Maximize c,G
P
N n
( u [ c n
] − a nn
+ b n v [ G ]) subject to: E + pG +
P
N n
( c n
− n ) ≤ 0 u [ c n
] − a nn
+ b n v [ G ] ≥ u [ c m
] − a mn
+ b n v [ G ] for m < n for all n
(1)
Forming a Lagrangian, we have
1

L =
X
N n
( u [ c n
] − a nn
+ b n v [ G ]) − λ
³
E + pG +
X
N n
( c n
− n )
´
(2)
+
X X
µ nm
( u [ c n
] − a nn
+ b n v [ G ] − ( u [ c m
] − a mn
+ b n v [ G ]))
The FOC for public good provision is
X
N n b n v 0 [ G ] = λp (3)
2

c n
= x n
− (1 − s n
) pg n
Utility can be written:
(4) u [ x n
− (1 − s n
) pg n
] − a nn
+ b n v [ G
˜ n
+ g n
]
The contributions for workers who make positive contributions satisfy
(5)
(1 − s n
) pu 0 [ c n
] = (1 − s n
) pu 0 [ x n
− (1 − s n
) pg n
] = b n v 0 [ G
˜ n
+ g n
] = b n v 0 [ G ]
(6)
Social welfare maximization
Maximize c,G,g
P
N n
( u [ c n
] − a nn
+ b n v [ G ]) subject to: E + pG +
P
N n
( c n
− n ) ≤ 0 u [ c
2
] − a
22
+ b
2 v [ G ] ≥ u [ c
1
] − a
12
+ b
2 v [ G − g
2
+ g
1
]
G ≥
P
N n g n
; g n
≥ 0 for all n
(7)
The optimum will have one of two forms - either both have the same consumption and (generically) the incentive compatibility constraint is not binding, or the optimum will have the incentive compatibility constraint binding.
3

Note that the subsidy rate for type 1 can exceed the subsidy rate for type
2 and still support the optimum. Using the maximal subsidy for type 1 that still leaves a zero contribution, we have:
(1 − s
1
) pu 0 [ c
1
] = b
1 v 0 [ G ]
(1 − s
2
) pu 0 [ c
2
] = b
2 v 0 [ G ]
(8)
(9)
With b
1
= b
2
, we have s
1
≥ s
2 since c
1
≤ c
2 with strict inequality if the incentive compatibility constraint is binding.
If the incentive compatibility constraint binds, the constraint becomes u [ c
2
] − a
22
+ b
2 v [ G ] = u [ c
1
] − a
12
+ b
2 v [ G − G/N
2
] (10)
The FOC for the public good satis fi es:
X
N n b n v 0 [ G ] = λp − µb
2
( v 0 [ G ] − v 0 [ G (1 − 1 /N
2
)] (1 − 1 /N
2
)) (11)
This may or may not satisfy the Samuelson rule and can deviate in either direction depending on the shape of the public good utility function, that is the sign of ( v 0 [ G ] − v 0 [ G (1 − 1 /N
2
)] (1 − 1 /N
2
)) . If v ” is small, this is close to the Samuelson rule.
4

With preferences u [ c n
] − a nn
+ b n v [ G ] + w [ g n
] , the FOC for individual donations for workers who make positive contributions satis fi es
(1 − s n
) pu
0
[ c n
] = (1 − s n
) pu
0
[ x n
− (1 − s n
) pg n
]
= b n v 0 [ G
˜ n
+ g n
] + w 0 [ g n
] = b n v 0 [ G ] + w 0 [ g n
]
(12)
The donation level with a zero subsidy (the minimum donation level), g [ c n
, G ] , satis fi es the individual FOC pu 0 [ c n
] = pu 0 [ x n
− pg n
] = b n v 0 [ G
˜ n
+ g n
] + w 0 [ g n
] = b n v 0 [ G ] + w 0 [ g n
] (13)
Di ff erentiating the implicit equation,
∂g
∂c
= pu 00 w 00
∂g
> 0 ,
∂G
=
− bv 00 w 00
< 0 .
(14)
5

