Download Pareto Optimality, the Core, and Competitive Equilibrium

Pareto Optimality, the Core, and Competitive
Equilibrium
Econ 2100
Lecture 16, October 28
Outline
1
Pareto E¢ ciency
2
The Core
3
Planner’s Problem(s)
4
Competitive (Walrasian) Equilibrium
Fall 2015
General Equilibrium
Consumers i = 1; :::; I are described by a consumption set Xi RL , preferences
J
%i 0, endowments ! i 2 Xi , and shares i 2 [0; 1] of the pro…ts of each …rm.
Firm j = 1; :::; J are described by a production set, Yj RL .
P
The aggregate endowment is ! = i ! i .
P
Firms are owned by someone: 1 = i ij with ij 2 [0; 1] for all j.
De…nitions
fXi ; %i gi =1
I
An economy is a tuple
An economy with private ownership is a tuple
fXi ; %i ; ! i ;
J
fYj gj =1
I
i gi =1
!
J
fYj gj =1
An allocation: (x; y ) = (x1 ; :::; xI ; y1 ; :::; yJ ) 2 RL(I +J ) where xi 2 Xi for each
i = 1; :::; I , and yj 2 Yj for each j = 1; :::; J.
De…nition
An allocation (x; y ) is feasible if
I
X
i =1
xi
!+
J
X
j =1
yj
Pareto E¢ ciency
Question
How should we think about e¢ ciency?
A minimal requirement is that there is no waste of resources.
De…nition
A feasible allocation (x; y ) is Pareto optimal if there is no other feasible allocation
(x 0 ; y 0 ) such that
xi0 %i xi
for all i = 1; :::; I
and
xi0
i
xi
for some i
Nothing is left on the table: an allocation is e¢ cient if we cannot …nd another
one that harms no one and bene…ts someone.
This only looks at preferences to de…ne e¢ ciency.
Firms do not have preferences, but they matter via what they produce.
Remarks
Nobody chooses anything here.
This is not about markets and/or prices.
O2
x
x’
O1
x and x’are not Pareto optimal
O2
ω
x
O1
x is Pareto optimal
O2
O1
The locus of all Pareto optimal allocations (contract curve)
Individual Rationality
Suppose we let the individuals exchange goods with each other.
Observation
A consumer should not want allocations that do not improve over her initial
conditions.
Therefore, she may not want outcomes that are worse than consuming her
initial endowment since she can always refuse to trade.
De…nition
A feasible allocation (x; y ) is individually rational if xi %i ! i
for all i:
An individually rational allocation represents a trade that improves everyone
position relative to their initial endowment.
Pareto optimal allocations are not necessarily individually rational.
O2
ω
x
x’
O1
x is individually rational, x’is not
The Core
A related concept requires that no group of consumers can gain by ‘seceding’
from the economy.
This is easier to de…ne for an exchange economy: let J = 1, and YJ = RL+ .
De…nition
A coalition is a subset of f1; :::; I g.
De…nition
A coalition S f1; :::; I g blocks the allocation x if for each i 2 S there exist
xi0 2 Xi such that
X
X
xi0 i xi for all i 2 S
and
xi0
!i
i 2S
i 2S
A blocking coalition can make all its members better o¤.
One can think of a weaker de…nition where a coalition bene…ts at least one of
its members strictly without hurting the others.
De…nition
A feasible allocation x is in the core of an economy if there is no coalition that
blocks it.
Pareto Optimality, Individual Rationality, and the
Core
Easy to Prove Results
Any allocation in the core of an economy is also Pareto optimal.
Obvious since the ‘whole’(sometimes called ‘grand coalition’S = f1; :::; I g) is
not a blocking coalition.
Not all Pareto optimal allocations are in the core.
Slightly less obvious: xi = ! (consumer i gets everything) is Pareto optimal but
not in the core.
Any assumptions requred here?
In an Edgeworth box (I = 2), the core is the set of all individually rational
Pareto optimal allocations.
This is an (easy) homework problem.
With more consumers this result does not hold.
As the number of consumers grows, there are more possible coalitions, and more
allocations will be blocked. So the core is typically much smaller than the set of
Pareto optimal allocations that are individually rational.
De…nitions With Utility Functions
Suppose individuals’preferences are represented by a utility function.
Consumer i’s utility function is denoted ui (xi ).
We can rewrite Pareto e¢ ciency and individual rationality as follows.
De…nitions with utility functions
A feasible allocation (x; y ) is Pareto optimal if there is no other feasible
allocation (x 0 ; y 0 ) such that
ui (xi0 )
ui (xi ) for all i
and
ui (xi0 ) > ui (xi ) for some i:
A feasible allocation (x; y ) is individually rational if
ui (xi )
ui (! i )
for all i:
Pareto E¢ ciency and Utility Possibility Set
De…nition
The utility possibility set is
U=
(u1 ; :::; uI ) 2 RI :
there exists a feasible (x; y )
such that ui u(xi ) for i = 1; :::; I
The utility possibility frontier is the boundary of U, and a Pareto optimal
allocation must belong to the frontier.
Draw a picture.
