G-S Theorem
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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
for all h  1, 2,  H

max U h x h , Z h

s.t. x  qx  I h   a hf 
h
1
h
f
F
where


x h  x1h , x h  consumptio n vector of household h,
x1h is consumptio n of the numeraire good,


x h  x2h ,, x Nh is consumptio n of the N  1 nonnumerai re goods,
z h  vector of N h other vari ables that affect the utility of h
(e.g., pollution, average quality of a good consumed, etc),
q  vector of prices of N  1 nonnumerai re goods (price of x1 is p1  1),
 f  profits of firm f (note : initial endowment  0),
a hf  fractional holding of household h in firm f , H a hf  1,
I h  lump sum government transfer to household h,


I  I 1 ,, I H , vector of lump sum transfer.


Let E h q, z h , u h be the minimum expenditure required to attain a level of utility u h at
h
prices q and z the level of externality goods. Thus,

xˆ q; z , u
h
k
h
h

E h

q z h , u h
(1)
is the compensated demand for good k by h.


x h q, I , z h = uncompensated demand function of household h.
G-S Theorem
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B. Firms
Firms maximize the profit function,
 f  y1f  p  y f


subject to y1f  G f y f , z f  0


where y f  y1f , y f  production vector of firm f
p ≡ vector of producers’ prices for the N – 1 nonnumeraire goods
G f  a production function
z f  vector of other N f variables affecting firm f
The firm’s maximum profit function,
 *f  p, z f 
has the property that
 *f
 y kf ,
pk z f
k  1, N
(2)
(Which theorem in duality?)
And

  
 

y f p, z f  y1f p, z f , y f p, z f  supply function of firm f.
C. Government
The government produces nothing, collects taxes, and distributes transfers. Its net income
is
R  t  x I h
H
where the tax is just the difference between consumer and producer prices,
t  q  p 
and x 
xh

H
(i.e., the sum of nonnumeraire consumption).
G-S Theorem
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D. Equilibrium
We assume an initial equilibrium with no taxes and I h  0 for all h to exist. Thus
t  0  p  q and
x q, I , z    y f  p, z   0
F
(4)
x q, I , z    y f  p, z 
F
where we dropped superscript of z.
* This equilibrium is not Pareto if there exists t that
(a) leave households’ utilities unchanged, and
(b) increase 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
max R  t  x   I h
t ,I
subject to I 
h
has a solution at t  0.
a
h
hf
 E h q, z h ; u h 
f
(5)
NOTE: The constraint means that the households can still afford
their previous consumption levels and their previous utilities
u h  h . Note that u h , z h , z f ,  f , p, q are functions of t and I.
If on the other hand, if  a set t  0 , that can make at least one
agent better off (in this case, government), the equilibrium is NOT
Pareto.
Along the constraint equation (5), we have

dI h
dz f
dp 
dq
dz h
  Eqh
  a hf   zf
  pf
 E zh
,
dt
dt
dt
dt
dt
F


h  1, H
(6)
where
dI h
 change in lump sum income per unit change in tax required to maintain u h ,
dt
G-S Theorem
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  *f   G f 
   f    f , a vector with N f elements,
 z   z 
 E h   u h 
h
E z   h    h , an N h element vector.
 z   z 
f
z
Note:
dt  dq  dp

I N 1 
dq dp

dt dt
where I N 1 is an identity matrix.
dq
gives
dt
f
h
dI h
dp 
hf  f dz
f dp 
h N
h  dz 


  a  z 
 p
 Eg I    Ez 

dt
dt
dt 
dt 

F

 dt 
Substitution into (6) (text is wrong!) for

E   E qh   a hf  pf
f

h
q
f
h
 dp dI h 
hf
f dz
h dz 

 dt  dt   a  z dt  E z dt 
F


Pecuniary effect of t
(7)
Externality effect - nonpecuniary
Substituting (1) and (2) into (7):
f
h
 h

dI h 
hf k  dp
hf
f dz
h dz 

xˆ q; z , u   xˆ k   a y f 

  a  z
 Ez

dt
dt
dt
dt 
f
F



h
k

h
h

Summing over all households we have:
x  x  y 
dp
dI h 
dz f
dz h 

  zf
  E zh

dt H dt  F
dt
dt 
H
Therefore we have,
f
h

dI h
f dz
h dz 


x



E

  z dt  z dt 
H dt
H
 F

since x  y  0 in market equilibrium.
(8)
G-S Theorem
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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.
dR *
dx
dI h
 x  t  
dt
dt
H dt
(9)
f
h

dx
f dz
h dz 


 x   t   x   z
  Ez
dt
dt
dt
F
H



dR * dx

 t   t  t
dt
dt

(10)
where
dz f
  
,
dt
F
t
f
z
dz h
  E
dt
H
t
h
z
This is the derivative of R in directions which satisfy the compensation constraint.
For the initial equilibrium to be Pareto optimal,

dR
 0 at t  0 . But at t  0 ,
dt

dR *
  t  t
dt
t
t
For this to be zero,     0 , i.e., no z f and z h are affected by t!
(Note: p.236, wrong text. “pecuniary” should be “nonpecuniary”.)

dR *
 0 normally and the original equilibrium is not Pareto!
dt
This is the GREENWALD-STIGLITZ THEOREM!
G-S Theorem
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Thus, the optimal tax level (i.e. tax level such that
t

dR *
 0 is
dt
dx
   t  t
dt
Marginal deadweight
loss


Marginal benefit from
reduction in externalities

 dx 
t*       
 dt 
t
t
1