ECON 4910 Spring 2007
Environmental Economics
Lecture 2 Chapter 6
Lecturer: Finn R. Førsund
Environmental Economics
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The Coase theorem


Provided some assumptions are fulfilled, the
initial property right to an environmental
resource does not matter for the social
efficiency of the utilisation of the resource
Maximising profit or utility bargaining between
the parties will lead to the same use of the
environmental resource independent of the
alocation of property rights
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
Consider the basic model
B  b( P), b '  0, b ''  0
D  d ( P), d '  0, d ''  0


Benefit function is associated with a steel mill
emitting pollutants
Damage function is associated with a
downwind laundry drying clothes outdoors
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

Behind the benefit function we have the
multiple output production function
y  f ( x)
P  g ( x)
Pollution without regulation
Max  py   qi xi s.t. y  f ( x)

pollution corresponding to private profit
maximisation

Pπ = g(x*)
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Interpretation of the benefit function


The benefit function reflects the costs of
reducing the amount of pollution, P,
generated in steel production, i.e. by
purification and/or output reduction, from a
level equal to the private profit maximisation
level Pπ to a lower level
Notice that it is also costly for the steel mill to
increase the level of pollution beyond Pπ. The
b’(.) – function has a discontinuity
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Allocating property right to clean air


The steel mill has the right to pollute
The laundry has the right to clean air
d’(P)
b’(P)
P
PMin

P*
Pπ
Pmin is found by solving d’(P) = 0
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Asumptions for the Coase theorem






No transaction costs
Perfect information
Agents price-takers
Maximize profit (utility)
Costless enforcement of agreements
No income and wealth effects
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Elements of general equilibrium models
of the environment

Basic stages:


(1) Extracting/harvesting resources
(2) Transformation of resources into products



Residuals as by-products (Primary/secondary residuals if addon purification is an option)
(3) Environmental impacts of absorption of residuals
(4) Evaluation of changes in environmental indicators


Evaluation through utility functions, environment as public
good, or damage functions.
Products are also evaluated either through utility functions or
benefit functions
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Strands of models according to level of
aggregation

Aggregated macro level:


Micro level of the firm:


General equilibrium and interdependencies most important,
skip details about resource creation, production structure,
general abatement possibilities.
Detailed production function and purification possibilities,
household production functions
Classical externalities models:

Short- circuits the environment, direct representation of
residual-generating activities in production and/or utility
functions
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Why general equilibrium models





Overview of complexities, interdependencies in the
economy and economic - environmental
interactions, expose weaknesses of partial analyses
Establish concepts or variables fruitful for empirical
work.
Show the type and extension of information needed
to implement any optimal solution
Show possibilities for decentralised decision-making
Framework for study of distributional aspects of
pollution and pollution abatement
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An aggregated general equilibrium model

Model without resource extraction and inputs



Multiple outputs that generate pollutants
Environmental services influences by pollutants
Environment as a public good
F( y 1,.., y g )  0 , F i ' > 0
Z = s( y 1,.., y g ) , s i '  0
M = m(Z) , m  0
U j U j( x1j ,.., x gj ,M) , U ij > 0,U Mj > 0
N
x
ij
= yi
j=1
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Formulating the social optimisation
problem
n
Max  w jU j( x1j ,.., x gj ,M)
j=1
subject to
F( y 1,.., y g ) 0
M  m(s( y 1,.., y g ))
x  y
n
ij
i
j=1
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Solving the model


Simplifying the model by eliminating variables
The Lagrangian function
n
L =  w jU j( x1j ,.., x gj ,M)
j=1
-  F( y 1,.., y g )
- (-m(s( y 1,.., y g )) + M)
g
n
-   i(  x ij
i=1
-y)
i
j=1
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Differentiating wrt all endogenous
variables


Endogenous variables yi,xij,M
Necessary first –order conditions
L
= -  F i '   m ' si ' +  i  0
 yi
L
= w jU ij(x, M) -  i  0
 x ij
n
L
=  w jU Mj (x, M) -   0
M j=1
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Interpreting shadow prices

The marginal effect in the optimal solution of
changing constraints



μi : shadow price on the distribution constraint, if
the shadow price is zero, then giving more of it to
a person has no opportunity cost
γ: shadow price on the transformation function, if
the shadow price is zero, then there is no cost of
production
λ: shadow price on the environmental service
function, if the shadow price is zero, then there is
no environmental cost of pollution
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Interpreting inequalities

If the first-order conditions hold with
inequalities then the endogenous variable in
question should be set to zero

The supply of good i condition


inequality means that the production and pollution cost
is higher than the shadow price  the good should not
be supplied
The allocation condition of a good to an individual

inequality means that the weighted social value of the
good is lower than the shadow price  the good should
not be supplied to the individual
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Interpreting inequalities, cont.

The environmental service condition

inequality means that the weighted social value of the
environmental good summed over all individuals is lower
than the shadow price of the environmental good
the environmental good should not be supplied
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Standard procedure of solution


Assume interior solutions, i.e. all first-order
conditions hold with equality (back to
standard Lagrange with equality constraints)
Eliminating two of three Lagrangian
parameters
n
 i Fi ' w jU Mj (x, M)m ' si '  w jU ij(x, M)
j=1

Marginal costs equal to marginal benefits
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Forming MRT and MRS
n
 Fd ' Fd '


 Fe ' Fe '
wkU M (x, M)m ' sd ' w jU d(x, M )
j
j
k=1
n
wkU M (x, M)m ' se ' w jU e(x, M )
j

j
k=1
n
wkU M (x, M)m ' sd '
j
k=1
j
w jU e(x, M )
j
(x, M )
U
 dj
U e(x, M )
n
w U
k
j
M
(x, M)m ' sg '
k=1
j
e
w jU (x, M )
1
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Implementing the solution via competitive
markets

Profit maximisation to prices pi
g

Max  pi yi s.t. F( y 1,.., y g )  0 
 i 1


Necessary first-order conditions
pi   Fi '  0 , i  1,.., g

MRT assuming interior solutions
Fd ' pd

Fe ' pe
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Implementing the solution, cont.

Consumers maximising utility given income,
Ri

g
 j

Max U ( x1 j ,.., xgj , M )s.t. pi xij  Ri  0
i 1


Necessary first-order conditions
U i j ( x, M )   pi  0, i  1,.., g , j  1,.., n
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Implementing the solution, cont. 2

MRT = MRS
Fd ' pd U dj ( x, M )

 j
, j  1,.., n, d , e  1,.., g
Fe ' pe U g ( x, M )

A competitive market system will not realise a
Pareto optimum. The environmental costs are
left out of the market system. M is a public
good, no producer is providing this good and
selling it to consumers
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