De…nition
The Pareto frontier is de…ned as
UP =
(u1 ; :::; uI ) 2 RI :
there is no (u10 ; :::; uI0 ) 2 U such that
ui0 ui for all i and ui0 > ui for some i
Question 6, Problem Set 9
Show that a feasible allocation (x; y ) is Pareto optimal if and only if
(u1 (x1 ); :::; uI (xI )) 2 UP.
Pareto E¢ ciency and Social Welfare
De…nitions
A (linear) social welfare function is a weighted sum of the individuals’utilities:
W =
I
X
i ui
=
u
with
i =1
The social welfare maximization problem is max
u2U
P
i
i
0
i ui
This maximization problem is sometimes called the “planner’s problem”.
Proposition
P
If u 2 RI solves maxu2U i i ui with i > 0 for all i, then u 2 UP.
Moreover, if U is convex then for any u~ 2 UP there exists
0 with
that
u~
u for all u 2 U.
6= 0 such
An allocation that solves the social welfare maximization problem is Pareto
optimal; morever, if the utility possibility set is convex, any Pareto optimal
allocation solves the social welfare maximization problem.
Convexity of U follows from convexity of production and consumption sets and
concavity of utility functions.
The proof uses the separating hyperplane theorem.
Proof.
First show that
u 2 arg max
u2U
X
i ui
with
i
i
> 0 for all i ) u 2 UP
By contradiction.
If u is not Pareto optimal, then there exists u 2 U with u u and u 6= u ;
since i > 0 for all i .
This contradicts that u solves the social welfare maximization problem
( u>
u ).
Now show that
U convex )
for any u~ 2 UP
there exists
0 with
u~
6= 0
for all u 2 U
Since u~ belongs to the frontier of U, a convex set, by the separating
hyperplane theorem there exists a vector 6= 0 such that
u~
Verify that we also have
u
u
such that
for all u 2 U
0 to complete the proof.
Pareto E¢ ciency and Constrained Planner Problem
Another planner’s problem aims to maximizie the utility of one consumer
subject to everyone else achieving a prespeci…ed utility level.
Consider, for simplicity, an exchange economy.
Proposition
Suppose that, for each i = 1; :::; I , %i can be represented by a strictly increasing
and continuous utility function ui (xi ). Then, an allocation x is Pareto optimal if
and only if it solves the following
ui (xi ) ui for all i 6= j,
max uj (xj )
x
subject to:
I
P
xi = ! and
i =1
xi
0 for all i
for some choice of fui gi 6=j .
An allocation is Pareto optimal if and only if it maximizes one consumer’s
utility subject to all others obtaining some given utility level.
Notice that this does not say for any fui gi 6=j (easy to …nd a counterexample).
Proof is an homework problem.
Pareto E¢ ciency: An Example
Finding Pareto Optimal Allocations: An Example
Consider an economy: where
consumer 1
! 1 = (! 11 ; ! 21 )
u1 (x1 ) = u1 (x11 ; x21 )
consumer 2
! 2 = (! 12 ; ! 22 )
u2 (x2 ) = u2 (x12 ; x22 )
To …nd the set of Pareto optimal allocations for an Edgeworth box economy in
which consumers’utility functions are
u1 (x11 ; x21 ) = (x11 ) (x21 )
1
and
1
u2 (x12 ; x22 ) = (x12 ) (x22 )
We must solve the following planner’s problem:
1
max
x11 ;x21 ;x12 ;x22
(x11 ) (x21 )
1
subject to
(x12 ) (x22 )
u
x11 + x12 = ! 11 + ! 12
x21 + x22 = ! 21 + ! 22
x11 ; x21 ; x12 ; x22 0
for some u.
In this case, the utility functions are di¤erentiable, so we can write the
Lagrangean, write the …rst order conditions, and then solve.
Competitive (Walrasian) Equilibrium
De…nition
Given an economy fXi ; %i ; ! i ; i gi =1 ; fYj gj =1 , a competitive (Walrasian)
equilibrium is formed by an allocation (x ; y ) 2 RL(I +J ) and a price vector p 2 RL
such that:
I
yj
p
yj
for all xi 2 fxi 2 Xi : p
xi
p
1
For each j = 1; :::; J:
2
For each i = 1; :::; I :
xi %i xi
3
p
I
X
i =1
1
2
3
J
xi
I
X
i =1
!i +
J
X
for all yj 2 Yj
!i +
yj
PJ
j =1
ij
p
yj g
j =1
Firms’choices maximize pro…ts given the equilibrium prices;
Consumers’choices are optimal in the budget set at the equilibrium prices; and
In all markets, demand cannot exceed supply .
Remark
Equilibrium prices make all optimization problems ‘mutually compatible’.
