# THE GREENWALD-STIGLITZ THEOREM

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Oct 10, 2013 (4 years and 7 months ago)

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THE GREENWALD
-
STIGLITZ THEOREM

“Externalities in Economies with Imperfect Information and Incomplete Markets”

Bruce C. Greenwald and Joseph E. Stiglitz

in The Quarterly Journal of Economics (May, 1986).

I.

THE MODEL

A.

Households

whe
re

Let

be the minimum expenditure required to attain a level of utility

at
prices
q
and

the level of externality goods. Thus,

is the
compensated demand for good
k

by
h.

= uncompensated demand function of household
h
.

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B.

Firms

Firms maximize the profit function,

subject to

where

production ve
ctor of firm
f

p

≡ vector of producers’ prices for the
N

1 nonnumeraire goods

a production function

vector of other

variables affecting firm
f

The firm’s maximum profit function,

has the property that

(Which theorem in duality?)

And

supply function of firm
f
.

C.

Government

The government produces nothing, collects taxes, and distributes transfers. Its net income
is

where the tax is just the difference between consumer and producer prices,

and

(i.e., the sum of nonnumeraire consumption).

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D.

Equilibrium

We assume an initial equilibrium with no taxes
and

for all
h

to exist. Thus

and

where we dropped superscript of
z
.

* This equilibrium is not Pareto if there exists
t

that

(a) leave households’ utilities unchanged, and

(b) incr
ease government revenues.

REMARK: Government is an agent. If a tax raises its revenue
(utility) without reducing households’ utility, then the original
equilibrium is NOT Pareto.

Let us formalize this idea:

If the original equilibrium is Pareto, then the

problem

subject to

has a solution at

NOTE: The constraint means that the households can still afford
their previous consumption levels and their previous utilities
. Note that

are functions of
t
and
I
.

If on the other hand, if

a set
, that can make at least one
agent better off (in this case, government), the equilibrium is NOT
Pareto.

A
long the constraint equation (5), we have

where

change in lump sum income per unit change in tax required to maintain
,

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a vector with

elem
ents,

an

element vector.

Note:

where
is an identity matrix.

Substitution into (6) (
text is wrong!)

for

gives

Substituting (1) and (2) into (7):

Summing over all households we have:

Therefore we have,

since

in market equilibrium.

Pecuniary effect of
t

Externality effect
-

nonpecuniary

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We
are now ready for final push.

E.

Pareto or Not!

Take
R
* of max
R

subject to (5). It is a function of
t
.

where

This is the derivative of
R

in directions which satisfy the compensa
tion constraint.

For the initial equilibrium to be Pareto optimal,
. But at
,

For this to be zero,
, i.e., no

and

are affected by
t
!

(Note: p.236, wrong text. “pecuniary” should be “nonpecuniary”.)

normally and the original equilibrium is not Pareto!

This is the GREENWALD
-
STIGLITZ THEOREM!

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Thus, the optimal tax level (i.e. tax level su
ch that

is