Considering the two-types model with one public good, we can write a formulation allowing warm glow to enter ( θ = 1 ) and not enter ( θ = 0 ) social welfare.
Maximize c,G
P
N n
( u [ c n
] − a nn
+ bv [ G ] + θw [ g n
]) subject to: E + pG +
P
N n
( c n
− n ) ≤ 0 u [ c
2
] − a
22
+ bv [ G ] + w [ g
2
] ≥ u [ c
1
] − a
12
+ bv [ G − g
2
+ g
1
] + w [ g
1
]
G ≥
P
N n g n
; g n
≥ g [ c n
, G ] for all n
(15)
Di ff erentiating we have lots of FOC:
∂L
∂c
1
= N
1
( u 0 [ c
1
] − λ ) − µu 0 [ c
1
] − ξ
1
∂g [ c
1
, G ]
∂c
= 0 (16)
∂L
∂G
=
∂L
∂c
2
X µ
N n bv 0
= N
2
( u
0
[ c
2
] − λ ) + µu
0
[ c
2
] − ξ
2
∂g [ c
2
, G ]
∂c
[ G ] − ξ n
∂g [ c n
, G ]
∂G
¶
− λp + µb ( v 0 [ G ] − v 0
= 0 (17)
[ G − g
2
+ g
1
])+ ν = 0
(18)
∂L
∂g
1
= N
1
θw 0 [ g
1
] − µ ( bv 0 [ G − g
2
+ g
1
] + w 0 [ g
1
]) − νN
1
+ ξ
1
= 0 (19)
∂L
∂g
2
= N
2
θw 0 [ g
2
] + µ ( bv
0
[ G − g
2
+ g
1
] + w 0 [ g
2
]) − νN
2
+ ξ
2
= 0 (20)
We consider the FOC separately for the values of θ of 0 and 1.
6

= 0
Increasing the donation of the high type while lowering the donation of the low type weakens the incentive compatibility constraint while having no other e ff ects, until the lower limit on g
1 is hit. Similarly, donations by the high type dominate public provision. Thus we know that g
1
= g [ c
1
, G ] . Assuming that public good supply is less than optimal without any subsidization, we also have no binding minimum donation constraint for the high type, ξ
2
= 0 .
Thus we can write the FOC (16) and (17) for the case θ = 0 as
∂L
∂c
1
= N
1
( u 0 [ c
1
] − λ ) − µu 0 [ c
1
] − ξ
1
∂g [ c
1
, G ]
∂c
= 0
∂L
∂c
2
= N
2
( u 0 [ c
2
] − λ ) + µu 0 [ c
2
] = 0
(21)
(22)
Turning to the other FOC, with g
1 at its minimum, (18) and (20) become
∂L
∂G
= bv 0 [ G ]
X
N n
− ξ
1
∂g [ c
1
, G ]
∂G
− λp + µb ( v 0 [ G ] − v 0 [ G − g
2
+ g
1
]) + ν = 0
(23)
∂L
∂g
2
= µ ( bv 0 [ G − g
2
+ g
1
] + w 0 [ g
2
]) − νN
2
= 0 (24)
It is plausible for all but very large donors that v 00 is small enough that v 0 [ G ] − v 0 [ G − g
2
+ g
1
] is very small. Then the FOC for public good supply
7

is approximately
∂L
∂G
≈ bv 0 [ G ]
X
N n
− ξ
1
∂g [ c
1
, G ]
∂G
− λp + ν = 0 (25)
We can express the deviation from the Samuelson condition as bv 0 [ G ]
X
N n
− λp ≈ ξ
1
∂g [ c
1
, G ]
∂G
− ν = ξ
1
∂g [ c
1
, G ]
∂G
µ
−
N
2
( bv
0
[ G − g
2
+ g
1
] + w 0 [ g
2
])
(26)
Using again the assumption of a small v 00 , and so approximate constancy of v 0 , and the FOC for individual donations, we can write this as bv 0 [ G ]
X
N n
− λp = ξ
1
∂g [ c
1
, G ]
∂G
−
µ
N
2
(1 − s
2
) pu 0 [ c
2
] (27)
From the FOC for consumption, this can be written as bv 0 [ G ]
X
N n
− λp ≈ ξ
1
∂g [ c
1
, G ]
∂G
+ (1 − s
2
) p ( u 0 [ c
2
] − λ ) (28) or bv 0 [ G ]
X
N n
≈ ξ
1
∂g [ c
1
, G ]
∂G
+ s
2 pλ + (1 − s
2
) pu 0 [ c
2
] (29)
8

= 1
The FOC (16) - (20) become
∂L
∂c
1
= N
1
( u 0 [ c
1
] − λ ) − µu 0 [ c
1
] − ξ
1
∂g [ c
1
, G ]
∂c
= 0 (30)
∂L
∂G
=
∂L
∂c
2
X µ
N n bv 0
= N
2
( u 0 [ c
2
] − λ ) + µu 0 [ c
2
] − ξ
2
∂g [ c
2
, G ]
∂c
[ G ] − ξ n
∂g [ c n
, G ]
∂G
¶
− λp + µb ( v 0 [ G ] − v 0
= 0 .
(31)
[ G − g
2
+ g
1
])+ ν = 0
(32)
∂L
∂g
1
= N
1 w 0 [ g
1
] − µ ( bv 0 [ G − g
2
+ g
1
] + w 0 [ g
1
]) − νN
1
+ ξ
1
= 0
∂L
∂g
2
= N
2 w 0 [ g
2
] + µ ( bv 0 [ G − g
2
+ g
1
] + w 0 [ g
2
]) − νN
2
+ ξ
2
= 0
(33)
(34)
Donations of the low type may be subsidized. The dominance of donations of the high type over public provision remains true.
If donations of both types are subsidized ( ξ
1
, ξ
2
= 0 ), the private consumption FOC have the same form as with fully public provision.
∂L
∂c
1
= N
1
( u 0 [ c
1
] − λ ) − µu 0 [ c
1
] = 0
∂L
∂c
2
= N
2
( u
0
[ c
2
] − λ ) + µu
0
[ c
2
] = 0
(35)
(36)
9