Individual Demand
Consumer’s demand depend on prices and initial endowment
O2
ω21
Supply of
good 2
x121
x1
x221
x321
O1
x2
ω11 x1
11
Demand of
good 1
x211
x3
x311
Excess Demand
The sum of consumers’s demands does not equal the initial endowment
ω1j
ω
ω2i
x 1j
Oj
Supply of
good 1
ω2j
Demand
of good 2
x2j
Supply of
good 2
x2i
Demand
of good 1
Oi
ω1i
x 1i
There is excess demand of good 1 (and excess supply of good 2)
Equilibrium
ω12
Supply of good 1
x*12
O2
ω
ω21
ω22
Supply of
good 2
Demand
of good 2
x*
x*21
Demand of good 1
O1
ω11
x*11
Excess demands of good 1 and good 2 are ZERO
x*22
Competitive Equilibrium in an Edgeworth box with Cobb-Douglas utilities
An equilibrium is given by a price vector p and an allocation x such that:
xi = arg
max
x 2fx 2R2 :p
x
p
!1 g
(x1i ) (x2i )
1
xj = arg
the Walrasian demands are:
p ! +p !
x1i = 1 1ip 2 2i
1
p ! +p !
x2i = (1
) 1 1ip 2 2i
2
an equilibirum must also solve
x1 i + x1 j = ! 1 i + ! 1 j
max
x 2fx 2R2 :p
x1j =
and
x2j = (1
x
p
!2 g
(x1j ) (x2j )
1
p 1 ! 1j +p 2 ! 2j
p1
p ! +p !
) 1 1jp 2 2j
2
x2 i + x2 j = ! 2 i + ! 2 j
We can …nd an equilibrium solving these six equations in six unknowns, but...
... Walrasian demand is homogeneous of degree zero, so if one multiplies all
prices by some constant demand does not change.
Thus, we can only pin down the relative prices
p1
p2 .
So we have six equations but …ve unknowns: are we in trouble?
Walras’law to the rescue: if the market for one good clears (supply equals
demand for that good) then the market for the other good does also clear (we
will prove this later).
One of the equations linearly depends on the others (it is redundant).
Econ 2100
Fall 2015
Problem Set 9
Due 3 November, Monday, at the beginning of class
1. Suppose there are two goods and the consumer has a vNM utility index over consumption bundles
of v(x) = f (x1 + x2 ) where f : R ! R is a strictly increasing function. Let p = (1; 3) and p0 = (3; 1).
Let q = :5p + :5p0 = (2; 2). In Regime 1, there is a 1=2 probability that the realized price will be p
and a 1=2 probability that the realized price will be p0 , so her ex-ante utility is
:5 max f (x1 + x2 ) + :5 max f (x1 + x2 ):
x2Bp;w
x2Bp0 ;w
In Regime 2, the price is q with certainty, so her ex-ante utility is
max f (x1 + x2 ):
x2Bq;w
Prove that, for any strictly increasing f , the consumer is ex-ante better o¤ in Regime 1.
2. Kreps 6.4
3. Kreps 6.14.
4. Suppose a consumer’s utility for CDFs is
U (F ) =
R
where V ar(F ) = (x
2
F ) dF
F
V ar(F )
is the variance of F .
(a) Prove that if > 0 then the preference corresponding to
corresponding to 0 .
is more risk averse than the one
(b) Prove that U violates expected utility.
5. Prove that in an exchange economy with I = 2, the core is the set of all Pareto optimal allocations
that are individually rational. Hint: enumerate all possible coalitions.
6. Show that a feasible allocation (x; y) is Pareto optimal if and only if (u1 (x1 ); :::; uI (xI )) 2 U P .
7. Show that in an exchange economy where all consumption sets are convex and each consumer’s utility
function is concave then the utility possibility set is convex.
8. Consider an exchange economy. Suppose that %i can be represented by a strictly increasing and
continuous utility function ui (xi ) for each i = 1; :::; I. Prove that an allocation x is Pareto optimal
if and only if it solves the following
X
max uj (xj ) subject to: ui (xi ) ui for all i 6= j,
xi = ! and xi 0 for all i
i
for some choice of fui gi6=j . Hint: …rst prove the case I = 2 …rst, and then think about how to extend
it.
1
9. For each Edgeworth Box economy below do the following: (i) …nd the Pareto optimal allocations,
(ii) …nd the Pareto optimal allocation that gives consumers equal utility (ui = uj ), (iii) …nd the
competitive equilibria, (iv) draw the Edgeworth box and clearly show the initial endowment, the
equilibrium allocation and equilibrium prices, and the allocation you found in (ii). HINT: You can
solve most of these without calculus (just draw pictures).
(a) ! i = (1; 0) and ui (xi ) =
p
x1i x2i ; ! j = (0; 1) and uj (xj ) =
p
x1j x2j .
(b) ! i = (0; 1) and ui (xi ) = 2x1i + x2i ; ! j = (1; 0) and uj (xj ) = x1j + 2x2j .
(c) ! i = (2; 0) and ui (xi ) = x1i + x2i ; ! j = (0; 1) and uj (xj ) = x1j + x2j .
(d) ! i = (1; 0) and ui (xi ) = min f2x1i ; x2i g; ! j = (0; 1) and uj (xj ) = min fx1j ; 2x2j g.
(e) ! i = (1; 0) and ui (xi ) = min fx1i ; x2i g; ! j = (0; 2) and uj (xj ) = min fx1j ; x2j g.
2