ECON 4910 Spring 2007
Environmental Economics
Lecture 6, Chapter 9
Lecturer: Finn R. Førsund
Environmental Economics
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Illustration of spatial dimension


Spatial configuration: transport from source
to receptor
Key variable: transfer coefficient aij
i
Source i
(Point,
Mobile,
Diffuse)
Transfer coefficient
akij
Environmental
receptor, j
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Spatial dispersion of pollutants

Non-uniform dispersion
ei  e1i ,., eki ,.., eKi  , i  1,.., N
M sj  msj (e1 ,..., eN )
  0 , 0  M sj  M max
msjki
sj  m sj (0 ,.., 0)
j = 1,.., R , s = 1,..,S , k = 1,..,K , i = 1,..,N



ei = vector of secondary or remaining
discharges of pollutants from source i
eki = discharge of pollutant of type k from
source i
Msj = environmental service of type s
measured by indicator at receptor j
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Non-uniform dispersion of pollutants

Introducing transfer coefficients
N
N
i= 1
i= 1
msj (e1,..,e N )  m sj (a 1ij e 1i , ..., a Kij e Ki ) ,
s = 1,..,S , j = 1,..,R , k = 1,..,K , i = 1,.., N

The unit transfer coefficient akij is a pure
reorganisation of the environmental function
summing up the amount of a pollutant k
reaching the environmental receptor j from
source i
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Non-uniform dispersion of pollutants,
cont.

Marginal impact on environmental services is
depending on the location of the source i
 m sj( e1,..,e N )
 e ki


= msj akij
The transfer coefficient may also depend on level of
emission of other substances if physical interactions
Special case of constant transfer coefficient over
time, may be calculated as averages over several
time periods
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One-directional diffusion of pollutant


Pollution of a river, simplifying to one
pollutant and fixed transfer coefficients
Ordering the sources along the river starting
upstream of receptor j
a1 j  a 2 j  .....  a ij  ....  a N j ,
receptors j  1,.., R ,
sources i  1,.., N j (no. of upstream sources)

The transfer coefficient of source most
upstream must be the smallest due to
retention
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Illustration of river pollution
Source i
Downstream
Estuary
Receptor j
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One-directional diffusion of single
pollutant, cont.

Marginal impact from source i to the same
receptor j gets successively larger for
sources downstream located above the
receptor
m j (e1,.., e N )  m j (  a ij ei ) ,
iN j
m j (e1,.., e N )
 mj aij
 ei
j  1,.., R , i  1,.., N j
Environmental Economics
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The social solution, non-uniform
dispersion

The social optimisation problem adopting monetary
evaluation of environmental services
R
N
N
j 1
i= 1
i 1
Max { E j (m j (a ij ei ))   ci (ei )} ( E j  0, mj  0)


Assuming the monetary evaluation of the same
service level is independent of receptor
First-order condition
R
E mj aij  ci (ei )  0, i  1,.., N
j 1

Marginal purification cost equal to marginal
evaluation of the environment
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Implementation using a Pigou tax

Firms minimise costs plus tax payment
Min {ci (ei )  ti ei }

First-order condition
ci(ei )  ti

Comparing the social solution and the market
solution yields the optimal tax
R
ci (ei )  ti  ti   E maij , i  1,.., N
j 1
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The case of polluting a river

Must look at pollution caused by source i:
 m ( a e  ...)
jRi


j
ij i
Ri is the set of receptors downstream polluted
by source i
Load in receptor j is also coming from all
upstream sources, but by assuming additivity
of load these effects can be neglected when
investigating marginal effect of source i
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River pollution, cont.

The first-order condition in the social solution
 E j mj aij  ci (ei )  0
jRi

Simplifying to the same biological effect and
monetary evaluation of the environmental
service
E m  aij  ci (ei )  0
jRi
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Implementing using a Pigou tax

Finding the optimal tax rate
ci (ei )  ti   E m  aij
jRi


The tax rate is source-specific
The tax rate becomes smaller the further
downstream the location of the source
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Uniform dispersion of pollutants

Uniform dispersion implies that all transfer
coefficients are equal and typically equal to 1
N
N
i= 1
i= 1
msj (e1,.., e N )  mˆ sj (a e 1i , ..., a e Ki )
N
N
N
N
i= 1
i= 1
i= 1
i= 1
 mˆ sj (a (e 1i , ..., e Ki ))  m sj (e 1i , ..., e Ki )
s = 1,..,S , j = 1,..,R , k = 1,..,K , i = 1,.., N
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Uniform dispersion of pollutants, cont.

Marginal effects
M sj
eik

N
N
i=1
i=1
 (e 1i , ..., e Ki )
 msjk
The marginal effect is independent of source,
i.e. location, but depends on type of pollutant
and receptor
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The social solution, uniform dispersion

The social optimisation problem (simplifying to one
pollutant and one environmental service)
R
N
N
j 1
i= 1
i 1
Max { E j (m j (ei ))   ci (ei )}

First-order condition
R
 Em  c  (e )  0, i  1,.., N
j 1

j
j
i
i
Marginal purification cost equal to total marginal
evaluation of change in environmental service
independent of location of source
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Implementing the social solution using a
Pigou tax

Finding the optimal tax rate
R
ci(ei )  t   Ej mj
j 1

The tax rate is independent of source
implying the same marginal purification cost
for all sources and equal to the total marginal
monetary evaluation of the environmental
service
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Tradable emission permits

Trade in permits can be used when



The social solution is derived from setting
environmental standards because the damage
function is not known
Damage function known, but certainty of
achieving the desired pollution level is preferred
Trade in permits to a common trading price
can only be socially optimal if the pollutant is
uniformly dispersed
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Tradable emission permits, cont.



Modelling one receptor, one pollutant ,
multiple sources
Policy problem: how to distribute emission
permits on the sources in order to achieve
the environmental standard to least cost
Policy options


Auction the permits
Giving them free, following e.g. a grandfathering
principle
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Tradable emission permits, cont.

Finding the restriction on total emission
N
M sj  m sj (ei )  msj (e)
i=1

If the dose-response functions are known,
goals for environmental services, M sjT , will
determine the total emission restriction
e  Min {e : msj (e) 
R
T
M sj
for all s, j )
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Tradable emission permits, cont.

How to set the firm-specific quotas

Grandfathering: uniform reduction with factor a
N
a  (e /  eio )  1
R
i 1

Least cost allocation
N
Min  ci (ei )
i 1
N
s.t.  ei  e R
i 1
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Least cost allocation, cont.

The Lagrangian
N
L   ci (ei )
i 1
N
 (  ei  e )
R
i 1

First-order condition
L
 ci (ei )    0
ei
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Least cost allocation, cont.


The least cost solution: Marginal purifications
costs should be equal for all firms
Comparison with uniform reduction solution
ci (aeio )  ci (ei* )  const.

Marginal costs of uniform reductions will in
general differ from common marginal cost of
the optimal solution
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Efficiency of tradable permits
-c1’, -c2’
-c2’
-c1’
e1
e2
e1o
e1*
e2*
e2o
eR = a(e1o +e2o)
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Market implementation of emission
permits

Giving quotas free, allowing free trade


A firm can keep a permit or sell it to other firms
Assume a market with a price q for quotas
Min {ci (ei )  q(aeio  ei )} 
ci (ei )  q  0 for all i


Analogy with the Coase theorem
Assume an auction ending with a competitive
price q:

min sum of purification cost and outlay on quotas,
same solution as above
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