14.123
Spring 2005
Peter Diamond
page 1 of 6
Handout on Externalities
1.
Two roads, mandatory trips.
Fi ( Ni ) is the cost for each person using road i if there are N i people on road i , assumed to
be nondecreasing in
N , and increasing for some of the analysis.
There is a continuum of households of measure H .
Utility maximization:
Max u h = I h − Min ⎡⎣ F1 ( N1 ) , F2 ( N 2 ) ⎤⎦
(1.1)
Equilibrium assuming both roads are used in equilibrium:
F1 ( N1 ) = F2 ( N 2 )
N1 + N 2 = H
(1.2)
If only one road is used, then
F1 ( H ) ≤ F2 ( 0 )
(1.3)
Consider conditions for Pareto optimality for the subset of Pareto optima where everyone has
positive income at the optimum.
Note that in this case, income distribution does not matter for optimal road use and we can
derive the optimal road use by maximizing the sum of utilities.
There are other Pareto optima where these conditions do not hold – where all of the income
goes to a subset of the people who get all the income and may also get a “better” road.
Max
∑h I h − N1F1 ( N1 ) − N 2 F2 ( N 2 )
subject to N1 + N 2 = H
(1.4)
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Spring 2005
Peter Diamond
page 2 of 6
If both roads are used, we have the FOC:
F1 ( N1 ) + N1F1′( N1 ) = F2 ( N 2 ) + N 2 F2′ ( N 2 )
(1.5)
F1 ( H ) + HF1′( H ) ≤ F2 ( 0 )
(1.6)
If just one road is used:
In general the equilibrium is not Pareto optimal.
However, it can be Pareto optimal, for example, when only one road is used and when there is
symmetry –
F1 ( H / 2 ) = F2 ( H / 2 ) and F1 ( H / 2) + F1′( H / 2) H / 2 = F2 ( H / 2 ) + F2′ ( H / 2 ) H / 2
(1.7)
When equilibrium is not Pareto optimal, optimality can be restored either by taxing one of the
roads or by subsidizing the other, with the revenue (or subsidy cost) covered by lump-sum taxes.
When there are taxes on each road the equilibrium becomes
F1 ( N1 ) + t1 = F2 ( N 2 ) + t2
N1 + N 2 = H
(1.8)
For the optimal N i equation 1.8 gives us a continuum of values of the two taxes that achieve
the equilibrium. This does not extend to the next example.
14.123
Spring 2005
2.
Peter Diamond
page 3 of 6
Two roads, voluntary trips.
h distributed uniformly on [ 0,1]
utility of trip for household
h is g h , decreasing in h .
Utility maximization:
u h = I h + Max ⎡⎣0, g h − F1 ( N1 ) , g h − F2 ( N 2 )⎤⎦
(2.1)
Equilibrium assuming both roads are used in equilibrium:
g N1 + N2 = F1 ( N1 ) = F2 ( N 2 )
N1 + N 2 ≤ H
(2.2)
Conditions for Pareto optimum for optima where everyone has positive income at the optimum.
1
Max
N1 + N 2
∫ I + ∫0
h
g h dh − N1F1 ( N1 ) − N 2 F2 ( N 2 )
0
(2.3)
subject to N1 + N 2 ≤ H
If both roads are used and some people do not make the trip, we have the FOC:
g N1 + N2 = F1 ( N1 ) + N1F1′( N1 ) = F2 ( N 2 ) + N 2 F2′ ( N 2 )
(2.4)
The optimum can be achieved by particular taxes on each road:
g N1 + N2 = F1 ( N1 ) + t1 = F2 ( N 2 ) + t2
N1 + N 2 ≤ H
(2.5)
14.123
Spring 2005
3.
Peter Diamond
page 4 of 6
Production Economy with consumption externalities.
Assume a linear technology.
To derive conditions for Pareto optimality:
Max
∑h α hu h ( xh , x~ h )
s.t p ⋅ ∑ x h = A
(3.1)
h
FOC:
∑α h
h
∂u h
= λ pj
∂x ij
∨ i, j
(3.2)
Rearranging terms:
h
∂u i
h ∂u
α i = λ p j − ∑α
∂xij
∂x j
h ≠i
i
(3.3)
Assume good 1 has no externalities and is the numeraire:
∂u h
α
= λ p1 = λ
∂x1h
h
∀h
(3.4)
Dividing 3.3 by 3.4:
∂u i / ∂xij p j
∂u h / ∂xij
=
−
h
h
∂u i / ∂x1i p1 ∑
h ≠i ∂u / ∂x1
(3.5)
Introducing taxes on transactions, with taxes potentially being different for different individuals:
14.123
Spring 2005
Peter Diamond
page 5 of 6
∂u h / ∂xij
h
h
h ≠i ∂u / ∂x1
t ij = −∑
(3.6)
With these taxes, we can decentralize. We can see this by comparing FOC for individual choice
with conditions for PO.
Max u h ( x h , x − h )
∑ j ( p j + t hj )xhj = I h
s.t
(3.7)
FOC:
∂u / ∂x = λ
h
h
j
h
(p
j
+t
h
j
)
⎛
∂u i / ∂x hj ⎞
= λ ⎜ pj − ∑ i
i ⎟
⎜
⎟
∂
u
/
∂
x
i
h
≠
1
⎝
⎠
h
∂u h / ∂x1h = λ h p1
(3.8)
(3.9)
Using (3.3) and (3.9), income levels must then be chosen so that
λ = α hλ h
∀h
(3.10)
14.123
Spring 2005
4.
Peter Diamond
page 6 of 6
Pure public good.
Now consider government controlled expenditures,
G , which potentially enter every utility
function. We continue to assume a linear technology. Conditions for Pareto optimality:
∑ α hu h ( x h , G )
p ⋅ ∑ x h + pG G = R
Max
s.t
αh
∂u h
= λ pi
∂xih
(4.1)
(4.2)
∂u h
= λ pG
∂G
(4.3)
∂u h / ∂G pG
∑h ∂u h / ∂xih = pi
(4.4)
∑α h
h
Dividing 4.3 by 4.2: