The Fundamental Welfare Theorems

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Oct 8, 2013 (3 years and 9 months ago)

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The Fundamental Welfare Theorems
The so-called Fundamental Welfare Theorems of Economics tell us about the relation between
market equilibrium and Pareto eciency.
The First Welfare Theorem:Every Walrasian equilibrium allocation is Pareto ecient.
The Second Welfare Theorem:Every Pareto ecient allocation can be supported as a
Walrasian equilibrium.
The theorems are certainly not true in the unconditional form in which we've stated them
here.A better way to think of them is this:\Under certain conditions,a market equilibrium
is ecient"and\Under certain conditions,an ecient allocation can be supported as a
market equilibrium."Nevertheless,these two theorems really are fundamental benchmarks
in microeconomics.We're going to give conditions under which they're true.One (very
stringent) set of conditions will enable us to prove the theorems with the calculus (Kuhn-
Tucker,Lagrangian,gradient) methods we've used very fruitfully in our analysis of Pareto
eciency.Additionally,for each theoremwe'll provide a much weaker set of conditions under
which the theorem remains true.
Assume to begin with,then,that the consumers'preferences are\very nice"| viz.,that
they're representable by utility functions u
i
that satisfy the following condition:
u
i
is continuously dierentiable,strictly quasiconcave,and 8x
i
2 R
l
+
:u
i
k
(x
i
) > 0 ()
The First Welfare Theorem:If

b
p;(
b
x
i
)
n
1

is a Walrasian equilibrium for an economy
E = ((u
i
;x
i
))
n
1
that satises (),then (bx
i
)
n
1
is a Pareto allocation for E.
Proof:
Because

bp;(bx
i
)
n
1

is a Walrasian equilibriumfor E,each bx
i
maximizes u
i
subject to x
i
= 0
and to the budget constraint
b
p  x
i
5
b
p x
i
.Therefore,for each i 2 N there is a 
i
= 0 that
satises the rst-order marginal conditions for i's maximization problem:
8k:u
i
k
5 
i
bp
k
;with equality if bx
i
k
> 0;(1)
where of course the partial derivatives u
i
k
are evaluated at bx
i
k
.In fact,because each u
i
k
> 0,
we have 
i
> 0 and bp
k
> 0 for each i and each k.The market-clearing condition in the
denition of Walrasian equilibrium therefore yields
n
X
i=1
bx
i
k
=
n
X
i=1
x
i
k
for each k (because each bp
k
> 0):(2)
For each i we can dene 
0
i
=
1

i
,because 
i
> 0,and then we can rewrite (1) as
8k:
0
i
u
i
k
5 bp
k
;with equality if bx
i
k
> 0;(3)
for each i 2 N.But (2) and (3) are exactly the rst-order conditions that characterize the
solutions of the Pareto maximization problem (P-max).And since each u
i
k
> 0,a solution of
(P-max) is a Pareto allocation.
In the proof,we could have instead expressed the rst-order marginal conditions for the
individual consumers'maximization problems in terms of marginal rates of substitution:
8k;k
0
:MRS
i
kk
0 =
bp
k
bp
k
0
(4)
(written for the interior case,to avoid a lot of inequalities),which yields the (Equal MRS)
condition
8i;j;k;k
0
:MRS
i
kk
0
= MRS
j
kk
0
:(5)
The conditions (5) and (2) characterize the Pareto allocations.
Now let's see if we can prove the First Welfare Theorem without any of the assumptions in
().It turns out that we can't quite do that;however,the only thing we need to assume is
that each u
i
is locally nonsatiated.If we replace the utility functions with locally nonsatiated
preferences,the proof is exactly the same.
The First Welfare Theorem:If

bp;(bx
i
)
n
1

is a Walrasian equilibrium for an economy
E = ((u
i
;x
i
))
n
1
in which each u
i
is locally nonsatiated,then (bx
i
)
n
1
is a Pareto allocation for
E.
Proof:
Suppose (bx
i
)
n
1
is not a Pareto allocation |i.e.,some allocation (ex
i
)
n
1
is a Pareto improve-
ment on (bx
i
)
n
1
:
(a)
P
n
1
ex
i
5
P
n
1
x
i
(b1) 8i 2 N:u
i
(ex
i
) = u
i
(bx
i
)
(b2) 9i
0
2 N:u
i
0
(
e
x
i
0
) > u
i
0
(
b
x
i
0
):
Because