Assuming that the incentive compatibility constraint is binding ( µ > 0 ), we have u 0 [ c
1
] u 0 [ c
2
]
=
1 + µ/N
2
1 − µ/N
1
> 1
If donations of both types are subsidized ( ξ
1
, ξ
2
= 0 ):
(37)
∂L
∂g
1
= N
1 w 0 [ g
1
] − µ ( bv 0 [ G − g
2
+ g
1
] + w 0 [ g
1
]) − νN
1
= 0
∂L
∂g
2
= N
2 w 0 [ g
2
] + µ ( bv 0 [ G − g
2
+ g
1
] + w 0 [ g
2
]) − νN
2
= 0
Solving for w 0 [ g n
] we have
µ
1 −
µ
¶ w 0 [ g
1
] = ν +
N
1
µ
µ
1 +
N
2
¶ w 0 [ g
2
] = ν −
µ
N
1 bv 0 [ G − g
2
+ g
1
]
µ
N
2 bv 0 [ G − g
2
+ g
1
]
(38)
(39)
(40)
(41)
The total utility gain from a donation by the lower type is bv 0 [ G ] + w 0
µ
[ g
1
] = bv 0
=
[
µ bv 0
G
[
] +
G ] +
ν
ν
+
+
µ
N
µ
N
1
1
( bv 0
( bv 0
[ G − g
2
+ g
1
])
¶
/
µ
µ
1 −
¶ 1
[ G − g
2
+ g
1
] − bv 0 [ G ]) /
¶
µ
1 −
N
1
(42)
¶
µ
We can evaluate the subsidy level.
(1 − s
1
) = bv 0 [ G ] + w 0 [ g
1
] pu 0 [ c
1
]
= bv 0 [ G ] + ν +
µ
N
1
( bv 0 [ G − g
2
+ g
1
] − bv 0 [ G ]) pλ
(43)
10

Similarly, for the high type
(1 − s
2
) = bv 0 [ G ] + w 0 [ g
2
] pu 0 [ c
2
]
= bv 0 [ G ] + ν − µ
N
2
( bv 0 [ G − g
2
+ g
1
] − bv 0 [ G ]) pλ
(44)
Taking the ratio
1 − s
1
1 − s
2
= bv 0 [ G ] + ν + bv 0 [ G ] + ν −
µ
N
1
µ
N
2
( bv 0 [ G − g
2
+ g
1
] − bv 0 [ G ])
( bv 0 [ G − g
2
+ g
1
] − bv 0 [ G ])
> 1 (45)
Thus the high type receives a higher subsidy. If, as is plausible, v 00 is small relative to individual donations, then (1 − s
1
) / (1 − s
2
) ≈ 1 .
If donations of both types are subsidized ( ξ
1
, ξ
2
= 0 ) and v 00 is small, the
FOC for public good level becomes approximately: bv
0
[ G ]
X
N n
− λp + ν ≈ 0 (46) bv 0 [ G ]
X
N n
≈ λp − ν = λp − w 0 [ g
2
] −
µ
N
2
( bv 0 [ G ] + w 0 [ g
2
]) (47)
= λp − w 0 [ g
2
] −
µ
N
2
(1 − s
2
) pu 0 [ c
2
]
= λp − w 0 [ g
2
] + (1 − s
2
) p ( u 0 [ c
2
] − λ )
= s
2
λp + (1 − s
2
) pu 0 [ c
2
] − w 0 [ g
2
]
If w 0 is much larger than v 0 ,then the last two terms approximately cancel and the FOC is as if the only cost of public goods were the marginal cost to
11

the government of the marginal contribution by the high type. Similarly, in terms of the marginal utilities of type 1.
bv 0 [ G ]
X
N n
≈ s
1
λp + (1 − s
1
) pu 0 [ c
1
] − w 0 [ g
1
] (48)
12

n skill index (equal to productivity)
N n x n c n number of workers of skill n compensation at job of skill n consumption of worker holding job of skill n u [ c n
] utility of private good consumption a mn disutility of labor for a worker of skill n holding a job of skill m
G public good supply s n
λ
ξ n
θ
µ n
ν
G
˜ n p
E b n v [ G ] utility of public good consumption g n donation of worker holding a job of skill n w [ g n
] warm glow utility public good provision net of donation of a worker holding a job of skill n cost per unit of the public good other public expenditures donation subsidy rate for worker in job of skill n
Lagrangian on the resource constraint
Lagrangians on the incentive compatibility constraints
Lagrangian on the adding up constraint for public good provision
Lagrangians on the minimum donation constraints index to distinguish SWF with and without warm glow utility
13