14.123
Spring 2005
Peter Diamond
page 1 of 6
Handout on taxing externalities with uniform taxes
Let A , mnemonic for atmosphere, be the only route for externalities.
A = ∑ a i x1i
(1)
i
Note A is the same for all consumers. Note linearity is not important. This could be done with a
vector of different atmospheres.
I.
Quasi-linear preferences, 2 goods, additive externality
u h = x0h + f h ( x1h ) + g h ( A) ;
f ′h > 0, f ′′h < 0
(2)
x0h + qx1h = I h
Consumers ignore atmospheric feedback.
Interior solution
f ′h ( x1h ) = q → x1h ( q ) , x1′h ≤ 0
(3)
Social Welfare assuming linear technology with producer price p for good one.
∑h u h =∑h x0h + ∑h f h ⎡⎣ x1h ( q )⎤⎦ + ∑h g h ⎡⎣ A⎤⎦
⎡
⎤
= ∑ x0h + ∑ f h ⎡⎣ x1h ( q ) ⎤⎦ + ∑ g h ⎢ ∑ ai x1i ( q ) ⎥
h
h
h
⎣ i
⎦
(4)
⎡
⎤
⎣
⎦
= ∑ e0h − p ∑ x1h ( q ) + ∑ f h ⎡⎣ x1h ( q ) ⎤⎦ + ∑ g h ⎢∑ ai xi ( q )⎥
h
h
h
h
i
14.123
Spring 2005
Peter Diamond
page 2 of 6
First order condition from differentiating (4) with respect to q :
⎛
⎞⎛
⎞
0 = − p ∑ x1′h + ∑ f ′h x1′h + ⎜ ∑ g ′h ⎟ ⎜ ∑ a i x1′i ⎟
⎠⎝ i
⎠
⎛
⎞⎛
⎞
= − p ∑ x1′h + q ∑ x1′h + ⎜ ∑ g ′h ⎟ ⎜ ∑ a i x1′i ⎟
⎝ h
⎠⎝ i
⎠
h
h
h
⎝
h
h
(5)
∑ a x′
⎛
⎛
h ⎞⎛
i i⎞ ⎛
i⎞
h⎞
q − p = t = − ⎜ ∑ g ′ ⎟ ⎜ ∑ a x1′ ⎟ / ⎜ ∑ x1′ ⎟ = − ⎜ ∑ g ′ ⎟
⎝ h
⎠⎝ i
⎠ ⎝ i
⎠
⎝ h
⎠ ∑ x′
i
i
1
i
i
(6)
1
i
II.
Quasi-linear preferences, 2 goods, externality not necessarily additive
u h = x0h + f h ( x1h , A)
∂f h ( x1h , A)
∂x1h
=q
→ x1h ( q, A )
Solve (8) and (1) simultaneously. Note ∂x1 / ∂q | A different from
h
(7)
d h
x1 ( q, A ( q ) ) .
dq
(8)
14.123
Spring 2005
III.
Peter Diamond
page 3 of 6
Quasi-linear preferences, 3 goods, additive externality
⎛
⎞
u h = x0h + f h ( x1h , x2h ) + g h ⎜ ∑ ai x1i ⎟
⎝
i
⎠
(9)
f1h ( x1h , x2h ) = q1
f 2h ( x1h , x2h ) = q2
(10)
xih ( q1 , q2 )
Social welfare function
∑h eh − p1 ∑h x1h ( q1, q2 ) − p2 ∑h x2h ( q1, q2 ) + ∑h f h ⎡⎣ x1h ( q1, q2 ) , x2h ( q1, q2 )⎤⎦ + ∑h g h ( A)
First order conditions with respect to
q1 and q2
i
∂x1h
∂x2h ⎛
h ⎞⎛
i ∂x1 ⎞
0 = t1 ∑
+ t2 ∑
+ ⎜ ∑ g′ ⎟ ⎜ ∑ a
⎟
∂q1 ⎠
h ∂q1
h ∂q1
⎝ h
⎠⎝ i
i
∂x1h
∂x2h ⎛
h ⎞⎛
i ∂x1 ⎞
′
+ t2 ∑
+ ⎜∑ g ⎟⎜∑ a
0 = t1 ∑
⎟
∂q2 ⎠
h ∂q2
h ∂q2
⎝ h
⎠⎝ i
If, at the optimum we have
∑ ai
i
(11)
∂x1i
∂x i
≠ ∑ a i 1 , then at the optimum t2 ≠ 0 .