bp;(bx
i
)
n
1

is a Walrasian equilibrium for E,each bx
i
maximizes u
i
on the budget set
B(bp;x
i
):= f x
i
2 R
l
+
j bp  x
i
5 bp  x
i
g.Therefore,(b2) implies that
b
p 
e
x
i
0
>
b
p 
b
x
i
0
;(6)
2
and since each u
i
is locally nonsatiated,(b1) implies that
bp  ex
i
= bp  bx
i
for each i:(7)
((7) follows from a duality theorem that we'll state and prove below | in the nal three
pages of this set of notes.) Summing the inequalities in (6) and (7) yields
n
X
i=1
bp  ex
i
>
n
X
i=1
bp  bx
i
;(8)
i.e.,
b
p 
n
X
i=1
e
x
i
>
b
p 
n
X
i=1
b
x
i
:(9)
Since bp 2 R
l
+
,it follows from (9) that there is at least one k for which
bp
k
> 0 and
n
X
i=1
ex
i
k
>
n
X
i=1
bx
i
k
:(10)
Since bp
k
> 0,the market-clearing equilibrium condition yields
P
n
i=1
bx
i
k
=
P
n
i=1
x
i
k
,and (10)
therefore yields
P
n
i=1
ex
i
k
>
P
n
i=1
x
i
k
| i.e.,(ex
i
)
n
1
does not satisfy (a).Our assumption that
(ex
i
)
n
1
is a Pareto improvement has led to a contradiction;therefore there are no Pareto
improvements on (bx
i
)
n
1
,and it's therefore a Pareto allocation.
3
The Second Welfare Theorem:Let (bx
i
)
n
1
be a Pareto allocation for an economy in which
the utility functions u
1
;:::;u
n
all satisfy () and in which the total endowment of goods is
x 2 R
l
++
.Then there is a price-list bp 2 R
l
+
such that
for every (x
i
)
n
1
that satises
n
X
i=1
x
i
=x and 8i:bp  x
i
= bp  bx
i
;(11)

bp;(bx
i
)
n
1

is a Walrasian equilibrium of the economy E = ((u
i
;x
i
))
n
1
.
|i.e.,

bp;(bx
i
)
n
1

is a Walrasian equilibrium of the economy in which each consumer has the
utility function u
i
and the initial bundle x
i
.
Proof:
Let (bx
i
)
n
1
be a Pareto allocation for the given utility functions and endowment x.We rst
show that the conclusion holds for (x
i
)
n
1
= (bx
i
)
n
1
| i.e.,if each consumer's initial bundle is
bx
i
.Since (bx
i
)
n
1
is a Pareto allocation,it is a solution of the Pareto maximization problem
(P-max),and it therefore satises the rst-order marginal conditions
8i;k:9
i
;
k
= 0:
i
u
i
k
5 
k
;with equality if bx
i
k
> 0:(12)
Each 
i
is the Lagrange multiplier for one of the utility-level constraints in (P-max),and
since u
i
k
> 0 for each i and each k,it's clear that each constraint's Lagrange multiplier must
be positive:relaxing any one of the constraints will allow u
1
to be increased.
For each k,let bp
k
= 
k
.Now (12) yields,for each i,
9
i
:8k:u
i
k
5
1

i
bp
k
;with equality if bx
i
k
> 0:(13)
These are exactly the rst-order marginal conditions for consumer i's utility-maximization
problem at the price-list bp.Therefore,since each consumer's initial bundle is bx
i
,each con-
sumer is maximizing u
i
at bx
i
.And since bp
k
= 
k
> 0 for each k,the constraint satisfaction
rst-order condition for (P-Max) yields
P
n
i=1
bx
i
=x,so the market-clearing condition for a
Walrasian equilibrium is satised as well.Thus,we've shown that all the conditions in the
denition of Walrasian equilibrium are satised for

bp;(bx
i
)
n
1

.
If each consumer's initial bundle is some other x
i
instead of bx
i
,but still satisfying (11),then
the second equation in (11) ensures that each consumer's budget constraint is the same as
before (the right-hand sides have the same value),and therefore the inequalities in (13) again
guarantee that each bx
i
maximizes u
i
subject to the consumer's budget constraint.And as
before,the constraint satisfaction rst-order condition for (P-Max),
P
n
i=1
bx
i
= x,coincides
with the market-clearing condition for a Walrasian equilibrium.
4
Just as with the First Welfare Theorem,the Second Theorem is true under weaker assump-
tions than those in () | but in the case of the Second Theorem,not that much weaker:
we can dispense with the dierentiability assumption,and we can weaken the convexity
assumption and the assumption that utility functions are strictly increasing.
The Second Welfare Theorem:Let (bx
i
)
n
1
be a Pareto allocation for an economy in
which each u
i
is continuous,quasiconcave,and locally nonsatiated,and in which the total
endowment of goods is x 2 R
l
++
.Then there is a price-list bp 2 R
l
+
such that
for every (x
i
)
n
1
that satises
n
X
i=1
x
i
=

x and 8i:
b
p  x
i
=
b
p 
b
x
i
;(14)

bp;(bx
i
)
n
1

is a Walrasian equilibrium of the economy E = ((u
i
;x
i
))
n
1
.
The proof of the Second Theorem at this level of generality is not nearly as straightforward
as the proof of the First Theorem.The proof requires a signicant investment in the theory
of convex sets,as well as some additional mathematical concepts.This is a case in which
(for this course) the cost of developing the proof outweighs its value.
5
Duality Theorems in Demand Theory:
Utility Maximization and Expenditure Minimization
These two Duality Theorems of demand theory tell us about the relation between utility max-
imization and expenditure minimization | i.e.,between Marshallian demand and Hicksian
(or compensated) demand.We would like to know that the utility-maximization hypothesis
ensures that any bundle a consumer chooses (if he is a price-taker) must minimize his expen-
diture over all the bundles that would have made him at least as well o.And conversely,
we would like to know that a bundle that minimizes expenditure to attain a given utility
level must maximize his utility among the bundles that don't cost more.The following two
examples show that the two ideas are not always the same.
Example 1:A consumer with a thick indierence curve,as in Figure 1,where there are
utility-maximizing bundles that do not minimize expenditure.
Example 2:A consumer whose wealth is so small that any reduction in it would leave
him with no aordable consumption bundles,as in Figure 2,where there are expenditure-
minimizing bundles that do not maximize utility.
The examples suggest assumptions that will rule out such situations.
Utility Maximization Implies Expenditure Minimization
It's clear in Example 1 that the reason bx fails to minimize p  x is because the preference
is not locally nonsatiated.If a consumer's preference is locally nonsatiated,then preference
maximization does imply expenditure minimization.
First Duality Theorem:If % is a locally nonsatiated preference on a set X of consumption
bundles in R
`
+
,and if bx is %-maximal in the budget set fx 2 X j p  x 5 p  bxg,then bx
minimizes p  x over the upper-contour set fx 2 X j x % bxg.
Proof:(See Figure 3)
Suppose bx does not minimize p  x on fx 2 X j x % bxg.Then there is a bundle ex 2 X
such that p 
e
x < p 
b
x.Let N be a neighborhood of
e
x for which x 2 N ) p  x 5 p 
b
x
| i.e.,N lies entirely below the p-hyperplane through bx.Since % is locally nonsatiated,
there is a bundle x
0
2 N that satises x
0
 ex,and a fortiori x
0
 bx.But then we have both
p x
0
5 p bx and x
0
 bx |i.e.,bx does not maximize % on the budget set,a contradiction.

6
Expenditure Minimization Implies Utility Maximization
It seems clear that the reason bx does not maximize utility among the bundles costing no
more than it does is that there are no bundles in the consumer's consumption set that cost
strictly less than bx.If there were such a bundle,we could move\continuously"along a line
from that bundle to any bundle ex costing no more than bx but strictly better than bx,and
we would have to encounter a bundle that also costs strictly less than bx and that also is
strictly better than bx | so that bx would not maximize utility among the bundles costing
no more than bx.Note,though,that in addition to having some bundle that costs less than
bx,this argument also requires both continuity of the preference and convexity of the set of
possible bundles.The proof of the Second Duality Theorem uses exactly this argument,so it
requires these stronger assumptions,which were not needed for the First Duality Theorem.
Can you produce counterexamples to show that the theoremdoes indeed require each of these
assumptions |continuity and convexity |in addition to the counterexample in Figure 2 in
which there is no cheaper bundle than bx in the consumer's set of possible consumptions?
Since we have to assume the preference is continuous,it will therefore be representable by
a continuous utility function,so we state and prove the theorem in terms of a continuous
utility function.We could instead do so with a continuous preference %.
Second Duality Theorem:Let u:X!R be a continuous function on a convex set X of
consumption bundles in R
`
+
.If bx minimizes p x over the upper-contour set fx 2 X j u(x) =
u(bx)g,and if X contains a bundle x that satises p x < p bx,then bx maximizes u over the
budget set fx 2 X j p  x 5 p  bxg.
Proof:(See Figure 4)
Suppose that bx minimizes p  x over the set fx 2 X j u(x) = u(bx)g,but that bx does
not maximize u over the set fx 2 X j p  x 5 p  bxg | i.e.,there is some ex 2 X such that
p ex 5 p bx but u(ex) > u(bx).Since u is continuous,there is a neighborhood N of ex such that
each x 2 N also satises u(x) > u(bx).We've furthermore assumed that there is a bundle
x
0
2 X that satises p x
0
< p
b
x.Since X is convex,the neighborhood N contains a convex
combination,say x
00
,of x
0
and ex.Since p ex 5 p bx and p x
0
< p bx,we have p x
00
< p bx.
But since x
00
2 N,we also have u(x
00
) > u(bx).Thus,bx does not minimize p  x over the set
fx 2 X j u(x) = u(bx)g,a contradiction.
7
Figure 1 Figure 2
Figure 3 Figure 4
8