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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
Marginal deadweight
lo
ss
Marginal benefit from
reduction in externalities
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