∂q1
∂q2
i
(12)
14.123
Spring 2005
IV.
Peter Diamond
page 4 of 6
General preferences, additive externality, lump sum redistribution, two goods
Max
⎧⎪
⎡
⎩
p ⋅ ∑ x ( q, I
h
s.t.
⎤ ⎫⎪
∑h ⎨⎪vh ( q, I h ) + g h ⎢⎣∑i ai x ( q, I i )⎥⎦ ⎬⎪
h
h
i
1
⎭
)=B
(13)
Normalizations: p0 = q0 = 1; p1 = p; q1 = q
Alternative writing of the resource constraint:
∑ { x ( q, I h ) + px ( q, I h )} = ∑ e
h
h
1
0
h
h
0
(14)
h
FOC:
h
i
⎧⎪ ∂x0 h
∂v h
∂x1 ⎫⎪ ⎛
h ⎞⎛
i ∂x1 ⎞
′
−
+
g
a
λ
+
p
∑h ∂q1 ∑h ⎪⎨ ∂q1 ∂q1 ⎪⎬ ⎝⎜ ∑h ⎠⎟ ⎜ ∑i ∂q1 ⎟ = 0
⎝
⎠
⎩
⎭
(15)
Using Roy’s identity and moving the last term:
h
i
⎛ ∂x0 h
⎛
∂v h
∂x1 ⎞
h ⎞⎛
i ∂x1 ⎞
′
− λ∑⎜
−∑ x1
+p
⎟ = −⎜ ∑ g ⎟⎜∑ a
⎟
∂I
∂q1 ⎠
∂q1 ⎠
h
h ⎝ ∂q1
⎝ h
⎠⎝ i
h
(16)
From the individual budget constraints we have:
x0h (q, I ) + qx1h (q, I ) = I
x1h (q, I ) +
∂x0h
∂x h
+q 1 =0
∂q
∂q
(17)
∂x0h
∂x h
+ q 1 =1
∂I
∂I
With
q = p + t , we have:
∂x0h
∂x h
∂x h
+ p 1 = −t 1 − x1h (q, I )
∂q
∂q
∂q
∂x0h
∂x h
∂x h
+ p 1 = 1− t 1
∂I
∂I
∂I
Substituting in (16) we have:
(18)
14.123
Spring 2005
Peter Diamond
page 5 of 6
−∑ x1h
h
⎛
⎛
⎞⎛
∂x h ⎞
∂xi ⎞
∂v h
+ λ ⎜ ∑ x1h + t ∑ 1 ⎟ = − ⎜ ∑ g ′h ⎟ ⎜ ∑ ai 1 ⎟
∂I
∂q1 ⎠
h ∂q1 ⎠
⎝ h
⎠⎝ i
⎝ h
(19)
h
We have FOC for each I :
h
∂v h
∂x0h
∂x1h ⎛
h ⎞ h ∂x1
′
+p
)+⎜∑ g ⎟a
− λ(
=0
∂I
∂I
∂I
∂I
⎝ h
⎠
(20)
Substituting from the budget constraints:
∂v h
∂x h
= λ − λt 1
∂I
∂I
⎛
⎞
− ⎜ ∑ g ′h ⎟ a h
⎝
h
⎠
∂x1h
∂I
(21)
The Slutsky equation
s1h =
∂x1h
∂x h
