
........,.......
.............
...........
............................
A
B
x
xB
Can a point like x be supported as a general equilibrium? The budget line is a set of prices,separating the two
indifierence curves,A is maximising,B is not | they could do better,by choosing xB.
xB maximises utility given these prices for consumer B,x maximises utility for consumer A,This is not a general
equilibrium since the market for good 2 is not clearing.
Market | Welfare 6
Implications of the Theorem
Notice that convexity is only required for the second theorem,The flrst theorem holds for any preferences.
What are the implications of the second theorem?
Distributional issues can be separated from e–ciency issues,A government,operating in such a world can \simply"
transfer endowments to achieve any distributional goals they might have and leave the market to attain e–ciency.
Prices play two roles in the market,(i) allocative | relative scarcity of the two goods and (ii) distributive | how
much agents can afiord,These can be separated,Do not use prices to attain distributive goals,use endowments.
Of course,such a transfer of endowments is not so simple | it may be impossible.
Note,Although only pure exchange economies have been considered so far,everything goes through in an
analogous way for a production economy,The assumptions need to apply both to consumers and flrms.
Market | Welfare 7
E–ciency and Welfare
The contract curve is the set of allocations that are Pareto e–cient.
By applying the flrst and second welfare theorems a general equilibrium will lie on this curve and moreover a
government could transfer endowments to achieve any of these points,The question is,Which one?
Pareto e–ciency says nothing about \fairness" or \justice",Indeed,the allocation where consumer A gets
everything and consumer B gets nothing is Pareto e–cient,It is probably not fair however.
Suppose the government could rank the various allocations available,simply attaching a number to each outcome.
The one with the highest number would then be the best allocation from societies point of view.
How should the government construct their ranking over difierent allocations? In other words,how should the
government turn individual preferences into social welfare? How can preferences be aggregated?
Market | Welfare 8
Aggregating Preferences
Suppose there are three agents (1,2 and 3) in the economy and three possible allocations,a,b and c.
Suppose the following table illustrates the preferences the three individuals have over the various allocations.
Agent 1 Agent 2 Agent 3
a b c
b c a
c a b
How should their preferences be aggregated? Suppose a majority voting mechanism is proposed,Given a choice
between a and b,a would win (agents 1 and 3 would vote for a),Given a choice between b and c,b would win
(agents 1 and 2 would vote for b),Given a choice between a and c,c would win (agents 2 and 3 would vote for c).
So this social ordering is not transitive even though the utility function of each agent is.
Is there a better way to rank the three alternatives? What properties should such a social welfare function have?
Market | Welfare 9
Arrow’s Impossibility Theorem
Suppose the aggregation method (or social welfare function) had three properties:
1,Given a set of complete,transitive and re exive preferences,aggregation should result in social preferences that
are complete,transitive and re exive (unlike majority voting).
2,If everyone prefers some allocation a to another,b,then a should be socially preferred to b.
3,Social preferences between a and b should only depend on the way that agents rank a and b and not on the
relative rank of any other \irrelevant" allocation c.
Theorem,Arrow’s impossibility theorem states that any mechanism for the aggregation of preferences (social
welfare function) which satisfles the above three properties is a dictatorship.
In other words,the social rankings correspond exactly with one individual’s rankings | the social welfare function
is simply the utility function of a particular agent.
There is no \perfect" way to rank allocations,How can the government choose between allocations?
Market | Welfare 10
Social Welfare
Obviously,they use a less than \perfect" social welfare function,Either one of the properties is not satisfled or they
use a dictatorship welfare function,Surely this is bad? Perhaps not.
Formally,a social welfare function (SWF) maps individual utility functions to a number | social welfare.
Consider the Rawlsian and Utilitarian social welfare functions which are given respectively by:
WR(u1;,,,; un)=min
i
ui and WU(u1;,,,; un)=
nX
i=1
ui
Consider the Rawlsian SWF,It satisfles property 1 | since it is simply a utility function and hence complete,
transitive and re exive,It satisfles property 2,since if everyone prefers a to b then the minimum utility individual
prefers a to b and hence the SWF ranks a above b.
It is independent of irrelevant alternatives,c,since all individuals (including the one with the smallest utility) rank
a and b independently of c,Hence,there must be a dictator,There is,The least well ofi agent,Is this so bad?
The Utilitarian SWF also either breaks one of the three conditions or is a dictatorship,Exercise,Which?
