Stock pollution 1 ECON 4910 Spring 2007 Environmental Economics Lecture 9: Stock pollution Perman et al. Chapter 16 Lecturer: Finn R. Førsund
Jan 11, 2016
Stock pollution 1
ECON 4910 Spring 2007 Environmental Economics Lecture 9: Stock pollutionPerman et al. Chapter 16
Lecturer: Finn R. Førsund
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Effects of pollution Two ways
Environmental effects of pollution within a time period as function of pollutants discharged during the same period
Environmental effects within a time periods a function of accumulated amounts of pollutants from earlier periods
The time dimension of accumulation effects Real time: accumulation over short time periods, e.g.
accumulation of organic waste over a few hours in a river, important when ecosystems are highly vulnerable to extreme values in real time, threshold values for when “bad things” happen, e.g. day variation in oxygen in rivers
Discrete time with longer time periods, day, week , month , year: total load is the determining the pollution effect, not variation within the chosen time period, e.g. death of fish when snow melts in spring with accumulated acidity.
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Waste accumulation model
Stock of pollutants at time t from previous emissions eo ,e1……. et
Environmental impact from the stock of pollutants At and not from the current emission et
Damage function( ), 0t t t tD D A D
0
t
t tA e dt
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Accumulation as entropy The long-run situation: the Haavelmo
predictament
The only solution is to stop accumulating with At < Ao
A more general situation with depreciation of stocks due to natural processes: decay of materials, decomposition due to bacteria, sunlight, carbon sinks, chemical reactions Generation of pollution balanced against decay
( )limo
tA A
D A
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Waste accumulation model with decay Decay due to natural processes in Nature
α ”radioactive” decay coefficient Decay balancing current emissions
Critical loads: current emission corresponding to the decay of a stock that does not yield significant damages in the ecosystem
, 1 0t t tA e A
0t t tA e A
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A two-period problem Most simple model of waste dynamics without
decay damage function Dt(.) in accumulated waste benefit function in current emissions
Social planning problem faced in period 1
β discount factor: 0 < β < 1 Must know the emission in period 2 to decide on
emission in period 1 implying that the whole path of emissions must be decided in period 1
1 1 1 1 2 2 2 1 2{ ( ) ( ) ( ( ) ( ))}Max B e D e B e D e e
( ), 0, 0t t t t tB B e B B
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A two-period problem, cont. First-order conditions
Period 1
Period 2 (as decided in period 1)
Assumption: no restriction on emission in period 2 Solving simultaneously for e1 and e2.
1 1 1 1 2 1 2( ) ( ) ( )B e D e D e e
2 2 2 1 2( ) ( )B e D e e
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Illustration of the two-period case for period 1
B1’
βD2’
D1’
D1’+βD2’
e1
B’,D’
e1*(e2)
Infinite horizon model for waste accumulation in continuous time The planning problem with decay of
accumulated waste
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0
[ ( ) ( )]
0,
0
t
rte t t
t t t
o
t
Max B e D A e dt
subject to
A e A
A
e
The mathematical method
The Hamiltonian plays same role as the Lagrange function Current value Hamiltonian: variables are not
discounted to the time of the planning decisions; time zero, but refer to time t
The Hamiltionan consists of the objective function and a constraint expressing the change in the stock variable over time
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( ) ( ) ( )t t t t tH B e D A e A
Rules for using the Hamiltonian to get first-order conditions The static first-order condition for the control
(flow) variable
The dyamic first-order condition for the state (stock) variable
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( ) 0t tt
HB e
e
( )t t t t tt
Hr D A r
A
Interpreting the first-order conditions Rearranging the static condition
The shadow price on the stock-accumulation equation is negative: more waste reduces the objective function
Balance between marginal benefit of emission at time t and damage created by this emission from t to infinity expressed by the shadow price at t
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( )t tB e
Interpreting the first-order conditions, cont. Interpreting the dynamic first-order condition
Shadow price decrease (increase) if marginal damage is less (greater) than the rental value of the shadow price
The sum (α+r) can be interpreted as a ”gross rate of discount”:
future damages are discouted with r, and the decay coefficient also reduces the damage by reducing the stock with a fixed rate
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( ) ( )t t tD A r
Interpreting the first-order conditions, cont. Combining the static and the dynamic
conditions introducing marginal benefit to facilitate interpretation
The shadow price increases (decreases) when marginal damage of the stock is higher (lower) than the “gross interest” on the marginal benefit of emission.
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( ) ( )( )t t tD A B e r
Steady state
In steady state the stock of waste and the shadow price on the stock are constant
Inserting the static condition into the dynamic condition and setting the change in the shadow price equal to zero (dropping index t)
Marginal damage is set equal to the gross interest rate on the stock shadow price
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0 ( ) ( )( ) ( ) ( )
( ) ( )
D A B e r D A r
D A r
Steady state, cont. From the growth equation for waste
The flow of emission is equal to the amount of decay of the accumulated stock taking place
Interpretation of the shadow price on the stock of waste in steady state
Shadow price equal to present value of the marginal damage,using gross rate of interest
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0 , /e A e A A e
( )D A
r
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Steady state, cont.
The present value of damages shall be equal to the marginal benefit in steady state
Two equations to determine the variables A and e in steady state, eliminating A
( )( )
( )
D AB e
r
( ) ( / ) ( / )( ) ( )
( ) ( )
D A D e D eB e B e
r r r
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Studying steady state using a phase diagram Variables:, the stock A, and the control
variable (the instrument) e. The differential equations governing the
development of these variables
Growth equation for the stock of waste
( , ) , ( , )t t t t
dA def A e g A e
dt dt
t t tA e A
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Phase diagram, cont.
Second equation: start with differentiating w.r.t time the static first order condition
Inserting into the dynamic condition yields
0t tde dBdt dt
( )tt
deB D rdt
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Phase diagram, cont.
Substituting for the shadow price from the static first order condition
( ) ( )
( )( ) ( )
( )
t
t t t
t
deB D r D B rdt
de B e r D A
dt B e
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Phase diagram, cont.
Solving
From the steady state solution
The curve for no change in A is a straight line through the origin with a positive slope of α
, 0t tA e
/A e
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Phase diagram, cont.
Finding the location of curve where
Solving for A
A falling curve in (A, e) space with convex damage function and concave benefit function
( )0 ( )( ) ( )t
B r De B e r D A
B
1[ ( )( )]A D B e r
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The phase diagram
1/α
A
Ao
eo
b
d
ca
e
de/dt = 0
dA/dt = 0
e*
A*