+ x1h 1
∂q1
∂I
(22)
Substituting from the Slutsky equation in (19):
h
i ⎞
⎛
⎛
⎞
⎛
⎛ h
∂v h
h
h⎞
i⎛ i
h ∂x1 ⎞
i ∂x1 ⎞
−∑ x
+ λ ⎜ ∑ x1 + t ∑ ⎜ s1 − x1
⎟ ⎟ = − ⎜ ∑ g ′ ⎟ ⎜⎜ ∑ a ⎜ s1 − x1
⎟⎟
∂I
∂I ⎠ ⎠
∂I ⎠ ⎟⎠
h
h ⎝
⎝
⎝ h
⎠⎝ i
⎝ h
h
1
(23)
Rearranging terms:
i
⎛
⎛ h ∂x1h ⎞
⎛
∂v h
h
h⎞
h ⎞⎛
i i⎞ ⎛
h ⎞⎛
i i ∂x1 ⎞
−∑ x
+ λ ⎜ ∑ x1 + t ∑ s1 ⎟ = − ⎜ ∑ g ′ ⎟⎜ ∑ a s1 ⎟ + ⎜ ∑ g ′ ⎟ ⎜ ∑ a x1
⎟ + λt ∑ ⎜ x1
⎟
∂I ⎠
∂I
∂I ⎠
h
h
h ⎝
⎝ h
⎠
⎝ h
⎠⎝ i
⎠ ⎝ h
⎠⎝ i
h
1
(24)
Substituting from (21):
−∑ x1h
h
⎛ ⎛ ∂v h
⎛
⎞⎛
⎞
⎞⎞
⎛
⎞
∂v h
+ λ ⎜ ∑ x1h + t ∑ s1h ⎟ = − ⎜ ∑ g ′h ⎟ ⎜ ∑ a i s1i ⎟ − ∑ ⎜ x1h ⎜
− λ ⎟⎟
∂I
h
⎝ h
⎠
⎠⎠
⎝ h
⎠⎝ i
⎠ h ⎝ ⎝ ∂I
(25)
14.123
Spring 2005
Peter Diamond
page 6 of 6
Simplifying, we have:
⎛ ∑ g ′h ⎞ ⎛ ∑ a h s1h ⎞
⎟
⎟⎜ h
t = − ⎜⎜ h
h ⎟
⎜
⎟
⎜ λ ⎟ ⎜ ∑ s1 ⎟
⎝
⎠⎝ h
⎠
(26)
h
∂v h
∂x1h ⎛
h ⎞⎛
h ∂x1 ⎞
∑h ∂I − λ H = −λt ∑h ∂I − ⎜ ∑h g ′ ⎟ ⎜ ∑h a ∂I ⎟
⎝
⎠⎝
⎠
(27)
Adding (21) over h:
Substituting for
λt :
λH − ∑
h
⎛
⎞⎛
⎛
⎞⎛
∂x h ⎞
∂v h
⎞ ⎛ ∂x h ⎞
= − ⎜ ∑ g ′h ⎟ ⎜ ∑ a h s1h ⎟ ⎜ ∑ 1 ⎟ / ∑ ( s1h ) + ⎜ ∑ g ′h ⎟ ⎜ ∑ a h 1 ⎟
∂I
∂I ⎠
⎠ ⎝ h ∂I ⎠ h
⎝ h
⎠⎝ h
⎝ h
⎠⎝ h
(28)
or
⎧ ⎛ h ∂x1h ⎞ ⎛ ∂x1h ⎞ ⎛
⎛
∂v h ⎞ ⎛ ∂x1h ⎞ ⎛
⎞ ⎛
⎞ ⎫⎪
h ⎞⎪
′
λ
H
−
=
g
∑h ∂I ⎟ ⎜ ∑h ∂I ⎟ ⎜⎝ ∑h ⎟⎠ ⎨⎪∑h ⎜ a ∂I ⎟ / ⎜ ∑h ∂I ⎟ − ⎜⎝ ∑h a h s1h ⎟⎠ / ⎜⎝ ∑h s1h ⎟⎠ ⎬⎪
⎜
⎝
⎠ ⎝
⎠
⎠ ⎝
⎠
⎩ ⎝
⎭
(29)
h
⎧
⎫
h ∂x1
a
a h s1h ⎪
∑
∑
h
h
⎪
⎛ ∂x ⎞ ⎛
∂v
⎞⎪
⎪
∂I
+ ∑ ⎜ 1 ⎟ ⎜ ∑ g ′h ⎟ ⎨ h
− h h ⎬
λH = ∑
h
∂x1
h ∂I
h ⎝ ∂I ⎠ ⎝ h
⎠⎪
∑h s1 ⎪
∑
⎩⎪ h ∂I
⎭⎪
(30)
or