Market | Welfare 11
Welfare Maximisation
The government wishes to choose the best allocation according to the SWF | it wishes to maximise welfare.
Not all allocations are possible,The utility possibilities set is the set of feasible allocations.
.......,......,......,......,......,.....
......
.
......
.
......
.
......
.
......
.
.....
.......,......,......,......,......,......,......
......
.
......
.
......
.
......
.
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
....
.............................................................................................................................................,......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
.
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
...
......................................................
......
......
......
......
......
......
......
......
......
...................................................................
......
......
......
......
......
......
.....
......
......
......
......
.
......
......
......
......
......
........
...
.........
......
......
......
......
......
........
...
.........
.......
.......
.......
.......
......
.....
.....
......
......
......
......
.........
................
......................................
.......
.......
.......
.......
......
.....
.....
.......
........
..............
.........................
.................
U U
u1 u1
u2 u2
u?1
u?2
u?1
u?2
0 0
The feasible allocations lie within the sets U in the graph above,The boundary of this set is called the Pareto
frontier,The isowelfare lines | lines of equal social welfare are illustrated for two examples.
The flrst example is for a Rawlsian SWF | it results in equality,u?1 = u?2,The second example is for a Utilitarian
SWF | \the greatest good to the greatest number" | it does not result in equality,u?1 > u?2.
Notice that every Pareto e–cient point (a point on the Pareto frontier) is the maximum of some SWF.
Market | Welfare 12
Fairness,Envy and Equity
Another approach might be to propose a particular type of allocation | a fair one for example.
What is a fair allocation? An envy-free or equitable allocation is one where consumer A does not prefer the bundle
consumer B gets and vice-versa,In the Edgeworth box the other consumer’s bundle is the \mirror image" bundle.
.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
....
.........................................................................................................................................................................................................
...........................................................................................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
.....
......
......
......
....
.......
.......
.......
.......
......................
......
.......
.......
.....
.....
......
........
..........
............
.............
...............
...............
................
...............
.............
............
..........
........
..
.....
.......
.....
.
A
B
x
y
!
x is a fair allocation,(if) it is equitable and e–cient,Neither consumer would prefer to be at y | that is,neither
consumer wants to swap bundles with the other,Notice that ! lies on this budget line also | equal endowments.
In fact if both consumers start with an equal amount of the two goods each they will trade to a fair allocation,A
competitive equilibrium from equal division must be a fair allocation.
The First Welfare Theorem
In the last lecture two concepts were introduced,Pareto e–ciency and general equilibrium,How do they relate?
Theorem,The flrst welfare theorem states that every general equilibrium involves a Pareto e–cient allocation.
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.,,,,,,,,,,,
............
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
...
...........................................................................................................................................................................................................
..........................................................................................................................................................................................................
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
...........
.
.......
..
......
......
......
......
......
......
......
......
......
......
......
......
...........
.
.......
..
......
.......
.......
.....
...
.......
..........
...........
...........
...............
................
...
................
...............
...........
............
..........
.......
.
.....
.......
.......
......
.A
B
xA
xBx2A
x2B
The proof works by contradiction,Suppose there was a non-Pareto e–cient allocation which was also a general
equilibrium,This point cannot lie on the contract curve (the dashed line) since it is not e–cient.
Suppose it was a point like xA,If it is a general equilibrium as well,consumer A must be maximising given prices
(illustrated by the budget line),But consumer B must be maximising also | say at point xB.
xA cannot be equal to xB as it is not on the contract curve,Therefore xB is a difierent point.
But now the market for good 2 does not clear,x2A + x2B > !2A + !2B | a contradiction (this is not an equilibrium).
Market | Welfare 2
Market Failure
Assumptions need to be made for this theorem to work,There are three crucial ones.
1,No Externalities,Each agent’s consumption decision does not afiect the utility of any other agent.
2,Price Taking Behaviour,Each agent in the economy behaves as a price taker.
3,Prices are Known,All the prices for each of the goods must be known to each of the agents,Importantly,the
consumers do not have difierent (asymmetric) information concerning the goods.
The flrst assumption is critical,The next lecture deals with the case of externalities in more depth.
The last is the minimal information requirement,Agents need only know price,They need not know the demand or
output decisions of others,or how much of a good is available,They behave \selflshly" given the known prices.
Market failure arises when any of these assumptions is not met,Externalities,market power and asymmetric
information are all examples of market failure.
