The Fundamental Welfare Theorems

The so-called Fundamental Welfare Theorems of Economics tell us about the relation between

market equilibrium and Pareto eciency.

The First Welfare Theorem:Every Walrasian equilibrium allocation is Pareto ecient.

The Second Welfare Theorem:Every Pareto ecient 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 ecient"and\Under certain conditions,an ecient 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

eciency.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 dierentiable,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 satises (),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

satises 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

denition 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 dene

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 satises

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 satises 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 satised as well.Thus,we've shown that all the conditions in the

denition of Walrasian equilibrium are satised 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 dierentiability 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 satises

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 signicant 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 indierence 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 aordable 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 satises 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 satises 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 satises u(x) > u(bx).We've furthermore assumed that there is a bundle

x

0

2 X that satises 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

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