14.472 – Spring
Optimal Public Goods Supply
page 1 of 4
A…Samuelson FOC
Notation
x
y
z
w
g
u [ x, y , g ]
fi
p
consumption
labor supply
earnings
wage
public good
utility
fraction of population type i
price of public good
Social welfare maximization:
∑ fiu i  xi , yi , g 
E + pg + ∑ f i ( xi − wi yi ) ≤ 0
Maximize x, y , g
subject to:
(1)
FOC:
fi u xi [ xi , yi , g ] = λ fi
(2)
fi u [ xi , yi , g ] = λ fi wi
(3)
∑ fiugi  xi , yi , g  = λ p
(4)
i
y
Implying:
u gi  xi , yi , g 
∑ uxi  xi , yi , g  = p


i
u x , y , g
∑ fi wi ugiy  xii , yii , g  = p


fi
(5)
(6)
14.472 – Spring
Optimal Public Goods Supply
page 2 of 4
B…Second-Best FOC
R. Boadway and M. Keen, “Public Goods, Self-Selection and Optimal Income Taxation,”
International Economic Review v34, n3 (August 1993), 463-78.
Maximize x, y , g
subject to:
∑ fiu i  xi , yi , g 
E + pg + ∑ f i ( xi − wi yi ) ≤ 0
(7)
u i  xi , yi , g  ≥ u i  x j , y j w j / wi , g  for all i and j
Assume two types. Assume the only binding moral hazard constraint is type 1 considering
imitating type 2.
Maximize x, y , g
subject to:
f1u1  x1 , y1 , g  + f 2u 2  x2 , y2 , g 
E + pg + ∑ f i ( xi − wi yi ) ≤ 0
(8)
u1  x1 , y1 , g  ≥ u1  x2 , y2 w2 / w1 , g 
FOC:
f1u1x  x1 , y1 , g  − λ f1 + µ u1x  x1 , y1 , g  = 0
(9)
f1u1y  x1 , y1 , g  − λ f1w1 + µ u1y  x1 , y1 , g  = 0
(10)
f 2u x2  x2 , y2 , g  − λ f 2 − µ u1x  x2 , y2 w2 / w1 , g  = 0
(11)
f 2u y2  x2 , y2 , g  − λ f 2 w2 − µ u1y  x2 , y2 w2 / w1 , g  = 0
(12)
f1u1g  x1 , y1 , g  + f 2u g2  x2 , y2 , g  + µ ( u1g  x1 , y1 , g  − u1g  x2 , y2 w2 / w1 , g  ) = λ p
(13)
14.472 – Spring
Optimal Public Goods Supply
page 3 of 4
Rearranging terms and adding and subtracting
 u g2  x2 , y2 , g  
 to the
 u x2  x2 , y2 , g  

 
µ u1x  x2 , y2 w2 / w1 , g  
last equation:

( f1 + µ ) u1g  x1, y1, g  +  f 2 − µ

u1x  x2 , y2 w2 / w1 , g   2
u x , y , g
u x2  x2 , y2 , g   g  2 2 
(14)

 u2  x , y , g   
− µ  u1g  x2 , y2 w2 / w1 , g  − u1x  x2 , y2 w2 / w1 , g   g2  2 2    = λ p
 u x  x2 , y2 , g   

 
 

Rearranging terms again:
u1  x , y , g 
u2  x , y , g 
g
g 2 2 
1
1


+ f u 2  x , y , g  − µu1  x , y w2 / w1 , g 
f + µ u1  x , y , g 
1
2 x 2 2 
x  1 1  u1  x , y , g 
x 2 2
 u2  x , y , g 
x 1 1 
x 2 2 
 u1  x , y w / w , g  u 2  x , y , g  
 g 2 2 2 1 
g  2 2 
=λp
− µ u1  x , y w2 / w1 , g  
−
x 2 2
 u1  x , y w / w , g  u 2  x , y , g  
 x 2 2 2 1 
x  2 2  

(
)
(
)
(15)
14.472 – Spring
Optimal Public Goods Supply
page 4 of 4
Using the other FOC:
 u1g  x2 , y2 w2 / w1 , g  u g2  x2 , y2 , g  
u1g  x1 , y1 , g 
u g2  x2 , y2 , g 
1
− µ u x  x2 , y2 w2 / w1 , g   1
λ f1 1
+ λ f2 2
−
=λp
 u x  x2 , y2 w2 / w1 , g  u x2  x2 , y2 , g  
u x  x1 , y1 , g 
u x  x2 , y2 , g 



 
(16)
This reduces to the Samuelson condition if
u1g  x2 , y2 w2 / w1 , g  u g2  x2 , y2 , g 
=
u1x  x2 , y2 w2 / w1 , g  u x2  x2 , y2 , g 
For example, this holds with everyone having the same utility function that is additive:
a  x  + b  y  + c  g 
or everyone has a utility function separable in labor and with the same subadditive form:
u i  h  x, g  , y  .
Otherwise, the deviation from the Samuelson condition depends on the MRS for the imitated
type compared with that for an imitator, that is, it depends on how the MRS varies with labor
supply.
And the conditions for the Samuelson rule to hold are different for different numeraires.
(17)