Market | Welfare 3
The Invisible Hand
What are the implications of this theorem?
In a general equilibrium everyone maximises utility \selflshly" given prices,Firms \selflshly" maximise proflts.
However,as a result of this selflsh behaviour (given the assumptions discussed earlier) a socially desirable outcome
arises,An allocation is achieved where no-one can be made better ofi without making someone else worse ofi.
This is a formalisation of the invisible hand argument of Adam Smith.
What role is there for government intervention in such an e–cient world? There are two possibilities:
1,Market failure,If one of the assumptions fail,the allocation may no longer be e–cient.
2,Distributive Goals,Pareto e–ciency says nothing about distributional fairness.
Market | Welfare 4
The Second Welfare Theorem
An equilibrium is e–cient,are e–cient allocations always part of an equilibrium?
Theorem,The second welfare theorem states that every Pareto e–cient allocation can be supported by a general
equilibrium set of prices given a suitable reallocation of the endowment.
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
.........................................................................................................................................................................................................
..........................................................................................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.........................................,
........
.......
.......
.......
......
....
........
.....
......
......
......
......
.......
.......
.....
....
.......
..........
..............
.................
...........
.................
...............
.............
........
.....
.....
.....
.......
.......
......
......
......
......
...
A
B
x
Equilibrium Prices
!
0
!?
The demonstration proceeds constructively,Which set of prices can support the above Pareto e–cient allocation
(x) as a general equilibrium? The answer is the given by the budget line that separates the two indifierence curves.
In order to support x the endowment would need to reallocated from ! to !0.
Market | Welfare 5
Convexity and the Theorem
The crucial assumption here is that of convexity,Preferences need to be well-behaved for the theorem to work.
What would happen if they were not? Consider the following case,A has well-behaved preferences,B does not.
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
...........................................................................................................................................................................................................
..........................................................................................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.....
......
......
......
......
.......
......
.....
.....
.......
..........
..............
.................
...........
.............
..............
........
.........
..........
..........
...
.............
...........
.......
.................
..................
........,.......
.............
...........
............................
A
B
x
xB
Can a point like x be supported as a general equilibrium? The budget line is a set of prices,separating the two
indifierence curves,A is maximising,B is not | they could do better,by choosing xB.
xB maximises utility given these prices for consumer B,x maximises utility for consumer A,This is not a general
equilibrium since the market for good 2 is not clearing.
Market | Welfare 6
Implications of the Theorem
Notice that convexity is only required for the second theorem,The flrst theorem holds for any preferences.
What are the implications of the second theorem?
Distributional issues can be separated from e–ciency issues,A government,operating in such a world can \simply"
transfer endowments to achieve any distributional goals they might have and leave the market to attain e–ciency.
Prices play two roles in the market,(i) allocative | relative scarcity of the two goods and (ii) distributive | how
much agents can afiord,These can be separated,Do not use prices to attain distributive goals,use endowments.
Of course,such a transfer of endowments is not so simple | it may be impossible.
Note,Although only pure exchange economies have been considered so far,everything goes through in an
analogous way for a production economy,The assumptions need to apply both to consumers and flrms.
Market | Welfare 7
E–ciency and Welfare
The contract curve is the set of allocations that are Pareto e–cient.
By applying the flrst and second welfare theorems a general equilibrium will lie on this curve and moreover a
government could transfer endowments to achieve any of these points,The question is,Which one?
Pareto e–ciency says nothing about \fairness" or \justice",Indeed,the allocation where consumer A gets
everything and consumer B gets nothing is Pareto e–cient,It is probably not fair however.
Suppose the government could rank the various allocations available,simply attaching a number to each outcome.
The one with the highest number would then be the best allocation from societies point of view.
How should the government construct their ranking over difierent allocations? In other words,how should the
government turn individual preferences into social welfare? How can preferences be aggregated?
Market | Welfare 8
Aggregating Preferences
Suppose there are three agents (1,2 and 3) in the economy and three possible allocations,a,b and c.
Suppose the following table illustrates the preferences the three individuals have over the various allocations.
Agent 1 Agent 2 Agent 3
a b c
b c a
c a b
How should their preferences be aggregated? Suppose a majority voting mechanism is proposed,Given a choice
between a and b,a would win (agents 1 and 3 would vote for a),Given a choice between b and c,b would win
(agents 1 and 2 would vote for b),Given a choice between a and c,c would win (agents 2 and 3 would vote for c).
So this social ordering is not transitive even though the utility function of each agent is.
Is there a better way to rank the three alternatives? What properties should such a social welfare function have?
Market | Welfare 9
Arrow’s Impossibility Theorem
Suppose the aggregation method (or social welfare function) had three properties:
1,Given a set of complete,transitive and re exive preferences,aggregation should result in social preferences that
are complete,transitive and re exive (unlike majority voting).
2,If everyone prefers some allocation a to another,b,then a should be socially preferred to b.
3,Social preferences between a and b should only depend on the way that agents rank a and b and not on the
relative rank of any other \irrelevant" allocation c.
Theorem,Arrow’s impossibility theorem states that any mechanism for the aggregation of preferences (social
welfare function) which satisfles the above three properties is a dictatorship.
In other words,the social rankings correspond exactly with one individual’s rankings | the social welfare function
is simply the utility function of a particular agent.
There is no \perfect" way to rank allocations,How can the government choose between allocations?
Market | Welfare 10
Social Welfare
Obviously,they use a less than \perfect" social welfare function,Either one of the properties is not satisfled or they
use a dictatorship welfare function,Surely this is bad? Perhaps not.
Formally,a social welfare function (SWF) maps individual utility functions to a number | social welfare.
Consider the Rawlsian and Utilitarian social welfare functions which are given respectively by:
WR(u1;,,,; un)=min
i
ui and WU(u1;,,,; un)=
nX
i=1
ui
Consider the Rawlsian SWF,It satisfles property 1 | since it is simply a utility function and hence complete,
transitive and re exive,It satisfles property 2,since if everyone prefers a to b then the minimum utility individual
prefers a to b and hence the SWF ranks a above b.
It is independent of irrelevant alternatives,c,since all individuals (including the one with the smallest utility) rank
a and b independently of c,Hence,there must be a dictator,There is,The least well ofi agent,Is this so bad?
The Utilitarian SWF also either breaks one of the three conditions or is a dictatorship,Exercise,Which?
Market | Welfare 11
Welfare Maximisation
The government wishes to choose the best allocation according to the SWF | it wishes to maximise welfare.
Not all allocations are possible,The utility possibilities set is the set of feasible allocations.
.......,......,......,......,......,.....
......
.
......
.
......
.
......
.
......
.
.....
.......,......,......,......,......,......,......
......
.
......
.
......
.
......
.
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
....
.............................................................................................................................................,......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
.
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
...
......................................................
......
......
......
......
......
......
......
......
......
...................................................................
......
......
......
......
......
......
.....
......
......
......
......
.
......
......
......
......
......
........
...
.........
......
......
......
......
......
........
...
.........
.......
.......
.......
.......
......
.....
.....
......
......
......
......
.........
................
......................................
.......
.......
.......
.......
......
.....
.....
.......
........
..............
.........................
.................
U U
u1 u1
u2 u2
u?1
u?2
u?1
u?2
0 0
The feasible allocations lie within the sets U in the graph above,The boundary of this set is called the Pareto
frontier,The isowelfare lines | lines of equal social welfare are illustrated for two examples.
The flrst example is for a Rawlsian SWF | it results in equality,u?1 = u?2,The second example is for a Utilitarian
SWF | \the greatest good to the greatest number" | it does not result in equality,u?1 > u?2.
Notice that every Pareto e–cient point (a point on the Pareto frontier) is the maximum of some SWF.
Market | Welfare 12
Fairness,Envy and Equity
Another approach might be to propose a particular type of allocation | a fair one for example.
What is a fair allocation? An envy-free or equitable allocation is one where consumer A does not prefer the bundle
consumer B gets and vice-versa,In the Edgeworth box the other consumer’s bundle is the \mirror image" bundle.
.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
....
.........................................................................................................................................................................................................
...........................................................................................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
.....
......
......
......
....
.......
.......
.......
.......
......................
......
.......
.......
.....
.....
......
........
..........
............
.............
...............
...............
................
...............
.............
............
..........
........
..
.....
.......
.....
.
A
B
x
y
!
x is a fair allocation,(if) it is equitable and e–cient,Neither consumer would prefer to be at y | that is,neither
consumer wants to swap bundles with the other,Notice that ! lies on this budget line also | equal endowments.
In fact if both consumers start with an equal amount of the two goods each they will trade to a fair allocation,A
competitive equilibrium from equal division must be a fair allocation.
